The Maximal and Free (∞, n)-Categories

of an (∞, n + 1)-Category

Kyle Ferendo

11 April 2016

Submitted to theDepartment of Mathematics

of Amherst Collegein partial fulfillment of the requirements

for the degree ofBachelor of Arts with honors

Faculty Adviser: Professor Michael Ching

Copyright c© 2016 Kyle Ferendo

Abstract

In this thesis, we describe left and right adjoints (which we call the maximal andfree (∞, n)-categories of an (∞, n + 1)-category respectively) to the inclusionfunctor of (∞, n)-categories into (∞, n+ 1)-categories which respect the homo-topy theory of the same. We model (∞, n)-categories with Rezk’s Θn-spaces,introduced in [Rez10], and use model categories as our framework for homotopytheory.

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Acknowledgements

My foremost acknowledgement must be of Professor Michael Ching, who hasproven not only an adept, patient, and encouraging thesis adviser, but who hascontributed more than any other individual to my development as a studentof mathematics. For as long as we have worked together, he has shown greatgenerosity (encompassing two special topics courses, support for summer research,two semesters of thesis work, vigilance for extra-curricular opportunities, andnumerous recommendation letters) and energetically encouraged intellectualcuriosity (indeed, in my first interaction with him, we, at his encouragement,supplanted my proposal for a standard course in algebraic topology with a courseon motivic homotopy theory, which served as my introduction to the infinitelyfascinating field of modern homotopy). I also thank Prof. David Cox, whosegeometry course was my first immersion in rigorous mathematics, and who hasalways, over numerous courses, pushed me to sharpen and clarify my thinking.

I thank Professor Rick Lopez, whose encouragement and friendliness helpedrealize an unfading passion for history. And I thank all the other facultymembers, current and former, including Professors Kearns, Castro Alves, Boucher,Drabinski, Sadjadi, Shandilya, Daniels, and many others, who have broadenedmy thinking in ways transcending the mere transmission of knowledge.

As this thesis symbolizes the approaching conclusion of one stage of my life,I am moved to recall all the friends on whom I have relied so heavily.

I thank Joy Brenner-Letich, “the terminal object in the category of people Ilike,” who has always challenged me to grow as a thinker and as a person, andwho has been with me through thick and thin. There was never a truer friendand comrade. I also thank Julia Vrtilek, Lucas Renique-Poole, Laura Merchant,Rebecca Boorstein, Stephanie Sosa, and Jordan Hugh Sam, my first, lovelycohort of friends in an alien environment, for whose warmth and kindness I amgrateful beyond words. I thank Elena Marione (and her delightful fascinationwith fiber bundles), Sylvia Hickman, Hannah Goodwillie, Charlie Newman, EmilyWillick, Sam Rosenblum, Sunil Suckoo, Dane Engelhardt, and Deme Shahmehri,whose friendly affections have brought me untold happiness in my last years atAmherst. I thank Cheyne Marshall, whose good humor has been a reliable andmuch appreciated jollity and restorative. I thank all the other friends and friendlyacquaintances too numerous to name who have provided pleasant companionshipthroughout my college years. Each of you has irreversibly introduced more loveinto my life, and to these delightful friendships and to my parents I attribute

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whatever virtues I possess. I cannot overstate my gratitude to each of you, andI regret that I lack the space to acknowledge and elaborate on each of youruniquely wonderful qualities. I will treasure these friendships forever.

Finally, I thank my parents, whose love, support, and compassion has alwaysbeen both unwavering and essential. You are phenomenal parents and phenome-nal friends, and I am so thankful. Time with you is always a joy. I am again ata loss for words.

As I look to what lies ahead, I am filled with gladness to recall so manypeople who have brought so much fulfillment to my life.

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Contents

1 Introduction and motivation 1

2 Simplicial sets and (∞, 1)-categories 62.1 Simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Categories as simplicial sets . . . . . . . . . . . . . . . . . . . . . 72.3 Spaces as simplicial sets . . . . . . . . . . . . . . . . . . . . . . . 82.4 Complete Segal spaces . . . . . . . . . . . . . . . . . . . . . . . . 92.5 The adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Higher categories 143.1 Strict n-categories . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 The categories Θn . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Θn-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 More model category theory . . . . . . . . . . . . . . . . . . . . . 243.6 The inclusion functor is left and right Quillen . . . . . . . . . . . 29

Bibliography 32

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Notation

We use the following conventions throughout the thesis.

• When defining a functor C → D, we often use the symbol “−” to stand infor an object or morphism of C. We do this only when both an object ora morphism could stand in meaningfully for “−”; in fact, we prefer thisnotation in order to emphasize that the same description of the functorapplies to both objects and morphisms. Multiple instances of “−” neednot all represent the same object or morphism.

• As is typical in category theory, we often use the article “the” to speak ofstructures defined uniquely up to unique isomorphism (e.g. “the” terminalobject of a category, “the” left adjoint of a functor, and so on).

• When we write lim followed by a diagram in some category C, we areindicating the limit of that diagram in C. Similarly for colim.

• Beware that there are two definitions of the word “pullback” in categorytheory: a limit of a certain shape, and a functor between functor categoriesinduced by a functor between their domain categories. We hope that itwill always be clear from context which is intended.

• D1 denotes the abstract category · → · (where we suppress identity mor-phisms) and D2 the abstract category · → · → · (again suppressing identitymorphisms).

• Set denotes the category with objects sets and morphisms functions.

• Cat denotes the category with objects small categories and morphismsfunctors.

• T op denotes the category with objects topological spaces and morphismscontinuous functions.

• ∆ denotes the simplex category, defined in Definition 2.1.1.

• Θn denotes an indexing category defined in Definition 3.2.3.

• For C and D two categories, DC denotes the category with objects functorsfrom C to D and morphisms natural transformations.

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• For V a monoidal category, a V-category is a category enriched in V.

• For C a (possibly enriched) category, X ∈ C denotes that X is an object ofC.

• For C a category, we denote its terminal object ∗ ∈ C.

• For C a category and X,Y ∈ C, X×Y denotes the Cartesian product of Xand Y in C. For complete precision, by “Cartesian product” we mean thelimit of a functor from the category with two objects and no non-identitymorphisms to C. Note that the term “product” is often used to mean“Cartesian product” in category theory; we too occasionally adopt thisusage, but also sometimes use the phrase “Cartesian product” in order toavoid confusion with any monoidal product that may be defined.

• For C and X,Y ∈ C, X q Y denotes the coproduct of X and Y in C.

• For C a category and X ∈ C, we denote by idX the identity morphism forX.

• For C a category and X,Y ∈ C, X ∼= Y indicates that X and Y areisomorphic in C. f : X ∼= Y indicates that f is an isomorphism from X toY .

• For C a (possibly enriched) category and X,Y ∈ C, C(X,Y ) denotes thehom-object of morphisms from X to Y . Note that for C a V-category,C(X,Y ) ∈ V.

• For C a (possibly enriched) category, Cop its opposite category; i.e. thecategory with the same objects as C and for all X,Y ∈ C, Cop(X,Y ) =C(Y,X).

• sSet denotes the category of simplicial sets, Set∆op

.

• For C a category, PC denotes the category sSetCop ; that is, the category ofpresheaves on C in sSet.

• For a category C, F : C → PC denotes the Yoneda embedding, F (A)(B)def=

C(B,A), where we regard C(B,A) as a discrete simplicial set.

• For V a category with monoidal structure (V,⊗, I) and C a V-category,f ∈ C(X,Y ) indicates that f is a morphism from X to Y in the underlying(Set-)category of C. In other words, f ∈ C(A,B) is shorthand for f ∈V(I, C(A,B)).

• For V a category with monoidal structure (V,⊗, I) and right unitor ρA :A ⊗ I → A, C a V-category, X,Y, Z ∈ C, and f ∈ C(X,Y ), C(f, Z) :C(Y, Z) → C(X,Z) denotes the morphism arising from the compositionC(Y, Z) ∼= C(Y,Z) ⊗ I → C(Y,Z) ⊗ C(X,Y ) → C(X,Z), where the firstmorphism is ρ−1

C(Y,Z), the second idC(Y,Z) ⊗ f , and the third composition inC.

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• When we have as above but with f ∈ C(Y,Z), C(X, f) : C(X,Y )→ C(X,Z)is defined similarly.

• For M a model category, −c :M→M denotes the cofibrant replacementfunctor, defined in Definition 3.4.1.

• We denote the composition of functors with juxtaposition, and when weevaluate a functor on an object or morphism, we usually enclose thatobject or morphism in parentheses. At moments, when the notation fora functor (or natural transformation) includes several symbols, we find ithelpful to enclose those symbols within parentheses. When we composethose parenthesized functors with other functors, we hope that it is alwaysclear from context that composition is what is intended. It should alwaysbe possible to dispel any ambiguity by checking to which category eachsymbol in question belongs.

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Chapter 1

Introduction andmotivation

The purpose of the thesis is to introduce the reader to (∞, n)-categories and togeneralize two adjunctions, which we will explain shortly, between groupoidsand categories to (∞, n)-categories and (∞, n+ 1)-categories.

On the part of the reader, we assume broad proficiency with category theoryas developed e.g. in [Mac98]. We also make use of concepts from enrichedcategory theory ([Kel05] is a standard reference), but we expect that the thesiswill be intelligible to a reader who is willing to accept on faith that constructionsexist with the indicated properties (e.g. right and left enriched adjoints, i.e.Kan extensions, to pullback functors MD → MC exist and are given by endand coend formulae as long as M is complete or cocomplete respectively). Weassume some familiarity with the homotopy theory of topological spaces, andseek to convey the pertinent ideas of modern homotopy theory over the courseof the thesis. We lack the space to reproduce many standard technical results,particularly regarding the theory of model categories, and often refer the readerto [Hir03] for these details.

We begin with the observation that spaces and categories are both gen-eralizations of groupoids in complementary ways (recall that a groupoid is acategory in which every morphism is an isomorphism).1 The way in which cate-gories generalize groupoids is immediately obvious, but the relationship betweengroupoids and spaces may be less clear, and so we take a moment to explain.Given a groupoid G, we construct a topological space (called its classifying

1A common theme in this paper, as well as in homotopy theory, is the use of mathematicalstructures as vessels for information. In general, we care about the information, and not thevessel. By this we mean to indicate that certain mathematical structures may encode theinformation we care about as well as some extraneous data which does not interest us. Whenwe say that structure A generalizes structure B, we only mean that structure A can encodeat least all of the relevant information held by B, even if it is impossible to move certainextraneous data of B into A. In these cases, it may not be true from some technical perspectivethat A generalizes B, but it is true for the purposes of the information we care about.

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space) B(G) in the following manner: we begin with 0-cells (i.e. discrete points)corresponding to the objects of G; for each non-identity morphism f : A→ Bof G, we attach a 1-cell (i.e. a closed line segment) from A to B as 0-cells(we regard the constant paths to the 0-cells as the identity morphisms); andwhenever gf = h in G, we attach a 2-cell (i.e. a closed, triangular subspaceof the real plane) with boundary the cycle gh−1f . We continue this processfor each sequence of n composable morphisms in G, attaching an n-cell (i.e. ahigher-dimensional closed triangle) to our space so that its (n− 1)-dimensionalhyperfaces are identified with the (n − 1)-cells corresponding to the sequenceof n− 1 composable morphisms resulting from composing a pair of consecutivemorphisms in our original sequence.2

An interesting fact is that we can try to move in the opposite direction: givenany space X, we can construct its fundamental groupoid $(X), in which theobjects are the points of the space and the morphisms are endpoint-preservinghomotopy classes of paths from one point to another. When we claim that spacesgeneralize groupoids, we have in mind the interesting fact that $(B(G)) ' G –that the fundamental groupoid of the classifying space of a groupoid is equivalentto that groupoid. But in general, we lose some information when we pass from aspace to its fundamental groupoid: namely, we lose information about the distincthomotopies which exist between paths, along with similar higher dimensionalinformation about homotopies between homotopies and so on. A concreteexample is this: the fundamental groupoid of the 2-sphere is equivalent to thefundamental groupoid of the point, but the constant path at the basepoint of a2-sphere has non-trivial endpoint-preserving self-homotopies (i.e. π2(S2) ∼= Z),while the same is not true of the point. Put more precisely, for X a space,B($(X)) is weakly homotopy equivalent to the 1-truncation of X.

The takeaway is that classifying spaces and fundamental groupoids suggesta comparison between spaces and groupoids, but spaces are somewhat morerich: they can contain “2-morphisms” (homotopies) between “1-morphisms”(paths), 3-morphisms between 2-morphisms, and so on, and composition (ofpaths, homotopies, homotopies between homotopies, etc.) is not given by somestrictly associative operation, but is only associative (and really even defined)up to higher homotopies. Because there need be no dimensional bound onhomotopically non-trivial k + 1-morphisms between k-morphisms, we identifyspaces with ∞-groupoids (with the ∞ indicating that we have 2-morphisms,k-morphisms, and so on), also called (∞, 0)-categories, where the 0 indicatesthat all k-morphisms are equivalences for k > 0 (since paths, homotopies, andso on can always be run in reverse). This notation immediately suggests thenext question: if groupoids are, in a sense, both spaces and categories, is therea connection between the two further up, as well as further down, the web ofgenerality? This would be a theory of “higher” categories with k-morphismsand so on, perhaps such that each k-morphism is an equivalence for k > 1, butsuch that non-invertible 1-morphisms may exist (since, unlike paths in a space,

2For the reader with a more sophisticated knowledge of category theory, this constructionis given by a coend formula.

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it need not be possible to run a morphism in a category in reverse). If so, thisjoint generalization ought to be called a theory of (∞, 1)-categories.

A fairly natural definition of an (∞, 1)-category is a T op-category – that is, acategory C in which for two objects X,Y ∈ C, C(X,Y ) is a topological space, andcomposition is defined by a continuous function C(Y,Z)× C(X,Y )→ C(X,Z).The appeal of this model is that we evidently retain all of the higher-orderhomotopical/categorical information that is present in spaces, since we havespaces of morphisms: we can think of 2-morphisms as paths in hom-spacesbetween points (1-morphisms) of those hom-spaces, and so on.

We can see how ordinary Set-categories fit into this model easily: we justinterpret hom-sets as discrete hom-spaces. But how do (∞, 0)-categories, orspaces, fit into the framework of topologically enriched categories? As a startingpoint, we can take the points of a space X to be the objects of a topologicallyenriched category X, and given two points A,B ∈ X, we define X(A,B) tobe the function space of paths from A to B with the compact-open topology.The problem that we encounter is that we don’t have a natural way to definecomposition of morphisms in this topological category. A pair of composablepoints in two hom-spaces corresponds to a pair of composable paths in X, butcomposing paths depends upon a choice of a continuous function f : [0, 1]→ [0, 2]such that f(0) = 0 and f(1) = 2. Although some such functions are moresymmetrical than others, any such choice is essentially arbitrary, and no choiceyields a strictly associative definition of composition. But on the other hand,any such choice yields a definition of composition which is associative up tohomotopy (since any two such endpoint-preserving functions f, g : [0, 1]→ [0, 2]are homotopic through endpoint-preserving homotopies).

It is possible to remedy the problem by passing to quotients of mappingspaces by the action of the group of order-preserving automorphisms of the closedinterval, and from such a topological category (in fact, a topological groupoid),it is possible to recover a space that is weakly homotopy equivalent to X. Butwe only really expect composition to be defined up to a contractible space ofchoices anyway (that is, for it to be “homotopy coherent”), and so we consider itpreferable for the purposes of studying the homotopy theory of (∞, 1)-categoriesto use a model which does not require such rigidity. We provide an extendedtreatment of a particularly suitable model for (∞, 1)-categories, complete Segalspaces, in Chapter 2, but the reader can find an overview of four common modelsin [Ber10].

But there is no need to stop generalizing now! Our notation leads theway: we can ask about (∞, n)-categories, categories with k-morphisms between(k−1)-morphisms, in which all k-morphisms need be equivalences only for k > n.We hope that the foregoing discussion begins to shed some light on the rolewhich spaces may play in (∞, n)-categories: that is, a space naturally capturesa system of equivalences of unbounded dimension. The mystery lies in howto simultaneously describe a system of not-necessarily invertible k-morphismsfor k ≤ n which is compatible with the structure of the space describing theequivalences in our imagined (∞, n)-category.

There is much more to say on this topic and on the nature of higher categories

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generally, but we postpone this discussion until later in the paper, and insteadturn to the construction which motivates the central question of the presentpaper.

The succinct statement is that the full subcategory of groupoids is bothreflective and coreflective in the category of categories.3 In other words, given acategory C, there exist groupoids FreeGrpd(C) and MaxGrpd(C) and functorsC → FreeGrpd(C) and MaxGrpd(C) → C such that every functor from C toa groupoid (or from a groupoid to C) factors uniquely through the functorC → FreeGrpd(C) (or MaxGrpd(C)→ C respectively). A synonymous statementis that the inclusion functor from the category of groupoids to the category ofcategories has both a left and a right adjoint.

The construction of MaxGrpd(C) is straightforward: it has the same objects asC and exactly the morphisms of C which are isomorphisms. Thus MaxGrpd(C)is the maximal subgroupoid of C, and we have an obvious inclusion functorMaxGrpd(C)→ C. Because functors preserve isomorphisms, MaxGrpd(C) hasthe desired universal property. The construction of FreeGrpd(C) is more difficultto describe. Conceptually speaking, we obtain FreeGrpd(C) by “freely inverting”all morphisms of C to obtain the “free groupoid” on C. This entails pasting inan inverse f−1 to each non-invertible arrow f of C. The axioms of categoriesthen determine certain other behavior of FreeGrpd(C): for instance, it is aneasy lemma of category theory that a one-sided inverse of an isomorphism isequal to the unique two-sided inverse of that isomorphism. Thus, if two distinctmorphisms g, h of C are both one-sided inverses of a third morphism f of C,the universal functor C → FreeGrpd(C) necessarily identifies g and h. We verymuch regret that we lack the time and space for a fully rigorous discussion ofthe functor FreeGrpd.

Another perspective on these two constructions comes from monoids: recallthat a monoid is a category with a single object, so for a monoid M we canconsider MaxGrpd(M) and FreeGrpd(M). Then MaxGrpd(M) is the group ofinvertible elements of M , while FreeGrpd(M) is what is called the Grothendieckgroup of M when M is commutative (but there is a generalized constructionwhich applies to non-commutative monoids as well).

Let us examine a very simple example in order to illustrate the free groupoidfunctor: Let C be the category illustrated by the following diagram:

A B

f

g

Then to obtain FreeGrpd(C), we freely adjoin arrows f−1 and g−1 and obtain atwo-object groupoid equivalent to the integers (generated at A by f−1g or by

3In general, the reader should assume that the objects of all categories we discuss in thispaper are small; i.e. built from small sets, strictly less than some strongly inaccessible cardinalwe postulate to exist. It is sometimes nice to imagine claims like “the category of all categoriesis a 2-category,” and for this to be coherent, we need to assume the existence of an ω+1-indexedsequence of strongly inaccessible cardinals, and move into larger and larger Grothendieckuniverses as we discuss higher categories. This is not an important point for this thesis.

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g−1f). We observe that B(FreeGrpd(C)) is homotopy equivalent to the circle.4

In this paper, we provide constructions which generalize these two adjunctionsbetween groupoids and categories (in the more general notation, we can callcategories and groupoids (1, 1)- and (1, 0)-categories respectively, to indicatethat they have no non-trivial k-morphisms for k > 1 and that the 1-morphismsof a category, but not a groupoid, are allowed to be non-invertible) to (∞, n)-and (∞, n + 1)-categories – that is, functors which forget or freely invert allnon-equivalence (n+ 1)-morphisms in an (∞, n+ 1)-category.

4This fact is related to the fact that the suspension of S0 is S1.

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Chapter 2

Simplicial sets and(∞, 1)-categories

2.1 Simplicial sets

We begin our exposition by introducing the simplex category ∆.

Definition 2.1.1. The simplex category ∆ is the category with objects totally

ordered finite sets of the form {0 < · · · < n} def= [n] ⊂ N and with morphisms

functions of sets respecting ≤. Note that [0] is the terminal object in ∆, butthat there is no initial object.

Definition 2.1.2. A simplicial set is a functor X : ∆op → Set. We denote the

category of simplicial sets sSet def= Set∆op

(with morphisms given by naturaltransformations). More generally, a simplicial object in a category C is a functorX : ∆op → C.

Given a simplicial set X : ∆op → Set, we denote Xndef= X([n]) and call this

the set of n-simplices of X. We say that a simplicial object is discrete when as afunctor it constantly evaluates to a fixed objects (on objects) and that object’sidentity morphism (on morphisms).

Remark 2.1.3. There are two classes of morphisms in ∆ which are particularlyimportant: we denote by δi : [n]→ [n+ 1] the functions (called “face maps” forreasons we’ll clarify soon) which “skip” i (note that we use the same notationregardless of the domain); that is,

δi(j)def=

{j j < i

j + 1 j ≥ i

And we denote by σi : [n]→ [n− 1] the functions (called “degeneracy maps”)which hit i twice:

σi(j)def=

{j j ≤ ij − 1 j > i

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Every morphism in ∆ can be described as the composition of a sequence ofδis followed by a sequence of σis. Naturally, there are several relations whichgovern the composition of these morphisms, but we need not describe them here(and we expect that the reader will be able to work them out for him or herselfshould she or he so desire).

2.2 Categories as simplicial sets

Let us now examine how we can encode the information of a category in asimplicial set.

Definition 2.2.1. Given a category C the nerve of the category C is a simplicialset N(C) which we now describe.

Let C : ∆→ Cat be the functor sending [n] to the category with n+1 objectscorresponding to the elements of the set [n] and with a single morphism fromi ∈ C([n]) to j ∈ C([n]) if and only if j ≥ i. Let C(f) : C([m])→ C([n]) be thefunctor acting on the objects of [m] exactly as f acted on the elements of [m].Observe that C embeds ∆ as a full subcategory of Cat.1

We define N(C)(−)def= Cat(C(−), C) (here we understand Cat as an ordinary,

rather than enriched, category).

Thus N(C)n is the set of sequences of n composable arrows in C. In particular,N(C)0 is the set of objects of C, N(C)1 is the set of morphisms, with an elementf ∈ N(C)1 pointing from δ0(f) to δ1(f), and N(C)2 records composition, since,as we can check from the definition, N(C)(δ1) : N(C)2 → N(C)1 sends a sequenceof composable arrows to their composite.

Definition 2.2.2. We say that a simplicial set X : ∆op → Set satisfies theSegal condition when it has the following property:

Let αi : [1]→ [n] be the morphism in ∆ defined by αi(0) = i and αi(1) = i+1.

Denote αidef= X(αi) (in general we’ll use subscripts rather than superscripts

to denote the image in a simplicial object of a morphism in ∆). Observe thatαiδ0 = αi+1δ1, so the universal property of pullbacks gives us a map

(α0, . . . , αn−1) : Xn → lim

(X1 X0 · · · X0 X1

δ1 δ0 δ1 δ0

)where X1 appears n times. The Segal condition requires that these maps beisomorphisms Xn

∼= X1 ×X0 · · · ×X0 X1.

Proposition 2.2.3. A simplicial set X is the nerve of a category if and only ifX satisfies the Segal condition.

Proof. Proof omitted.

1“Full” here means that every morphism C([m]) → C([n]) is the image under C of somemorphism [m] → [n] in ∆.

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2.3 Spaces as simplicial sets

There is a second reason why we care about simplicial sets, and this applicationwas historically their original motivation: they can be used to describe spaces ina neat, combinatorial way.

Definition 2.3.1. Just as we could take the nerve of a category, we can takethe singular simplicial set of a topological space. We define a functor T : ∆→T op sending [n] to the topological n-simplex ∆n, the (closed) n-dimensionaltetrahedron. We number the vertices of these tetrahedra with the elements of[n], so that T (δi) is the inclusion of ∆n as the hyperface of ∆n+1 not containingi as a vertex. Similarly, the degeneracy maps σi collapse the line segmentbetween i and i+ 1 while holding everything else rigid. More generally, T (f) forf : [m]→ [n] acts on the vertices of ∆m as f acts on the elements of [m], andextends linearly from the action on the vertices to the entirety of of the space.

We define Sing(Y )(−)def= T op(T (−), Y ), for Y any topological space.

Sing(Y ) provides the best approximation of Y as a simplicial set, and, inparticular, preserves its homotopy type. The reader familiar with singularhomology will observe that all of the singular homology (and cohomology)functors factor through Sing(−).

Definition 2.3.2. Sing(−) has a left adjoint, called the realization of a simplicialset and denoted | − |T op. As a coend, this functor has the form

|X|T opdef=

∫ [n]∈∆

X([n])× T ([n])

More informally, to realize a simplicial set X as a topological space, we glue in theapparent way: an element of Xn corresponds to a copy of ∆n, which we glue in tofill a collection of n−1-simplices arranged to form a boundary. Thus, a simplicialset provides us with instructions for constructing a CW-complex, and in fact,every topological space is weakly homotopy equivalent to the realization of somesimplicial set. Note also that B(C) ∼= |N(C)|T op, where B is the classifying spacefunctor mentioned in Chapter 1.

Since we use simplicial sets as a combinatorial model for the homotopy theoryof spaces, it is natural that we wish to carry over the same notion of homotopyequivalence.

Definition 2.3.3. A morphism f : X → Y of simplicial sets is called a weakequivalence of simplicial sets exactly when |f |T op is a weak homotopy equivalenceof topological spaces. Recall that a weak homotopy equivalence induces abijection of path components of |X|T op and |Y |T op and induces an isomorphismbetween the homotopy groups πn(|X|T op) ∼= πn(|Y |T op) for each n ∈ N and forany choice of basepoint in X.

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2.4 Complete Segal spaces

The model, called “complete Segal spaces,” that we use for (∞, 1)-categories inthis thesis employs simplicial sets in both of these ways, and in fact, completeSegal spaces are examples of simplicial simplicial sets, or bisimplicial sets, or,both more evocatively and ambiguously, simplicial spaces.

To help motivate the definition of a complete Segal space, we give a modifiedversion of the nerve of a category.

Definition 2.4.1. The groupoidal nerve2 of a category is the simplicial groupoid

N ′(C)([n])def= MaxGrpd(C[n])

which defines a functor N′ : Cat→ Grpd∆op

. This suggests a way of modelingcategories with simplicial groupoids.

It is clear that the groupoidal nerve of a category captures that categoryup to isomorphism, since the ordinary nerve is recoverable from the groupoidalnerve (by forgetting the morphisms in the groupoids at each level). Our reasonfor introducing this variation in the nerve functor is to suggest one approachto generalizing categories to (∞, 1)-categories: by replacing our simplicial setmodel for categories with simplicial ∞-groupoids (in analogy with the role ofgroupoids in the groupoidal nerve just defined), so that all of the equivalencesat each level are already built into the lowest level of the simplicial structure;that is, all of the higher equivalences come built into the ∞-groupoid of objects.Then, in order to complete our description of an (∞, 1)-category, we needonly describe the 1-morphisms in a compatible way, which we do by fillingout the simplicial structure and introducing some requirements to ensure thatthe simplicial structure describing sequences of not-necessarily-invertible 1-morphisms interacts nicely with the spatial/∞-groupoidal structure describingthe higher equivalences.

A complete Segal space is a simplicial presheaf on ∆ (i.e. a functor X :∆op → sSet, and equivalently, an object of P∆) satisfying certain conditions.Before we can proceed with the definition, we need the following observationand definition.

Proposition 2.4.2. The category of simplicial presheaves on ∆ is enriched oversSet.3

Proof. It is a standard result of enriched category theory that the categoryof V-enriched functors from one V-category to another is itself a V-category(assuming that V is complete). See §2.2 of [Kel05]. In the case at hand, we

2This is an ad hoc term which we will not use later in the paper, but we do wish todistinguish this construction from the previous, more standard definition of “nerve” whilesimultaneously stressing the similarity of this construction.

3It is also a pleasant and desirable fact that this category is Cartesian closed, but that isnot important for the discussion at hand.

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regard both ∆op and sSet as sSet-categories, with the hom-simplicial sets in∆op being discrete.

Definition 2.4.3. We say that an object C of an sSet-category C is local withrespect to a morphism f : A→ B of C when the morphism C(f, C) : C(B,C)→C(A,C) is a weak equivalence of simplicial sets. When we have a collection T ofmorphisms in an sSet-category C, we say that an object is T -local when it islocal with respect to each morphism in T .

Definition 2.4.4. In the remainder of this exposition, we use F (−) to denotethe Yoneda embedding.4 Let G([n]) denote the colimit of the following diagramin sSet

G([m])def= colim

(F ([1]) F ([0]) · · · F ([0]) F ([1])

Fδ0 Fδ1 Fδ0 Fδ1

)where F ([1]) appears in the diagram n times. Then let sen : G([n]) → F ([n])denote the map from the colimit determined by F (αi) : F ([1]) → F ([n]) for0 ≤ i ≤ n− 1 and αi as defined in §2.2.

The reader may notice that this diagram is dual to that which we used todefine the Segal condition in §2.2. In fact, by the Yoneda lemma, we couldhave defined the Segal condition in this way, using local objects, except withisomorphisms of hom-sets filling the role of weak equivalences of hom-simplicialsets (and thus necessitating that we tweak our definition of localness).5

When a simplicial presheaf is local with respect to sen for all [n] ∈ ∆, we saythat that presheaf satisfies the Segal condition.

Definition 2.4.5. Consider the simplicial set N(iso), where iso is the categorywith two objects and a unique isomorphism between them. Let E denote the

simplicial presheaf on ∆ defined by E([n])def= D(N(iso)([n])), where D(−) :

Set→ sSet is the functor which turns a set into a discrete simplicial set. Denotethe unique morphism E → F ([0]) ∼= ∗ by cpt. When a simplicial presheaf islocal with respect to cpt, we say that it satisfies the completeness condition.

Definition (2.4.6). A complete Segal space is a simplicial presheaf on ∆ whichsatisfies the Segal condition and the completeness condition.6

4See [Mac98] for more information on the Yoneda embedding and the Yoneda lemma.5The use of weak equivalences, rather than isomorphisms, to describe the Segal conditions in

the context of (∞, 1)-categories helps to solve a problem from higher category theory: one oftendesires that the axioms of a category hold only up to higher coherent equivalences; for example,we might accept that two ways of associating a sequence of composable morphisms yield twounequal results, so long as these two morphisms be isomorphic (or equivalent in some weakersense) to each other. But these weakened conditions, called “coherence laws,” are extremelytricky to describe for weak n-categories for n as low as 4 or 5. The homotopical approach tohigher categories sidesteps all of this trouble by introducing what are essentially the “modulispaces” of composable sequences of morphisms, requiring that these spaces coincide only up toweak equivalence, and thus that composition be defined only up to homotopy in the first place.

6We omit an especially technical condition from our definition of complete Segal spaces. This

10

The role of the completeness condition in the definition of a complete Segalspace may be unclear. We defer a precise explanation of this condition to §6of [Rez01], but the gist of it is the following: given a complete Segal space X,there is (as we would hope!) a notion of composition of morphisms (morphismsbeing points (0-simplices) in X([1])), and with it, a criterion for a morphismto be an equivalence. Then we can consider Xequiv ⊂ X([1]), the subspace ofweak equivalences, and it is Theorem 6.2 of [Rez01], proven in §11 of that paper,that P∆(E,X) ' Xequiv, so that we can think of P∆(E,X) as a representativefor the moduli space of weak equivalences in X. Then the localness conditionwe’ve just described requires that the moduli space of weak equivalences mustbe weakly equivalent to the moduli space of objects X([0]) ∼= P∆(F ([0]), X).

Lemma 2.4.7. A constant functor X : ∆op → sSet is a complete Segal space.

Proof. This follows from the localness conditions: the Segal conditions requirethat map from X([n]) to the pullback of n-copies of X([1]) over X([0]) be aweak equivalence of simplicial sets, but this is immediate, since all of thesespaces are identical, the limit of the diagram is again X([1]) ∼= X([n]), and theidentity map is certainly a weak equivalence. Although we have not definedcomposition, we hope that it is clear to the reader that every morphism inX([1]) corresponds to an identity morphism, since X([0]) ∼= X([1]). Thus wehave P∆(E,X) ' Xequiv

∼= X([1]) ∼= X([0]) ∼= P∆(F ([0]), X), so by the Yonedalemma, X is local with respect to cpt.

There is a notion of weak equivalence of complete Segal spaces which runssubstantially parallel to the corresponding notions for categories and spaces: thatthere should exist a morphism in the opposite direction so that each compositioncan be transformed into the identity morphism. Since the category of completeSegal spaces is enriched of sSet, this would mean that each composition shouldbe in the same connected component as the identity morphism. However, justas weak homotopy equivalences can only be described in an analogous waywhen one restricts consideration to CW-complexes, a similar problem arisesin this case: we must either give a less satisfying condition for a morphism tobe a complete Segal space (in analogy with the definition of a weak homotopyequivalence in terms of isomorphisms of homotopy groups) or we must furtherrestrict our definition of what a complete Segal space is. We can give neitherapproach the proper treatment without the machinery of model categories,which we postpone until the next chapter due to the technical details theyinvolve. In the meantime, we emphasize the following: there is a notion ofweak equivalence between complete Segal spaces, and we desire that a functor

condition is not necessary for understanding how complete Segal spaces model (∞, 1)-categories,but is important for ensuring that the category of complete Segal spaces exhibit the correcthomotopical behavior, and cannot be defined without the language of model categories. Weintroduce model categories in the following chapter, in which we treat our subject matter withgreater rigor, and prove that the functors which are the focus of our paper behave appropriatelywith respect to the homotopical structure of our models for (∞, n)-categories.

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between categories with weak equivalences preserve those weak equivalences,just as functors automatically preserve isomorphisms. In the following section,we introduce several functors between simplicial sets and complete Segal spaces;we claim that these functors preserve weak equivalences, and we will prove theclaim in the following chapter.

2.5 The adjunctions

We now return to the idea presented in the introduction of generalizing theinclusion of groupoids into categories and the two adjunctions to this inclusionfunctor. We generalize groupoids to ∞-groupoids, modeled by simplicial sets,and categories to complete Segal spaces. We would like to describe two adjoints(left and right) to a single functor. A very economical way of producing bothof these at once, when it applies, is with Kan extensions.7 This would involvedescribing the inclusion functor H∗0 : sSet→ sSet∆op

as a pullback, which seemplausible, since sSet and sSet∆op

are both functor categories. In fact, we dohave the following definition.

Definition 2.5.1. We define the inclusion functor H∗0 : sSet→ sSet∆op

as thepullback along the functor H0 : ∆op → ∗. This means that we turn a simplicialset X into a “discrete simplicial simplicial set,” so that H∗0 (X)([n]) = X for any[n] ∈ ∆.

Observe that the moduli space of objects X0 contains all the information ofthe original ∞-groupoid (i.e. the objects, the equivalences between them, andall the higher equivalences), and the complete Segal space (it’s evident that itmeets the conditions described in the preceding section) H∗0 (X) no contains newinformation beyond this.

That we can use Kan extensions to construct right and left adjoints (whichwe denote J∗0 and U0) to this pullback functor depends on the fact that thecodomain of the functor categories, sSet, is closed monoidal, complete, andcocomplete. Because this is the case, we can use ends and coends to constructthe desired adjoints. This is all we have to say about the left adjoint, which“freely inverts” the morphisms of a complete Segal space in a way that is difficultto describe in finer detail. However, the behavior of the right adjoint, whichsimply “forgets” the non-invertible morphisms of a complete Segal space, is fareasier to describe.

Before considering how J∗0 actually works out, let’s consider how we mightwish it to behave. The purpose of including a moduli space rather than set ofobjects is to encode the equivalences and higher equivalences of the objects of an(∞, 1)-category alongside its objects. The completeness condition tells us thatthis is whole story: there aren’t any morphisms introduced in X([1]) which windup being equivalences but which were not somehow indicated by the shape ofX([0]). So we’d hope that J∗0 (X) = X([0]), and in fact, this is precisely the case.

7See [Mac98], or [Kel05] for the enriched case.

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Proposition 2.5.2. For X a complete Segal space, J∗0 (X) ∼= X([0]).

Proof. It’s easy to see that this is how the right adjoint behaves because ∆op

has an initial object, [0]. In other words, for X ∈ sSet and Y ∈ sSet∆op

,sSet(X,Y ([0])) already completely determines P∆(H∗0 (X), Y ), since the re-striction of any map f : H∗0 (X) → Y to fn : X → Y ([n]) must commutewith the degeneracy map Y ([0]) → Y ([n]), so that each fn is determined byf0 : X → Y ([0]).

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Chapter 3

Higher categories

3.1 Strict n-categories

There is a particular class of enriched categories called “strict higher categories.”The inclusion of groupoids into categories has a certain generalization to thissetting, as do the constructions of the maximal and free groupoids of a category.These generalizations are the subject of this section. In subsequent sections ofthis chapter, we generalize strict n-categories to (∞, n)-categories; introduceadditional technical machinery from homotopy theory (i.e. simplicial modelcategories) to support a more rigorous treatment of our result; and finally,generalize the adjunctions we present in this section to the (∞, n)-categorysetting.

Definition 3.1.1. We define Cat0 to be the monoidal category Set with monoidalproduct given by the Cartesian product. Then we define the category Catn+1 ofstrict (n+1)-categories1 to be the monoidal category of small categories enrichedin Catn, again with the monoidal product given by the Cartesian product.2

Elaboration 3.1.2. Slightly less formally (but perhaps more descriptively), astrict n-category consists of a set of objects and a set of k-morphisms for each1 ≤ k ≤ n. 1-morphisms are objects of hom-(n− 1)-categories; 2-morphisms areobjects of hom-strict (n− 2)-categories inside of hom-strict (n− 1)-categories,and so on. Each k-morphism f has domain and codomain (k − 1)-morphisms.Between each pair of parallel k-morphisms (i.e. k-morphisms sharing the samedomain and the same codomain), we have a hom-strict (n− k − 1)-category, theobjects of which are (k + 1)-morphisms in our strict n-category.

Given a pair of k morphisms f : a→ b and g : b→ c, they can be composed,yielding a k-morphism g ◦ f : a → c. Composition at each level satisfiesidentity and associativity, and additionally is natural, so that, for example, given

1The word “strict” here indicates that these categories satisfy the axoms of categories upto equality, rather than only higher equivalence morphisms.

2That this is well defined, i.e. that these Cartesian products exist, follows from a standardresult of enriched category theory and the fact that Set is complete.

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x : f → h and y : g → j, we obtain y · x : g ◦ f → h ◦ j (in other words, thecomposition of k-morphisms between three fixed (k − 1)-morphisms is actuallygiven by a bifunctor of strict (n− k)-categories).

A functor of strict n-categories is a function which sends objects to objectsand k-morphisms to k-morphisms in a way which respects the domain andcodomain of k-morphisms and respects composition of k-morphisms for all1 ≤ k ≤ n. We can then define natural transformations in the normal way, butalso 3-transformations between natural transformations and so on, all the way upto n+ 1-transformations, so that Catn is itself a (large3) strict (n+ 1)-category.

Example 3.1.3. As an example of a 2-category arising in another field, wecan fix a commutative ring R and consider weak 2-category4 with objects R-algebras, morphisms from A to B given by A-B-bimodules, and 2-morphismsgiven by homomorphisms of bimodules. Composition of 1-morphisms is givenby tensor products: if M is an A-B-bimodule and N an B-C-bimodule, we candefine an A-C-bimodule structure on M ⊗B N by a(m ⊗ n) = (am) ⊗ n and(m⊗ n)c = m⊗ (nc). But note that in this case composition of 1-morphisms isassociative only up to isomorphism. This is really an example of a “bicategory”or “weak 2-category” rather than a strict 2-category. This weakened approach tohigher categories is quite important, but we do not develop the theory here, inpart because there are serious technical impediments to giving a combinatorialaccount of the higher coherence laws of the weakened versions of the categoryaxioms. Instead, we note that the theory of (∞, n)-categories which we developin the following sections naturally subsumes the theory of weak n-categories.

Next, we define strict (n+1, n)-categories. These are important because strict(n+ 1, n)-categories are to strict n+ 1-categories as groupoids are to categories.Thus, strict (n+ 1, n)-categories lie on the path to the full generalization of freeand maximal groupoid constructions.

Definition 3.1.4. We define Cat(1,0) as Grpd (again with the cartesian productas monoidal product) and Cat(n+1,n) the category of small categories enrichedover Cat(n,n−1).

In other words, a strict (n+ 1, n)-category is a strict (n+ 1)-category suchthat each (n+ 1)-morphism is an (n+ 1)-isomorphism (since it is a morphismbetween two objects (n-morphisms) in a hom-groupoid lying between two (n−1)-morphisms).

Remark 3.1.5. We have an inclusion functor Cat(n+1,n) → Catn+1. This functoris evident from the discussion of what strict n+1- and (n+1, n)-categories actuallyare, but we can also describe it by observing that the inclusion Grpd → Catis a monoidal functor (i.e. it preserves cartesian products), and thus induces

3So to think of it as an object, we need to move to a larger Grothendieck universe.4We leave this term undefined; informally, the difference between a weak 2-category and a

strict 2-category is that in a weak 2-category, the axioms of an enriched category need nothold as equalities, so long as there exist families of 2-isomorphisms filling in for the missingequalities.

15

a change-of-basis functor Cat(2,1) → Cat2, and this functor (again monoidal)induces a change of basis functor Cat(3,2) → Cat3, and so on. We also have rightand left adjoints to these functors, given by the right and left adjoints at thebottom level, discussed in Chapter 1. It is this relationship which we seek togeneralize to (∞, n)-categories and (∞, n+ 1)-categories.

3.2 The categories Θn

In the previous chapter, we saw how (∞, 1)-categories can be modeled byfunctors ∆op → sSet. In this section, we introduce categories Θn such that(∞, n)-categories can be modeled by functors Θop

n → sSet.

Definition 3.2.1. We denote an object of Θ2 by [m]([n1], . . . , [nm]) for some[m] ∈ ∆ and [n1], . . . , [nm] ∈ ∆. A morphism f : [m]([n1], . . . , [nm]) →[r]([s1], . . . , [sr]) of Θ2 is given by a morphism f∆ : [m] → [r] of ∆ alongwith a morphism fb : [na] → [sb] for each a ∈ [m] and each b such thatf∆(a) < b ≤ f∆(a+1) (note that there is at most one fb for each b ∈ [r], so our no-tation is well-defined). The identity morphism of an object [m]([n1], . . . , [nm]) ∈Θ2 is given by the identity map id[m] and the identity maps id[ni] for eachi ∈ [m]. Composition of morphisms f : [m]([n1], . . . , [nm]) → [r]([s1], . . . , [sr])

and g : [r]([s1], . . . , [sr]→ [q]([t1], . . . , [tq]) is defined so that (gf)∆def= g∆f∆ and

(gf)c : [na]→ [tc] is defined so that (gf)cdef= gcfb for g∆(b) < c ≤ g∆(b+ 1) and

f∆(a) < b ≤ f∆(a+ 1).

An object [m]([n1], . . . , [nm]) of Θ2 is meant to represent a 2-category withm+ 1 objects and [m]([n1], . . . , [nm])(i− 1, i) = [ni] (viewing the ordered set [ni]as a category) for 1 < i ≤ m. We can extend this to a full definition of a strict 2-category by letting [m]([n1], . . . , [nm]) = 0 for i < j, [m]([n1], . . . , [nm])(i, i) = ∗,and [m]([n1], . . . , [nm])(i, j) = [ni+1]× · · · × [nj ], and defining the compositionbifunctors to be the identity functors id : ([ni+1]× · · · × [nj ])× ([nj+1]× · · · ×[nk]) → ([ni+1] × · · · × [nk]). To help clarify the idea, we illustrate the strict2-category represented by [3]([1], [0], [2]) below (where we depict only morphismsbetween “sequential” objects)

0 1 2 3

This information illuminates the definition of morphisms in Θ2, since thedefinition we have given corresponds precisely to the data of a 2-functor ofstrict 2-categories. A functor f : [m]([n1], . . . , [nm])→ [r]([s1], . . . , [sr]) must beorder-preserving on objects, since there can be no functor from a non-emptyhom-category to an empty one, so the behavior of such a functor f on objects isdescribed by a morphism [m]→ [r] of ∆. By the universal property of products, amorphism [na] = [m]([n1], . . . , [nm])(a− 1, a)→ [r]([s1], . . . , [sr])(b, c) = [sb+1]×. . .× [sc] is given by a collection of morphisms [na]→ [sd] for b+ 1 ≤ d ≤ c.

16

The idea is that we could now describe some conditions which make apresheaf on Θ2 behave like a 2-category, and by replacing presheaves of setswith presheaves of spaces (and weakening the isomorphisms in those conditionsto weak equivalences) we can get a model for (∞, 2)-categories. But we areinterested in (∞, n)-categories generally, so we need categories Θn for eachnatural number n.

Definition 3.2.2. We define the categories Θn recursively beginning withΘ0 = ∗ and giving a functor Θ : Cat→ Cat.

Objects of the category ΘC have the form [m](c1, . . . , cm) where [m] ∈ ∆ andc1, . . . , cm ∈ C. A morphism [m](c1, . . . , cm)→ [q](d1, . . . , dq) of ΘC is given bya morphism f∆ : [m] → [q] of ∆ along with morphisms fb : ca → db for eacha ∈ [m] and f∆(a) < b ≤ f∆(a + 1). Composition is defined as in Definition3.2.1, except the fb are morphisms in whatever category C we are working withrather than ∆.

If L : C → D is a functor, ΘL : ΘC → ΘD sends [m](c1, . . . , cm) to[m](Lc1, . . . , Lcm) and sends a morphism in ΘC to the morphism in ΘD obtainedby using the same underlying morphism on ∆ and applying L to each of thecomponent morphisms in C.

Definition 3.2.3. We define Θn+1 to be ΘΘn. It is evident that Θ1∼= ∆ and

that this definition of Θ2 agrees with the more explicit definition we have justgiven.

3.3 Θn-spaces

We now need to describe the conditions on a simplicial presheaf on Θn ensuringthat it behaves like an (∞, n)-category. Essentially, these conditions are the sameas those for complete Segal spaces (the Segal condition and the completenesscondition), but applied at every level, so that e.g. the space of sequences ofj composable k-morphisms is equivalent to the j-fold pullback of the space ofk-morphisms over the space of (k − 1)-morphisms, and so that the space ofk-equivalences is equivalent to the space of (k − 1)-morphisms.

How do we translate these ideas into rigorous mathematical statements? Inorder to move our conditions up the ladder, we need a notion of “suspension.”

Definition 3.3.1. Given a simplicial presheaf X on Θn we define the simplicialpresheaf ΣnX on Θn+1, the suspension of X, as follows: given [q](ψ1, . . . , ψq) ∈Θn+1 we have

Σn(X)([q](ψ1, . . . , ψq))def= ∗0 q

(q∐i=1

X(ψi)

)q ∗1

Given a morphism f : [m](θ1, . . . , θm) → [q](ψ1, . . . , ψq) we need to defineΣnX(f) for each component of the coproduct ΣnX([q](ψ1, . . . , ψq)). We defineΣnX(f) to be the coproduct of id∗0 , id∗1 , X(fi) when f∆(0) < i ≤ f∆(m),X(ψi)→ ∗0 when i ≤ f∆(0), and X(ψi)→ ∗1 when i > f∆(m).

17

This definition extends naturally to morphisms of presheaves, so that wehave a functor Σn : PΘn → PΘn+1.5

The idea is to construct from an (∞, n)-category X an (∞, n+ 1)-categoryΣn(X) with two objects 0 and 1 such that the hom-(∞, n)-category between 0and 1 is X. This tool allows us to efficiently turn statements about the space of1-morphisms of an (∞, n)-category into analogous statements about the spaceof k-morphisms. We find this necessary as we seek to describe the localnessconditions for a presheaf on Θn to model an (∞, n)-category. The followingproposition confirms that Σn does its job.

Proposition 3.3.2. ΣnF (−) = F ([1](−)); i.e., the following diagram commutes:

Θn PΘn

Θn+1 PΘn+1

F

[1](−) Σn

F

Proof. The proof consists of the following calculation:

Σ(F (θ))([q](ψ1, . . . , ψq)) = ∗ q

(q∐

m=1

F (θ)(ψm)

)q ∗

∼= ∗ q

(q∐

m=1

Θn(ψm, θ)

)q ∗

∼= Θn+1([q](ψ1, . . . , ψq), [1](θ))∼= F ([1](θ))([q](ψ1, . . . , ψq))

We are now just a bit of notation away from having all the technology weneed to define the localness conditions for Θn-spaces.

Notation 3.3.3. Given φ = [m](c1, . . . , cm) ∈ Θn and 1 ≤ i ≤ m, denote byαiφ : [1](ci) → φ the map sending 0 ∈ [1] to i − 1, 1 to i, and comprising theidentity morphism idci .

Define the functor G : Θn → PΘn by

G[m](θ1, . . . , θm)def= colim

(F [1](θ1) F [0] · · · F [0] F [1](θm)Fδ1 Fδ0 Fδ1 Fδ0

)for θ1, . . . , θm ∈ Θn−1 (note that this restricts to our previous definition of G).

Definition 3.3.4. We denote by Tn the collection of morphisms with respectto which a Θn-space is local. T0 is empty. We define T1 to be the mapsfrom Definition 2.4.6 giving the Segal and completeness conditions for complete

5The definition we have given here corresponds to a restriction, denoted V [1], of Rezk’s“intertwining functor” V defined in §4.4 of [Rez10].

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Segal spaces; i.e. the morphisms which we denoted se[n] : G([n])→ F ([n]) andcpt : E → F ([0]) ∼= ∗. Recall that F denotes the Yoneda embedding. We definethe remaining collections inductively (note that the following inductive definitionapplies to T1, and agrees with the definition from Definition 2.4.6).

Let φ = [m](θ1, . . . , θm). Then denote by

seφ : Gφ→ Fφ

the morphism induced via the universal property of the colimit by the morphismsFαiφ : F [1](θi) → Fφ for 1 ≤ i ≤ m. Denote by Sen the set of morphisms seφ

for all φ ∈ Θn.6

Let Kn : P∆ → PΘn be the functor given by Kn(X)([m](ψ1, . . . , ψm))def=

X([m]). We define Tn+1def= Σn(Tn) ∪ Sen+1 ∪Kn+1(cpt).

Definition 3.3.5. A Θn-space is an object of PΘn which is local with respectto Tn.

All of these localness conditions should be interpreted via the Yoneda lemmaas conditions internal to the diagrams X : Θop

n → sSet themselves. For example,localness in PΘ2 with respect to se[2]([1],[0]) corresponds to the property that thenatural map X([2]([1], [0]))→ X([1]([1]))×X([0])X([1]([0])) be a weak equivalence.Put another way, the Segal conditions and their suspensions tell us that whenwe can decompose a diagram shape θ1 into component diagrams θ2 and θ3 (andperhaps so on), the space of θ1-shaped diagrams in a Θn-space X must agree(up to weak equivalence) with the pullback of spaces of in X of shape θ2 andθ3 (the pullback ensuring that the component diagrams agree on their overlap).Similarly, the completeness condition and its suspensions indicate that the spaceof k-equivalences must be equivalent to the space of (k − 1)-morphisms.

3.4 Model categories

In this section, we introduce simplicial model categories, model structures onfunctor categories, and (left) Bousfield localization. These tools allow us tostate and prove our main result in rigorous terms: that the inclusion functorsH∗n : Cat(∞,n) → Cat(∞,n+1) and their left and right adjoints all respect thehomotopical information of these categories, i.e. that they are functors of (∞, 1)-categories (since model categories provide another approach to the study of(∞, 1)-categories).7

Before giving the definition of a model category, we try to explain whysomeone interested in calculations in (∞, 1)-categories ought to care about model

6In fact, Sen considered in totality defines a natural transformation G → F .7In a certain sense, this is weaker than the result we would like to be true, that these functors

respect homotopies not only between 1-morphisms, but between not-necessarily-invertiblen-morphisms for all n – i.e. that they are homotopical functors not just of (∞, 1)-categories butof (∞,∞)-categories – but we content ourselves with the weaker statement for the purposes ofthe present exposition.

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categories. As the reader has (we hope) come to understand, an (∞, 1)-categoryis a category with hom-spaces rather than hom-sets and which satisfies “relaxed”versions of the axioms of a category, so that they must be satisfied only up to“coherent homotopy” rather than strict equality. This gives rise, in particular,to a weaker notion of equivalence: a morphism in an (∞, 1)-category should becalled an equivalence if there exists a morphism in the opposite direction so thatboth composites are homotopic to the corresponding identity morphisms. Thisidea is familiar both from homotopy equivalences in topology and equivalencesof normal categories (in which case the compositions of an equivalence with its“inverse” are naturally isomorphic to the identity functors).

At base, model categories move in the opposite direction: they start with acategory and some extra structure that includes a class of maps called “weakequivalences” and with this information seek to approximate an (∞, 1)-category.It is a perhaps surprising fact that from this state of affairs, it is actually possibleto reconstruct the original (∞, 1)-category uniquely up to equivalence of (∞, 1)-categories. It is this fact (which lies outside the scope of this thesis) which makesmodel categories work. We’ll elaborate a little more on the conceptual connectionbetween model categories and (∞, 1)-categories after giving the definition ofmodel categories.

The foregoing explains why we can use model categories to describe (∞, 1)-categories, but not why we should. There are several answers to the latterquestion, but the principal one is as follows. In many cases, the information wehave on hand lends itself more nicely to the structure of a model category thanto anything else. This is, for example, the case with the category of topologicalspaces: we’d like to consider two spaces equivalent when there exists a continuousmap from one to the other inducing isomorphisms on each homotopy group. Wecan work with this definition, but it doesn’t jive with our simultaneous desirethat a homotopy equivalence actually have a homotopy inverse unless we restrictour attention to CW-complexes. But the category of CW-complexes has its ownserious shortcomings: for example, it is not even closed under products in T op.

This situation in which we have a category with poor homotopical behaviorand a subcategory with good homotopical behavior but poor categorical behavioris a common one, and model categories, as we will see, provide a way of mediatingbetween these two aspects. Category theory provides us with a rich collectionof tools for studying and manipulating categories, and in most models for(∞, 1)-categories, those tools are absent or incomplete, but working with modelcategories demands no such sacrifice.8 But even if we were to choose a model for(∞, 1)-categories with a well-developed category theory, we would still need todeal with the problem of translating the category of interest into that model, forwe indeed often begin with an interest in the homotopy theory of a particularcategory. Because model categories entail no such translation, but only thespecification of some extra structure, they can be less laborious to work with.And finally, no truly synthetic theory of (∞, 1)-categories or (∞, n)-categories

8Notable exceptions to this statement are quasicategories, which have a significantlydeveloped version of category theory largely thanks to the work Jacob Lurie, and, moregenerally, ∞-cosmoi, thanks to the work of Emily Riehl and Dominic Verity.

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yet exists.Having extolled the merits of model categories, it may be unclear why we

did not introduce them as our preferred approach to (∞, 1)-categories from theoutset. The answer is straightforward: there is no “model category of modelcategories,” so we have no way of using model categories to actually study thehomotopy theory of (∞, 1)-categories, which is one of our main intentions. Aslightly more technical fact is that only (∞, 1)-categories with a property called“presentability” can be described by model categories. And finally, there is noclear extension of model categories to a theory of (∞, n)-categories for n > 1.For all these reasons, model categories are a tool or framework rather than objectof study for us.

Definition 3.4.1. A model category is a category M equipped with the fol-lowing extra structure: three wide subcategories (i.e. subcategories containingall the objects of M), the morphisms of which are called “cofibrations,” “weakequivalences,” and “fibrations.” The subcategory of weak equivalences is re-quired to contain the maximal subgroupoid of the category (equivalently, everyisomorphism is a weak equivalence).

A model category must also be equipped with two functors α, β :MD1 →MD2 where D1

def= · → · and D2

def= · → · → · and satisfying the following

axioms:

1. M is complete and cocomplete.

2. The 2-out-of-3 axiom: for any two composable morphisms f and g, if gfis a weak equivalence and either f or g is as well, then all three are weakequivalences. Because weak equivalences are a subcategory, we alreadyknow that if f and g are weak equivalences, then gf is as well.

3. Any retract of a cofibration, weak equivalence, or fibration is also a cofibra-tion, weak equivalence, or fibration, respectively. A retract of an object Ais an object B such that there exist morphisms p : A→ B and i : B → Asuch that pi = idB . When we refer to a retract of a morphism, we have inmind a retract as just described in the categoryMD1 , in which the objectsare morphisms of M and morphisms are commutative squares in M.

A (co)fibration is called acyclic if it is also a weak equivalence. A morphismp has the right lifting property with respect to a morphism i (and i has theleft lifting property with respect to p) if, for any commuting outer square ofthe following form, there exists a “lift” d which makes the following diagramcommute:

· ·

· ·i pd

21

4. All fibrations have the right lifting property with respect to all acycliccofibrations, and all cofibrations have the left lifting property with respectto all acyclic fibrations.

5. The final axiom tells us that we can always factor any morphism into anacyclic cofibration followed by a fibration or a cofibration followed by anacyclic fibration in a functorial way. Here is a picture:

A B

C

f

α1(f)

∼α2(f)

A B

C ′

f

β1(f) β2(f)

∼

That this factorization is functorial means first of all that if we have acommutative square (i.e. a morphism in MD1)

A B

D E

f

g

then applying either factorization to both arrows on opposite sides of thesquare induces an arrow in the same direction between the intermediateobjects in the factorization in the following commutative diagram:

A C B

D F E

f

α1(f)

∼α2(f)

g

α1(g)

∼α2(g)

and moreover, the factorization, being functorial, respects composition inMD1 . Lastly, the fact which all of our diagrams have indicated is true:that α sends morphisms to an acyclic cofibration followed by a fibration,and β sends morphisms to a cofibration followed by an acyclic fibration.

An object X of a model category M is called fibrant if X → ∗ is a fibrationand cofibrant if 0 → X is a cofibration. We define a functor −c :M→M byway of M →MD1 → MD2 → M by A 7→ 0 → A 7→ β(0 → A) 7→ Ac, whereAc is the codomain of the unique map β1(0 → A). We see that Ac is bothcofibrant and weakly equivalent to A. For this reason, −c is called the cofibrantreplacement functor. Beware that −c sends objects to cofibrant objects, butneed not send morphisms to cofibrations. Nevertheless, we will refer to f c as thecofibrant approximation of a morphism f . There is also a “fibrant replacement”functor with a dual role, but we have no need for it here.

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Remark 3.4.2. Here is a very brief and informal account of how a model categorygives rise to an (∞, 1)-category. Given a 1-category M with designated weakequivalences, we can consider M as an (∞, 1)-category. Then we freely adjoininverses to the weak equivalences of M, not as inverse isomorphisms, but asmorphisms such that both compositions are homotopic by 2-morphisms, ratherthan equal, to the identity morphisms. Other, higher morphisms also arise.Suppose that a single morphism f has three retracts, and that f is a weakequivalence. Then because are necessarily unique up to higher homotopy, we’llfind that our approach yields three 2- morphisms connecting these morphismspair-wise, as well as a 3- morphisms connecting these three 2- morphisms pairwise,and so on, so that the three inverses are identified up to coherent homotomopy.9

In general, there is a way of making this higher homotopical structure explicit:to every model category, we can construct an “equivalent” “simplicial modelcategory” enriched over simplicial sets, so that we in fact have hom-spaces ratherthan hom-sets. Although we do not describe the construction here, we do clarifythe meaning of these as-yet undefined terms.

The following lemma is very useful when working with model categoriesbecause it provides one of the most important ways of determining when amorphism is a (co)fibration.

Lemma 3.4.3. A morphism p is a fibration if (and only if, according to the liftingaxiom) it has the right lifting property with respect to every acyclic cofibration. Amorphism is an acyclic fibration (if and) only if it has the right lifting propertywith respect to every cofibration.

Proof. Proof omitted for brevity.

Example 3.4.4. sSet is a model category (or more precisely, has a modelstructure corresponding to the homotopy theory of spaces). We do not provethis fact, which is somewhat involved, nor do we even construct the requiredfactorization functors, but we do name the cofibrations, weak equivalences, andfibrations. A morphism f : X → Y of simplicial sets is a cofibration when it isa monomorphism. Because sSet is a functor category, this simply means thatf([n]) : Xn → Yn is an injection of sets. A morphism of simplicial sets is a weakequivalence exactly when it is a weak equivalence as defined in Definition 2.3.3.Lemma 3.4.3 then determines the fibrations, but we can give a slightly moreprecise description, since sSet has a set of generating cofibrations.

We omit the proof that this definition satisfies the remaining axioms of amodel category.

Next, a definition from enriched category theory:

Definitions 3.4.5. For a closed monoidal category V, a V-category C is saidto be tensored or copowered (the term we will use) over V if it comes equipped

9Caveat: this description treats 3- morphisms as morphisms between 2- morphisms, whichis a perfectly sensible approach to (∞, 1)-categories, but not actually the one used by a numberof models. However, because this is only an informal discussion, we do not concern ourselvestoo much with these details.

23

with a functor −�− : V × C → C such that there exists a natural isomorphismC(v � c, d) ∼= V(v, C(c, d)) for v ∈ V and c, d ∈ C.

Note that copowers generalize monoidal products in V to V-categories. Notealso that in ordinary categories, coproducts provide copowers when they exist.

Dually, we say that a V-category C is powered over V when it comes equippedwith a functor t (−,−) : Vop × C → C such that there exists a natural isomor-phism C(c,t (v, d)) ∼= V(v, C(c, d)) for v ∈ V and c, d ∈ C.

And, similarly, note that powers generalize the internal hom of V to otherV-categories, and that in ordinary categories, products provide copowers whenthey exist.

Definition 3.4.6. A simplicial model category is an sSet-category M that ispowered and copowered over sSet and is equipped with the structure of a modelcategory satisfying the following additional property.

Let i : X → Y be a cofibration of simplicial sets and j : A→ B a cofibrationofM. We can form the pushout of i�idA : X�A→ Y �A and idX�j : X�K →X �L. Then the maps idY � j : Y �A→ Y �B and i� idB : X �B → Y �Bdetermine a map (idY �j)q (i� idB) : Y �AqX�AX�B → Y �B. We requirethat this map, (idY �j)q(i�idB) be a cofibration ofM. We additionally requirethat if one of i and j is acyclic (and both are cofibrations), that (idY �j)q(i�idB)be an acyclic cofibration.

Remark 3.4.7. sSet can be given a canonical Cartesian closed structure (i.e.considered a monoidal category with monoidal product equal to the Cartesianproduct and then given an enrichment over itself) so that it is itself a simplicialmodel category.

3.5 More model category theory

We next address the question of morphisms between model categories.

Definitions 3.5.1. A Quillen pair is an adjunction L : C � D : R betweenmodel categories in which the left adjoint, called a left Quillen functor, sendscofibrations to cofibrations and acyclic cofibrations to acyclic cofibrations, andin which the right adjoint, called right Quillen functor, has the dual property.A Quillen pair is called a Quillen equivalence if the the adjunct of a weakequivalence in either model category is a weak equivalence in the other; that is, ifthe natural isomorphisms arising from the adjunction D(L(C), D) ∼= C(C,R(D))preserve weak equivalences in both directions.

Lemma 3.5.2. The right adjoint of a left Quillen functor is a right Quillenfunctor (and vice versa, by duality). Hence, any adjunction in which one functoris a Quillen functor is automatically a Quillen pair. The statement is true formodel categories in general as well as simplicial model categories (with enrichedadjunctions); our proof uses the notation of simplicial model categories becausethese are of greater interest to us.

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Proof. Let L : C � D : R be an adjunction with L a left Quillen functor. Recallthat this means that the following diagram commutes up to natural isomorphism(not necessarily the identity)

Cop ×D Dop ×D

Cop × C sSet

F op×id

id×G D(−,−)

C(−,−)

and denote this natural isomorphism γ : D(L(A), B) ∼= C(A,R(B)) for all objects(A,B) of Cop ×D.

We would like to show that for any fibration of D p : D1 → D2, R(p) hasthe right lifting property with respect to all acyclic cofibrations in C and istherefore also a fibration. To do this, we use the natural isomorphism betweenhom-functors γ to find a corresponding lift in D and to move this lift back intoC. Let i : C1 → C2 be an arbitrary acyclic cofibration in C and consider thefollowing diagrams:

L(C1) D1

L(C2) D2

a

Li ∼ pd

b

C1 R(D1)

C2 R(D2)

γ(a)

i ∼ Rpγ(d)

γ(b)

We can see that p has the right lifting property with respect to Li (which isan acyclic cofibration because L is a left Quillen functor) and the diagram onthe left (in D) induces the diagram on the right (in C, and which is commutativebecause γ is a natural isomorphism), which gives us the lifts we need to confirmthat R(p) is a fibration.

Remark 3.5.3. The realization and singular simplicial set functors form a Quillenequivalence between sSet and T op (which has a model structure in which theweak homotopy equivalences are the weak equivalences).

Further remarks 3.5.4. Each model category C has an associated “homotopycategory,” hoC constructed by freely adjoining inverses (as isomorphisms, ratherthan inverses up to higher homotopy) to all weak equivalences. If we view a modelcategory as encoding the information of an (∞, 1)-category, then the homotopycategory of a model category is the 1-truncation of that (∞, 1)-category, inexactly the way that a Postnikov tower gives the n-truncations of a space/∞-groupoid. We do not describe the construction here because we are not interestedin calculations in homotopy categories.

The definition of a Quillen pair gives us criteria for determining when anadjunction of model categories-as-1-categories induces an adjunction of model-categories-as-(∞, 1)-categories and therefore of their homotopy categories. Sim-ilarly, a Quillen equivalence induces an equivalence of (∞, 1)-categories and

25

therefore of homotopy categories, so we say that two Quillen equivalent modelcategories describe the same homotopy theory. Note that two model categorieswith equivalent homotopy categories need not be Quillen equivalent (i.e. thereneed not exist a zig-zag of Quillen equivalences between them). This statementis analogous to the fact that two spaces may have isomorphic π0 without beinghomotopy equivalent spaces: there is structure present beyond that detected bythe homotopy categories (or π0, respectively).

Some additional comments on the homotopy category of a simplicial modelcategory: first, the homotopy category of a simplicial model category M isequivalent to the category with objects the fibrant-cofibrant objects of M andwith morphisms the connected components of the hom-simplicial sets of M.Further, we can think of hoM as enriched over hosSet by defining each hom-homotopy type to the homotopy type of the corresponding hom-simplicial set ofM. Finally, every model category is Quillen equivalent to a simplicial modelcategory, so every homotopy category has an enrichment over hosSet.

More generally, there is a definition of a monoidal model category, and ifwe replace sSet with a general monoidal model category M in the definition ofa simplicial model category, we obtain the definition of an enriched M-modelcategory. If C is an M-model category, then hoC is a hoM-category.

As we have seen, it is useful for us to locate our models for (∞, n)-categoriesin the category of presheaves on Θn; i.e. in a category of functors taking valuesin the model category sSet. We would like to describe a model structure onthis category, and as a starting point, we would like morphisms of functors (i.e.natural transformations) which are objectwise weak equivalences in sSet to beweak equivalences in our model structure. Fortunately for us, there two ways forus to set this up. Both of them will prove useful later.

Definition 3.5.5. Given a model category M and an indexing category C,we define the projective model structure on MC to be the model structure,if it exists, in which the weak equivalences and fibrations are the objectwiseweak equivalences and fibrations respectively. We define the injective modelstructure to be its dual: the model structure in which the weak equivalences andcofibrations are defined objectwise.

Remark 3.5.6. Note that in both cases these model structures are unique if theyexist, since the weak equivalences and fibrations determine the acyclic fibrations,which in turn determine the cofibrations. Both of these model structures alwaysexist as long as M is a “combinatorial model category;” this is a technicalcondition which lies outside the scope of the present exposition, but the fullproof, as well as further details on much of what is to follow, can be found in[Hir03]. The relevant fact is that sSet is a combinatorial model category, sowe can assume that model structures with the indicated properties exist oncategories of functors into sSet.

Proposition 3.5.7. The identity functor is a Quillen equivalence of the injectiveand projective model structures on MC (the right Quillen functor goes from theinjective structure to the projective structure).

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Proof. This is immediate once we verify that the identity functor is Quillen,since the adjunct of a morphism is itself, and the weak equivalences are the

same in both model structures. Consider id :MCproj →MC inj. We claim that

this is a left Quillen functor, so we need to verify that it sends cofibrations to

cofibrations. A cofibration in MCprojhas the left lifting property with respect

to objectwise acyclic fibrations, which means in particular that they have the

left lifting property with respect to morphisms in MCprojwhich as natural

transformations are constant families of acyclic fibrations between constant

diagrams. Because the component morphisms in M of a cofibration in MCproj

have the left lifting property with respect to all acyclic fibrations inM, they are

objectwise cofibrations (by Lemma 3.4.3) and therefore cofibrations inMC inj.

Proposition 3.5.8. The projective and injective model structures on functorcategories with codomain sSet are naturally simplicial model categories.

Proof. Proof omitted.

The following theorem will be of great importance later on, and explains whywe need both the injective and projective model structures.

Theorem 3.5.9. Given a functor H : A→ B, we obtain a “pullback” functor

of functor categories H∗ : sSetB → sSetA given by H∗ : X(a)def= X(Ha) for

X : B → sSet.10 This functor is both a left and right adjoint thanks to theexistence of left and right Kan extensions (which in turn exist because sSet iscomplete and cocomplete). H∗ is a left Quillen functor when we give sSetB andsSetA the injective model structures, and is a right Quillen functor when we givesSetB and sSetA the projective model structures.

Proof. Proof omitted.

So far, we have seen that projective and injective model categories give usmodel category structures for Θn-spaces such that substituting weakly equivalentsimplicial sets (via natural transformations which are objectwise weak equiva-lences) yields weakly equivalent Θn-spaces. This is certainly desirable, but itdoes not tell us the whole story: after all, Θn spaces are more than just diagramsof simplicial sets, but should have their own internal structure capturing thestructure of an (∞, n)-category and reflected in additional weak equivalences(just as equivalent categories did not necessarily yield isomorphic simplicialsets in §2.2). We accomplish this (i.e. adding more weak equivalences whilerestricting our collection of “good,” fibrant objects to those with the propertieswe desire) by a process called Bousfield localization.

Localization is a common idea in mathematics: the “localization” of anobject R at some subobject S (using the term loosely) refers to a morphisml : R→ S−1R with the universal property that every morphism from R sending Sto a subobject of elements that are somehow invertible factors uniquely through

10We could use any other combinatorial simplicial model category here, but sSet is the onlyexample relevant to our present purpose.

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l. Perhaps one of the most common examples of localization is the localizationof a ring R at a multiplicative subset S, which produces a homomorphisml : R → S−1R in which l(S) consists entirely of units, and the constructionof S−1R is accomplished in the most “free” way possible. This operation isclosely related to the one which presently interests us: we can consider ringsas single-object categories enriched in Ab: just as we can localize rings, wecan localize categories by “freely inverting” some set of morphisms – that is,sending those morphisms to isomorphisms. But in model category theory, therelevant notion of “invertible morphism” is not the isomorphism but the weakequivalence.

Definition 3.5.10. Given a simplicial model category11 M with cofibrantreplacement functor denoted −c and a set of morphisms S, the left Bousfieldlocalization ofM at S is a new model structure on the same underlying category.We call a fibrant object A ∈M S-local if for all morphisms f ∈ S, M(f c, A) isa weak equivalence of simplicial sets. When S = {f} is a singleton, we may alsorefer to an object as being f -local. A morphism w of M is called an S-localequivalence if M(w,A) is a weak equivalence whenever A is S-local. This givesus a new model category structure with the same cofibrations (and therefore thesame acyclic fibrations), more weak equivalences (the S-local weak equivalences),and fewer fibrations. We omit the redefinition of the acyclic cofibration-fibrationfactorization functor α :MD1 →MD2 because it is not needed in what follows.Note that no redefinition of β is necessary because our new model structure hasthe same cofibrations and the same acyclic fibrations.

Remark 3.5.11. The intuition here is that fibrant objects are the “good” objectswhich “detect” weak equivalences, and so if we want to add S to our collectionof weak equivalences, we need to restrict our definition of fibrant objects tothose which perceive morphisms of S as such. This in turn gives us a way ofdescribing the other morphisms which must necessarily be weak equivalences(by the axioms of model categories) if those of S are.

As before, we refer the reader to [Hir03] for the proof that left Bousfieldlocalization in fact yields a model category structure (which requires again theassumption that M be a combinatorial model category).

Definition 3.5.12. We now define model categories Catinj(∞,n) and Catproj

(∞,n)

modeling the homotopy theory of (∞, n)-categories. We begin with the injectiveand projective model structures on PΘn, respectively. We then take the Bousfieldlocalization of these model categories with respect to the morphisms Tn as definedin Definition 3.3.4.

Proposition 3.5.13. Catinj(∞,n) and Catproj

(∞,n) are Quillen equivalent (with the

equivalence given by the identity functor), and are thus both acceptable modelsfor the homotopy theory of (∞, n)-categories.

11There is a definition of left Bousfield localization which applies to all model categories,not just simplicial model categories. However, the definition restricted to simplicial modelcategories is more easily stated and applied, and is also all that we require.

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Proof. This follows from Lemma 3.5.7

Before proving the main result of our paper, we have one last essential lemmato state.

Lemma 3.5.14. Suppose L :M� N : R is a Quillen pair and that S and Tare sets of morphisms in M and N respectively. Then L : S−1M� T−1N : Ris also a Quillen pair if either of the following equivalent conditions are satisfied:R sends T -local objects to S-local objects, or L sends the morphisms of S toT -local equivalences.

Proof. Proof omitted.

3.6 The inclusion functor is left and right Quillen

In this section, we describe two models for the inclusion functor of (∞, n)-categories into (∞, n+ 1)-categories (both are the same functor, but on differentmodel structures). This functor naturally has left and right adjoints. We showthat in the appropriate model structures, the inclusion functor is a left or aright Quillen functor, which by Lemma 3.5.2 establishes that its adjoints arealso Quillen functors, thus generalizing the functors MaxGrpd : Cat→ Grpd andFreeGrpd : Cat→ Grpd we introduced in Chapter 1 and establishing that theseadjunctions respect the homotopy theory of (∞, n+ 1)-categories.

Notation 3.6.1. In the following, we use Hn : Θn+1 → Θn to denote the functor

Hndef= Θ(n)(∆→ ∗) and Jn : Θn → Θn+1 the functor Jn

def= Θ(n)(∗ → ∆) where

Θ(n) denotes n-fold composition of the functor Θ : Cat→ Cat defined in Definition3.2.2, and ∗ → ∆ refers to the functor ∗ 7→ [0].

We see that Hn forgets the “deepest layer” of data in an object θ ∈ Θn+1 (e.g.H1 : [3]([1], [0], [6]) 7→ [3]); or, if we think of Θn+1 as a subcategory of Catn+1,Hn is the restriction of the left adjoint of the inclusion functor Catn → Catn+1.12

We see that Jn appends copies of [0] to the “tails” of an object of Θn, so thate.g. J1[2] = [2]([0], [0]), and, if we think of Θn ⊂ Catn, Jn is the restriction ofthe inclusion of strict n-categories into strict (n+ 1)-categories.

We can confirm that Jn is the right adjoint to Hn by noting that [0] is theterminal object of ∆, so that just as the deepest layer of an object θ ∈ Θn+1

is irrelevant in determining the morphisms of Θn+1(Hn(φ), θ) (because thefunctor Hn has forgotten this information), so is it irrelevant in determiningthe morphisms of Θn(φ, Jn(θ)) (because no matter what data lies in φ at the“deepest layer,” there is always only one morphism from an object of ∆ to [0]).

We use H∗n as our inclusion functor. As a pullback functor, it has left andright adjoints given by left and right Kan extension (since sSet is complete

12This functor, the left adjoint to the inclusion of strict n-categories into strict (n + 1)-categories, identifies a pair of n-morphisms in a strict (n + 1)-category whenever thosen-morphisms are connected by a zig-zag of (n + 1)-morphisms, and leaves the objects andk-morphisms for k < n untouched.

29

and cocomplete). In what follows, we prove that H∗n : Catinj(∞,n) → Cat

inj(∞,n) is

a left Quillen functor, and that H∗n : Catproj(∞,n) → Cat

proj(∞,n) is a right Quillen

functor. Our result follows from lemmas 3.5.9, 3.5.14, and 3.5.2 once we verifythe hypotheses of Lemma 3.5.14.

We note that when verifying that H∗n : Catinj(∞,n) → Cat

inj(∞,n+1) is left Quillen,

we can disregard the role of cofibrant replacement in the hypotheses of Lemma3.5.14. Recall that the cofibrations of sSet are exactly the monomorphisms.Because the cofibrations of the injective model structure are the object-wisecofibrations, and because left Bousfield localization preserves cofibrations, wehave that the cofibrations of Catinj

(∞,n) are still exactly the monomorphisms.

In particular, every object is cofibrant, and the cofibration-acyclic fibrationfactorization functor can be defined so that the resulting cofibrant replacementfunctor is the identity. This means that we can work directly with Tn as weverify the hypotheses of Lemma 3.5.14.

Theorem 3.6.2. H∗n : Catinj(∞,n) → Cat

inj(∞,n+1) is a left Quillen functor.

Proof. We use induction to show that H∗n(Tn) ⊂ Tn+1. The base case is trivial,since T0 is empty. To see that H∗nKn(cpt) ∈ Tn+1 and that H∗n(Sen) ⊂ Tn+1, wedo not actually need the inductive hypothesis. However, the inductive hypothesisthat H∗n−1(Tn−1) ⊂ Tn is necessary for us to prove that H∗nΣn−1(Tn−1) ⊂ Tn+1.

Observe that, according to our definition of Kn in Definition 3.3.4, H∗nKn =Kn+1, so that we have H∗nKn(cpt) = Kn+1(cpt) ∈ Tn+1 by definition.

We next show that H∗n sends morphisms of Sen to morphisms of Sen+1. Inparticular, given seφ ∈ Sen, we claim that H∗n(seφ) = seJn(φ). Recall fromDefinition 3.3.4 that seφ : G(φ)→ F (φ) is determined by the universal propertyof G(φ), which is in turn a colimit of a diagram lying in the image of theYoneda embedding F : Θn → PΘn. Because H∗n is a left adjoint functor, itpreserves colimits, so to conclude the proof of our claim, we just have to verifythe commutativity of the following diagram.13

Θn PΘn

Θn+1 PΘn+1

F

Jn H∗n

F

Using the fact that Hn and Jn are adjoints, we have

FJn(θ)(−) = Θn+1(−, Jθ) ∼= Θn(Hn−, θ) = F (θ)(Hn−) = H∗nF (θ)(−)

We finally need to show that H∗nΣn−1(Tn−1) ⊂ Tn+1. We now invoke our in-ductive hypothesis that H∗n−1(Tn−1) ⊂ Tn. Then by definition, ΣnH

∗n−1(Tn−1) ⊂

13More precisely, when we say that a morphism “is determined by the universal property ofa colimit,” what we are really saying is that it is the adjunct morphism to some morphism ina diagram category. Composition of adjunctions means that left adjoints preserve not justcolimits, but also the morphisms arising from their universal properties.

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Tn+1. But ΣnH∗n−1 = H∗nΣn−1 because H∗n is a left and right adjoint (and there-

fore commutes with both limits and colimits), while Σ is defined to be a colimitover a limit. Hence H∗nΣn−1(Tn−1) ⊂ Tn+1 and so H∗n(Tn) ⊂ Tn+1 for all n, and

so H∗n : Catinj(∞,n) → Cat

inj(∞,n+1) is left Quillen.

Theorem 3.6.3. H∗n : Catproj(∞,n) → Cat

proj(∞,n+1) is a right Quillen functor.

Proof. We need to show that if X ∈ PΘn is Tn-local, then H∗n(X) is Tn+1-local. As we have just seen in the proof of Theorem 3.6.2, a large portion ofthe morphisms in Tn+1 lie in the image of H∗n. Fortunately for us, there is astraightforward proof that if X is f -local for f any morphism in PΘn, thenH∗n(X) is H∗n(f)-local.

First, note that HnJn = idΘn , and so also J∗nH∗n = idPΘn . Suppose that

PΘn(f,X) is a weak equivalence of simplicial sets; then we need to show thatPΘn+1(H∗n(f), H∗n(X)) is as well. By the (enriched) adjunction H∗n a J∗n,PΘn+1(H∗n(f), H∗n(X)) is adjunct to PΘn(f, J∗nH

∗n(X)) = PΘn(f,X), which

is a weak equivalence by assumption.Now we must address localness with respect to the maps f ∈ Tn+1 not in

the image of H∗n. Reviewing the proof of Theorem 3.6.2, we see that these mapsare those seφ for φ not in the image of Jn, Σ(n)(se[m]), and Σ(n)(cpt), where

Σ(n) def= Σn · · ·Σ1.

Suppose we are given φ = [m](θ1, . . . , θm) ∈ Θn+1 such that φ is not in JnΘn.By the Yoneda lemma and the definition of H∗n, H∗n(X) local with respect toseφ is equivalent to

(X(α1φ), . . . , X(αmφ )) : X(Hn(φ))→

X(Hn[1](θ1))×X([0]) · · · ×X([0]) X(Hn([1](θm)))

a weak equivalence. But we can check from the definitions that Hnαiφ = αiHn(φ),

so this is equivalent to X local with respect to seHn(φ), which we have assumedto be true.

We next address localness with respect to the maps Σ(n)(se[m]) and Σ(n)(cpt).

Define the functor [1]n(−) : Θ1 → Θn+1 by [1]n([k])def=

n copies︷ ︸︸ ︷[1](· · · ([1]([k]) · · · ). Then

we also have the pullback functor [1]n∗

: PΘn+1 → PΘ1. By Proposition 3.3.2and the Yoneda lemma, H∗n(X) local with respect to Σ(n)f is equivalent to[1]n

∗H∗n(X) local with respect to f , so what we need to verify is that [1]n

∗H∗n(X)

is a complete Segal space. But by the definition of H∗n, we know that [1]n∗H∗n(X)

is the constant presheaf evaluating to X([1]n−1([1])). By Proposition 2.4.7,constant presheaves on Θ1 satisfy the localness conditions for complete Segalspaces, so we are done.

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