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What are the Opetopes?Formalizing the Definition
Notation and ImplementationThe Opetopes and Type Theory
Type Theory and the OpetopesHDACT - Ljubljana
Eric Finster
June 20, 2012
Eric Finster Type Theory and the Opetopes
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What are the Opetopes?Formalizing the Definition
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Outline
1 What are the Opetopes?
2 Formalizing the Definition
3 Notation and Implementation
4 The Opetopes and Type Theory
Eric Finster Type Theory and the Opetopes
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What are the Opetopes?Formalizing the Definition
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Shape Categories
Definitions of higher categories typically begin with the selection of a shape to represent higher dimensional cells:
For example, there’s the globe category Gop :
(0) (1) (2) (3)
We’ve got the simplicial category ∆op :
[0] [1] [2] [3]
But there’s also the category of opetopes O:
⇐???
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The Idea of Opetopes
The two main priciples behind the definition of the opetopes arethe following:
The Informal Version
1 Cells will be allowed to have many sources (input faces), butonly a single target (output face)
2 Cells of dimension n + 1 should be in bijection with pasting diagrams in dimension n, that is, all possible ways of
attaching cells by gluing compatible sources and targets
We think of the process of turning a given pasting diagram into acell as extruding it into the next dimension up.
Eric Finster Type Theory and the Opetopes
Wh h O ?
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What are the Opetopes?Formalizing the Definition
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Low Dimensions
In dimension 0, we have a point. It has no source and notarget.
The only way to arrange a family of points, gluing sources totargets is to simply have a single point. Points do not coherein any meaningful way.
Extending our unique 0-dimensional pasting diagram gives usthe unique 1-dimensional cell, the arrow.
Eric Finster Type Theory and the Opetopes
Wh t th O t ?
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Low Dimensions (cont’d)
Now in dimension 1, we have the arrow: it has a single sourceand a single target.
What are all the ways of coherently gluing sources to targetsin a collection of arrows?
There are an N’s worth:
0 1 2 3
Now we extrude each pasting diagram into the next
dimension, and give it an “appropriate” target. In the case athand, we have only one choice: the arrow.
So our two cells look like this:
⇐ ⇐ ⇐ ⇐
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What are the Opetopes?
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What are the Opetopes?Formalizing the Definition
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Low Dimensions (cont’d)
Here are some 2-dimensional pasting diagrams:
⇐ ⇐
⇐
⇐
⇐
⇐ ⇐
⇐ ⇐
And and example 3-dimensional cell:
⇐
⇐
⇐
⇐ ⇐
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What are the Opetopes?Formalizing the Definition
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Low Dimensions (cont’d)
And finally a 3-pasting diagram:
⇐
⇐
⇐
⇐ ⇐
⇐ ⇐ ⇐
⇐
⇐ ⇐
⇐
Eric Finster Type Theory and the Opetopes
What are the Opetopes?
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What are the Opetopes?Formalizing the Definition
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Opetopes from Polynomial Funtors
1 How can we make this intuitive definition precise?
2 One of the simplest ways to do this (due to Kock, Joyal,Batanin and Mascari) is to realize these shapes as a canonicalsequence polynomial functors
3 These have different names in the computer sciencecommunity: inductive families, indexed containers, indexed
W -types, . . .
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What are the Opetopes?
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Polynomial Functors
DefinitionA polynomial P is a diagram of sets
E
p
> B
I
t
< I
r >
Any polynomial dertermines a functor P : S et /I → S et /I (its
extension) defined for an I -Set X → I by the formula:
P (X ) =b ∈B
p ∈E b
X t (p )
(Lower subscripts indicate the fibers of appropriate maps.)
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What are the Opetopes?
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p pFormalizing the Definition
Notation and ImplementationThe Opetopes and Type Theory
Graphical Interpretation
It’s useful to represent the elements b ∈ B as corollas
b
i 0
i 1 i 2 i 3 i n
We can then picture the set P (X ) as the collection of suchcorollas labelled with elements from X of the correct type:
P (X ) =
s
i 0
x 1 x 2 x 3 x n
b ∈B
That is, t (x k ) = i k .
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What are the Opetopes?
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p pFormalizing the Definition
Notation and ImplementationThe Opetopes and Type Theory
Useful Special Cases
Write 1I for the terminal object of S et /I . Then it is easilyseen that P (1I ) = B . Graphically:
P (1I ) = b
i 0
i 1 i 2 i 3 i n
b ∈B
For the initial object, we have
P (∅) =
b
i 0
i.e., the set of constructors with no places.
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What are the Opetopes?
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Formalizing the DefinitionNotation and Implementation
The Opetopes and Type Theory
Useful Special Cases (cont’d)
By iterating the functor, we generate trees: for example,P 2(1I ) = P (B ) is the set of two leveled trees
P 2(1I ) =
b 0
b 1 b n
Eric Finster Type Theory and the Opetopes
What are the Opetopes?fi
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Formalizing the DefinitionNotation and Implementation
The Opetopes and Type Theory
Monads
When is the extension of an indexed container a monad?In particular, we would need to have a map
µ1 : P 2(1I ) → P (1I ) = B
We can view this as a way to compose two-leveled trees:
b 0
b 1 b n⇒ µ(b 0; b 1, . . . , b n)
We say the monad is cartesian if the places of the multipliedconstructor are in bijection with the leaves of the two-leveltree (and their types match)
Eric Finster Type Theory and the Opetopes
What are the Opetopes?F li i h D fi i i
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Formalizing the DefinitionNotation and Implementation
The Opetopes and Type Theory
The Free Monad
We can freely generate a monad from any polynomial, andmoreover, this functor is again the extension of a polynomial
Write
tr (P ) =
n→∞
(I + P )n(∅)
The elemenets are the finite tree’s built from constructors inP (plus some units)
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What are the Opetopes?F li i th D fi iti
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The Free Monad (cont’d)
For a tree t ∈ tr (P ) write L(t ) for its set of leaves
Then the free monad on P is given by the polynomial
t ∈tr (P )
L(t ) π
> tr (P )
I <
I
>
The multiplication in this monad is given by simply graftingtrees together at their leaves
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What are the Opetopes?Formalizing the Definition
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The Opetopes and Type Theory
The Slice Construction
Observe that when P is a (cartesian) monad, we have a map
µ∞ : tr (P ) → B
which “collapses” each tree to a corolla
Write N (t ) for the set of internal nodes of a tree t ∈ tr (P )
The slice construction P + on P is the polynomial
t ∈tr (P )
N (t ) π
> tr (P )
B <
B
µ∞
>
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Formalizing the DefinitionNotation and Implementation
The Opetopes and Type Theory
Multiplication in the Slice Construction
Theorem
The slice construction P + is again a (cartesian) monad
Multiplication is given by substitution of trees.
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What are the Opetopes?Formalizing the Definition
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Definition of the Opetopes
One useful monad is the identity functor on S et , represented bythe trivial polynomial:
∗ > ∗ = O(1)
∗ < ∗ = O(0)
µ∞
>
Definition
The set O(n) of n-dimensional opetopes is the indexing set of then-th slice of the identity functor on S et .
Eric Finster Type Theory and the Opetopes
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Notation for the Opetopes
Our picture of tree substitution above leads naturally to thefollowing graphical notation for depicting opetopes in alldimensions
A nesting is a configuration of non-intersecting cicles and dotsin the plane which corresponds to a tree
⇒
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gNotation and Implementation
The Opetopes and Type Theory
Notation (cont’d)
A constellation is a nesting and a tree superimposed so thatthe nodes of the tree are the dots in the nesting
These are subject to two rules1 There must be an outer circle containing all other dots and
circles, except possibly if the tree contains exactly one node2 Every circle must cut a subtree (no “hanging” circles)
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Notation (cont’d)
An opetope can now be represented by a sequence of suchconstellations, with the dimension given by the number of
terms in the sequenceThis is subject to an initial condition and a simple rule formoving to higher dimensions
You can play with this notation in a graphical editor here:
http://sma.epfl.ch/~finster/opetope/opetope.html
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Notation and ImplementationThe Opetopes and Type Theory
Notational Example
⇐
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What are the Opetopes?Formalizing the Definition
N i d I l i
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Notation (cont’d)
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What are the Opetopes?Formalizing the Definition
N t ti d I l t ti
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Notation and ImplementationThe Opetopes and Type Theory
Opetopes and Globs
Globular shapes are a special case of opetopes:
⇒
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Notation and ImplementationThe Opetopes and Type Theory
Implemetation
Opetopes can be represented by the following inductive type:
data MTree (A : Set) : N → Set where
obj : MTree A 0
drop : {n : N} → MTree n → MTree A (n + 2)
node : {n : N} → A → MTree (MTree A (n + 1)) n
→ MTree A (n + 1)
Elements of this type are “possible ill-typed A-labelled pastingdiagrams”
It is not hard to implement a “type-checker”
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Notation and ImplementationThe Opetopes and Type Theory
The Derivative
For implementing type-checking, the following “higher-dimensionalzipper” is extremely useful:
data Deriv (A : Set) : N → Set where
∂ : {n : N} → MTree (MTree A (n + 1)) n
→ Zipper A (n + 1) → Deriv A (n + 1)
data Zipper (A : Set) : N → Set where
Nil : {n : N} → Zipper A (n + 1)
Cons : {n : N} → A → Deriv (MTree A (n + 1)) n
→ Zipper A (n + 1) → Zipper A (n + 1)
Context : Set → N → Set
Context A n = Tree A n × Zipper A n
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Cells, Frames and Niches
When working with simplicial sets, we have three canonicalfamilies
1 Simplices: ∆n
2 Boundaries: ∂ ∆n
3
Horns: Λ
n
k
Opetopic sets have similar notations:
⇐
Cell Frame Niche
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Notation and ImplementationThe Opetopes and Type Theory
Opetopic “Identity” Types
Consider the formation rule for identity types:
Γ A : Type
Γ, x : A, y : A Id A(x , y ) : Type
Iteration gives a derived rule:
Γ A : Type
Γ, x : A, y : A, f : Id A(x , y ), g : Id A(x , y ) Id Id A(x ,y )(f , g ) : Type
In each case, the data required in the context is exactlycorresponds to a frame for a globular opetope.
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pThe Opetopes and Type Theory
Opetopic ”Identity” Types (cont’d)
Let π denote an abitrary opetope
Write Γ, F : Aπ · · · as shorthand for the assumption of avariable for every face of the frame associated to π
Example: for π the 2-frame below
x
f y g
h
k
z
w
we would have
Γ, x : A, y : A, z : A,w : A, f : Id A(x , y ), · · · · · ·
Similarly, Γ, [N : A]π
· · · means enough variables for thefaces of the niche associated to π
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pThe Opetopes and Type Theory
Opetopic Formation and Introduction
Γ A : Type O-Formation
Γ, F : Aπ Fill (F ) : Type
Γ [N : A]π O-composition
Γ comp (N ) : Fill (N |τ (π))
Γ [N : A]π
O-reflectionΓ refl (N ) : Fill (N comp (N ))
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The Opetopes and Type Theory
Introduction Examples
x
f
y g
h
z
w
x
comp (x )
refl (x )
comp (N )
refl (N )
When π is a glob, it contains a unique top dimensional sourceface, say x , and a new reduction rule says that comp (x ) → x in this case
This corresponds to the slogan “a nullary composition is anisomorphism”
Eric Finster Type Theory and the Opetopes
What are the Opetopes?Formalizing the DefinitionNotation and Implementation
Th O d T Th
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The Opetopes and Type Theory
A Generalized J-Rule
The J -Rule
Γ, x : A, y : A, f : Id A(x , y ) P (x , y , f ) : Type
Γ, x : A p (x ) : P (x , x , refl (x ))
Γ a : A Γ b : A Γ g : Fill (G )Γ J (a, b , g ) : P (a, b , g )
An Opetopic J -Rule:
Γ, F : Aπ, α : Fill (F ) P (F , α) : Type
Γ, [N : A]π p (N ) : P (N comp (N ), refl (N ))Γ G : Aπ Γ β : Fill (G )
Γ J (G , β ) : P (G , β )
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Th O t d T Th
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The Opetopes and Type Theory
The Opetopes as a Substitution Calculus
The opetopes come equipped with a natural substitutionoperation arising from the fact that they are constructors in apolynomial monad
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The Opetopes and T pe Theor
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The Opetopes and Type Theory
Substitution (cont’d)
By introducing binding, we can build a rewrite systemreminiscent of λ-calculus:
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The Opetopes and Type Theory
Opetopic Type Theory
The opetopes provide a natural framework for organizing higherdimensional type-theoretic concepts geometrically:
Dimension Terms
0 Contexts1 Types2 Proofs
3 Proofs w/ Metavariables· · · · · ·
Eric Finster Type Theory and the Opetopes