transfer functors and projective spaces

Math. Nachr. 118 (1984) 147-165

Transfer Functors and Projective Spaces

By iKmco GRANDIS of Genova

(Received March 30, 1983)

Introduction

This work studies the images and inverse images of subobjects (and quotients) in exact categories (in the sense of PUPPE), and related notions. Any exact category E is provided with a transfer functor SubE: E 3 Mlc sending E into the exact category of modular lattices and modular connections (that is, GALOIS connections satisfying certain conditions) ; this functor is exact and reflects exacts sequences (though not faithful, generally) : hence an obvious metatheorem for exact categories. The quotient Trn (E) of E modulo the congruence associated to SubE is a sort of “projective” (exact) category associated to E: actually, if E is the (abelian) category of vector spaces on a commutative field K , Trn (E) is the category of projective spaces on K , still exact but no more abelian (for K .$I Z2).

After a review of exact categories (n. l), in n. 2 the category Mlc is introduced, and in n. 3 the category of relations on Mlc is shown to have a concrete realization MI r. In n. 4 we define the transfer functor SubE of the exact category E, and show how most of the diagrammatic lemmas can be checked in Mlc and “extended” to every exact category E, via SubE. Transfer categories and projective spaces are examined in n. 6.

In n. 6 we consider distributive and boolean exact categories (i.e. those having distributive or boolean lattices of subobjects); the latter are proved to coincide with exact inverse categories, already studied in [22]. Last, n. 7 sketches other examples of transfer functors, outside the frame of exact categories.

1. Review of exact categories

1.1. General conventions. Let C be a category: u E C ( A , B) will mean that u is a morphism of C , having domain A (Dom u = A ) and codomain B (Cod u = B ) ; the identity of the object A is written lA.

In every class of equivalent monomorphisms of codomain A , one is chosen and is called a subobject of A ; in the class of isomorphisms of codomain A , we choose 1,; when existing, images and kernels will always be supposed to be subobjects. Analog- ously for quotients, coimages and cokernels.

(1) C s X ’ + D , F = F , . P , where F2 is a faithful functor and F1 is a quotient (i.e. a functor bijective on the objects and full) : take for X the quotient category C/-F, where u wF v if u and w are parallel

10*

Any functor F : C -+ D has a CAT-factorization:

148 Grandis, Transfer Functors and Projective Spaces

maps of C and F ( u ) = F(v). Such a factorization is uniquely determined, up to isomorphism of categories.

We choose, once for all, a reference universe @. A %-category has objects and morphisms belonging to 4; i t is small if the set of morphisms (and therefore also the set of objects) belongs to @.

1.2. A category E is exact (in the sense of PUPPE~)) if it has a zero-object 0 (both initial and terminal) and every map factorizes via a conormal epi followed by a normal monic ; we always suppose it is a well-powered @-category (hence also well-copowered). The dual of an exact category is exact too.

1.3. Thus the exact category E is provided with kernels and cokernels, and is normal, conormal and factorizing (i.e. has epi-monic factorizations, unique up to isomorphism). Products and sums, general pullbacks and pushouts, equalizers and coequalizers need not exist; however E does have pullbacks of converging arrows if one of them is monic, and dually.

1.4. The E-subobjects of an object A build a modular lattice (with 0 and l) , to be written SubE ( A ) or Sub (A) . Any morphism u: A --t B determines two transfer mappings of subobjects [12], the image via u (written u* or u,) and the inverse image via u (u* or us), verifying the usual properties:

(1)

(2)

(3)

(4)

u* : Sub ( A ) -+ Sub (B) preserves finite unions (hence 0)

u* : Sub ( B ) --f Sub ( A ) preserves finite intersections (hence 1)

u*u*(x) = z v ker u (z E Sub ( A ) )

u*u,(y) = y A in1 u (y E Sub (B) ) .

There is a canonical anti-isomorphism between the lattice of subobjects of A and that of quotients:

Sub ( A ) 5 Quo ( A )

which commutes with images and inverse images; therefore we shall mainly consider the subobjects. We write uq and uQ the transfer mappings of quotients via u.

1.6. An exact category is abelian (with a determined additive structure) iff i t has finite products and finite sums [18; Th. 1.20.11.

Among non-abelian exact categories we have: (small) cyclic groups; (small) vector spaces of dimension lower than a fixed integer; (small) sets and partial bijections; Mlc, Dlc, Blc (n. 2; 4.6; 6.1); the distributive or boolean expansions of any exact, non null category (n. 6); (generally) the transfer category of an exact category (n. 6), in- cluding categories of projective spaces.

Moreover any full, locally full and colocally full subcategory of an exact category is an exact subcategory (i.e. exact in its own right, and with exact embedding (1.8)).

l) These categories were introduced by PIJPPE as qwcei-wact ([20], 1962), and succesively called muct in MITOEELL’S book [18]; also studied in [3], [4], [S], [el, [7], [ll], [12], [17], [22]. BARR and others used this term in a different aense.

Granclis, Transfer Functom and Projective Spaces 149

1.6. The exact category E is canonically imbedded in its category of relations Re1 (E)

The latter has the same objects as E; a morphism a : A’ + A ” has an essentially (see [51, [61, [31, [41, [71).

unique bipuaternary factorization [7, p. 1671 :

(1) a = ngp6 = g’n’fi’p’

where the two squares are bicartesian in E; the composition can be described by limits and colimits existing in E. Re1 (E) is provided with an involution a H b (which is regular: aba = a ) and an order 5 on parallel maps, agreeing with composition and involution.

1.7. The relation u : A‘ +A“ (1.6.1) determines the following subobjects of its domain or codomain [7, p. 152, 1711:

(1)

(2)

(3)

(4)

(5)

def a : Def a H A’ ann a : Ann a ++ A’

val a : Val a * A” ind a : Ind a H A“

ann a. < def a ;

(def a N m) (ann a = ker p’)

(Val a N n)

(ind a = ker q‘)

ind a < V a l a .

Each of the following conditions defines a subcategory of Re1 (E) :

(6) def a = 1 (iff da 2 1)

(7) ann a = 0 (iff 6a 1)

(8) Val a = 1 (iff a6 2 1)

(9) ind a = 0 (iff ad 5 1 ) .

These subcategories and their intersections will be called fundamental subcategories of Re1 E ; among them there are: E itself (conditions (6) and (9), respectively imposing that a be “everywhere defined” and “single valued”), the opposite category E* ((7) and (8)), the category of monorelations ((6) and (7)), epirelations ((8) and (9)), monomorphisms of E ((6)) (7) and (9)), epimorphisms of E ((6), (8) and (9)), isomorphisms of E and Re1 E((6)-(9)).

1.8. An exact functor P : El --f E, is a functor between exact categories which preserves exact sequences, or equivalently short exact sequences, or also kernels and cokernels (up to equivalence of monics and epis).

A zero-preserving functor F : E, + E, (between exact categories) is exact iff [7, Th. 6.151 it extends to an involution preserving functor Re1 (P) : Re1 E, -+ Re1 E,; the latter is uniquely determined and preserves 5 and all fundamental subcategories; moreover Re1 F reflects all fundamental subcategories iff it reflects the isomorphisms : this certainly happens when P (or equivalently Re1 F ) is faithful.


1.9. Every exact functor F : E l + E , has an essentially unique EX-factorization (already considered in [ 171) :

(1) El % E % E , , F = F2F1 where E is an exact category, F2 a faithful exact functor and F , an exact quotient : that is an exact functor, bijective on the objects and spanning E (every morphism of E is to be a composition of reached morphisms and of inverses of reached isomorphisms).

The uniqueness (up to isomorphism of categories) is easily checked; as to the existence, the extended functor a = Re1 F : Re1 El --f Re1 E , has a CAT-factorization

(2) Re1 E l -% A % Re1 E,.

Now A has a unique (regular) involution a H ii and a unique order 5 agreeing with G1 and G2; i t is not difficult to verify that the subcategory E of proper w p h i s m s u of A (Gu 2 1 and uli 5 1) is exact; the restrictions Fi of Gi to proper morphisms

Obviously, if the functor F reflects the isomorphisms, F1 is an ordinary quotient supply ( 1).

(l.l), and (1) is also a CAT-factorization.

1.10. Let A be a category provided with a regular involution a H d (i.e. a contro- variant endofunctor, identical on the objects and involutory, verifying aiia = a for any morphism a). A projection of the object A is an endomorphism e : A 4 A which is idempotent and symmetrical (e = ee = a ) ; the projections of A build a set PrjA ( A ) , canonically ordered by:

(1) e < f if e = ef (iff e = f e )

and any morphism a : A’ +A” defines two order preserving transfer mappings of projections [8] :

(2)

(3)

a p : Prj (A’ ) 4 Prj ( A ” ) ,

up : Prj (A”) --f Prj ( A ’ ) ,

ap(e) = aeci:

aP(f) = Gfu.

The mapping

(4) SUbA ( A ) -3 PrjA ( A ) , 2 H Xz

is order preserving and injective; it is easy to see that it is an isomorphism of ordered sets iff A is factorizing ([7, Th. 3.181).

When A is the (factorizing) category of relations of the exact category E we have thus a biunivocal correspondence between the subobjects of A w.r.t. Re1 E (i.e. the subqzlotients of A w.r.t. E [7]) and the projections of A .

2. The category Mlc of modular lattices and modular connections

We build the exact category Mlc, which “simulates” the images and inverse images of subjects in exact categories (1.4) ; a niodular lattice will always be supposed t o have a least element 0 and a greatest element 1 (equal for the one-point lattice), and to be small.

Granilia, Transfer Functors and Projective Spaces 151

2.1. Lemma. Let X and Y be modular lattices, and u. : X --f Y , u' : Y -+ X mappings; the following cunditions a ) and b) are equivalent2) :

a ' ) u. : X --f Y is an increasing mapping a") u' : Y -+ X is a n increasing mapping a" ' ) u'u.(z) = x v u'(O), for every x E X a"") u.u'(y) = y A u.(1), for every y E Y b') u. : X -+ Y preserves finite unions (hence 0 ) b") u' : Y -+ X preserves finite intersections (hence 1) b"') u'(u.z v y) = z v u'(y), for every z E X and y E Y b"") u.(x A u'(y)) = u.(z) A y, for every x E X and y E Y .

the two mappings determines the other:

(1)

(2)

hence verifies b'), b") and :

W h n they hold, the pair (u., u') i s a GALOIS connection from X to F), hence each of

u'(y) = max {s E X I u.(x) 5 y)

u.(z) = min {y E Y I u'(y) 2 x}. Proof. Trivally, 6 ) implies a ) ; conversely, if a) holds, (u., u') is a GALOIS connection,

U'(U.Z V y) = U*U.U'(U.Z V y) = U*((U.Z V y) A U.1) = U'(?l.Z V (y A L L . 1 ) )

= u'(u.z v u.u'y) = u'u.(x v u'y) = (z v u 'y ) v u-0 = 5 v u'y.

2.2. Definition. Say Mlc the category of modular lattices and modular connections: the objects are the (small) modular lattices (with 0 and 1, possibly equal); a morphism u = (u., u') : X --f Y is a pair of mappings satisfying 2.1 a) (or equivalently 2. lb) ) ; this will be called a modular connection; the composition of u with v = (v., v') : Y -+ Z is obviously ou = (v.u., u'v') : X --f 2 (which is still a modular connection, by charac- terization 2.1 b)).

The morphism u = (u., u') : X 3 Y will also be written X g' Y . It is an iso- 2 morphism of Mlc iff u. and u' are reciprocal isomorphisms of lattices.

2.3. The category Mlc is selfdual since it has an involutory contravariant endofunctor:

(1) x H x* (2)

where X * denotes the lattice opposed to X .

to any lattice its underlying set, and to any connection (u., u') the mapping u..

(u = (u., 1 L ' ) : x -+ Y) H (u* = (u., u.) : Y* --f x*)

I t is also a concrete category, via the faithful functor P , : Mlc -+ Set associating

2.4. The category Mlc is exact. Actually, the one-point lattice X , is clearly initial (hence also terminal); the morphism u = (u., u') is a zero-morphisni (i.e. factorizes

?) The condition a) is the conjunction of a')-a"''). 3, That is, a pair of adjoint functors u. + PL* between the associated order categories (u.:

X + Y and u' : I' + X are increasing mappings and u'u. 2 ix, U.U. S ip). In MAC LANE'S termino- logy, one would say a GALOIS connection from X to Y* [16, p. 931.


through X,) iff u.(r) = 0 for every z E X , iff u'(y) = 1 for every y E Y ; the kernel and cokernel of u = (u., u') : X --f Y are respectively m and p :

( 1 )

(3) P*(Y) = Y" 4 1 ) ; P'(Y) = Y

{z E x I z 5 U'(O)} % x % Y E + {y E Y I y 2 U.(l))

(2) m.(Z) = 2; m'(Z) = Z A U'(0)

while a factorization of u via conormal epi, isomorphism, normal inonic, is given by: U X + Y

(4) 1- {z E x 15 2 U'O) % ( y E Y I y 5 u.l}

(6) p ( 2 ) = 5 v u'0; pyz ) = 2

(6) v.(z) = u.(z); v ' (y ) = u'(y)

(7) m.(y) = y; m'(y) = y A u.1.

2.6. Thus we can choose the subobjects of X in Mlc to be of the following form, for XOE x: ( 1 ) { z ~ X l z ~ z , ) % X ; m.( z )=z ; m * ( z ) = z ~ z , ,

and there is a lattice-isomorphism :

(2) IX: Sub ( X ) + X , m I+ m.(l)

which proves that Mlc is well-powered. Analogously, the quotients of X in Mlc will be chosen to be of the form (2, E X ) :

(3) x ?+ {z E x I 5 2 z,) ; p.(z) = 5 v 2,; p ' ( z ) = 2

and there is an anti-isomorphism of lattices :

(4) Quo (XI --f X , p H ~'(0).

2.6. Remark. By 2.5.2, any modular lattice X (with 0 and 1) can be realized as a lattice of subobjects of some object (actually, X itself) in a fixed exact category (Mlc).

2.7. Consider the category Mlh of modular lattices and lattice homomorphisms (preserving 0 and 1). It will be useful to link the latter with the category Mlc of modular connections in the double category Mhc of modular lattices, homomorphGm and modular connections, whose cells are the squares:

X h X '

Y k + Y'

(2) h, k E Mlh; U , v t Mlc; ku. = v.h; hu' = v'k.

The horizontal and vertical composition laws are obvious.

Grandis, Transfer Functors and Projective Spaces 153

2.8. Mhc is horizontally complete (i.e. a complete category w.r.t. horizontal com-

In particular the product of the modular connections X A+ Y , X ' Y' (objects position) and vertically exact.

w.r.t. the horizontal composition) :

(1) defines a (non Cartesian) monoidal structure [15] on Mlc, whose identity is the one- point lattice.

u x 2, = (u. x v., u' x v ' ) : x x Y --f X ' x Y'

3. The category of modular relations

We build here a concrete realization of the category of relations Re1 (Mlc).

3.1. Let A be the involutive category of modular lattices and pairs (a , , a') : X --f Y , where a. : X -+ Y and a' : Y + X are mappings, with the obvious composition, and involution (a., a')- = (a' , a.).

3.2. Lemma. The following conditions a) , b), c ) on a morphtkm a = (o. , u') : X -+ Y of A are equivalent:

a' ) a") U"')

b') b") b"')

c)

a. : X -+ Y and a' : Y -+ X are increasing mappings a'a.(z) = (z v a'O) A a'l = (z A a ' l ) v n'O a.a'(y) = (y v u.0) A a.1 = ( y A a.1) v n.0 a. : X -+ Y and a' : Y -+ X are increasing mappings a'( (yo v a . ~ ) A yl) = (a'yo v 2) A a'y,, for yo 5 y1 a.((s, v a'y) A q) = (a.zo v y) A a x l , for zo 5 z1 the morphtkm a factorizes in A as:

(1) a = n(i'bp& = $n'b&'p'

where m, n, m', n' (resp. p , q, p' , 4') are subobjects (resp. quotients) in Mlc (2.5), while b i s an Mlc-isomorphism: b.(s) = a.(z), b'(y) = a'(y) . When these conditions are satisfied:

(3) a. = a.a'a.; a' = a u.a . . .

Proof. a) 3 c ) : if a ) holds, property (3) follows easily; thus a.(z) = a.a*a.(z) = a.((z v a'O) A a ' l ) , a'(!/) = u'((y v a.0) A a.l), which means that the factorization ( 1 ) holds; b is easily seen to be an isomorphism.

c ) + 6 ) : each niorphism of Mlc satisfies condition b), which is stable for composition and involution in A.

b) 3 a ) : obvious.


3.3. Define MIr, the category of modular lattices and modular relations, as the (involutive) subcategory of A whose niorphisms (a., u’) : X --f Y satisfy the equivalent conditions 3.2 a)-c).

The inclusion Mlc + Mlr supplies the category of relations of Mlc: just verify the axioms (S 5-10) [7]. We shall always use this concrete realization Mlr of Re1 (Mlc).

3.4. Similarly, the vertical symmetrization of the (vertically exact) double category Mhc (2.7) has an obvious realization Mhr, with horizontal maps in Mlh and vertical ones in Mlr.

4. Transfer functors and a metatheorem for exact categories

Every exact category E “nearly embeds” in Mlc, via an exact functor SubE : E -+ Mlc which reflects exact sequences; thus Mlc is somewhat universal within exact categories.

4.1. Let E be an exact category: it5 transfer functor

( 1 ) S U b E : E + Mk

associates to any object A the modular lattice of subobjects of A , and to any niorphism u : A 3 B the modular connection (1.4)

(2) SUbE ( U ) = (U*, U*) : SubE ( A ) -+ SUbE ( B ) .

It is an exact functor.

isomorphism : In particular, for the exact category Mlc, the isomorphisnis 2.5.2 supply a functorial

(3) 1 : SubMi, 3 l M l c : MIC 4 Mlc.

-1.2. Generally, the functor SubE is neither faithful (5.3) nor full (5.7); however, i t d m rejlect zero-objects, monomorphisms (if (u*, u*) is monic in Mlc, then ker u = u*(O) = 01, epiniorphisms, isomorphisms, kernels (if A + B % C are in E and (k*, k*) - ker (u*, u*) in Mlc, then k is monic and k N k,( 1) = u*(O) = ker u), cokernels, exact sequences.

k

4.3. The exact functor SubE : E -+ Mlc extends to the categories of relations (1.6), supplying the involution preserving functor:

(1)

(2)

S = Re1 (Sub,) : Re1 E -+ Mlr

S(ngp5) = (n*, n*) - (q*, q * ) - . (P*, P*) - (m*, m*)’ = (?L*q*zwL*, m*p*q*n*)

nhow action on the E-relation a will be written, for short

(3) S(a) = (as, as) : Sub ( A ) 3 Sub (B) .

For z E Sub ( A ) and y E Sub (B) it can be easily checked, by writing down the com- positions ax and iiy, that:

(1) as(z) = val (as ) , aS(y) = val (by).

The functor S, by 4.2 and 1.8, preserves and reflects all fundamental subcategories, as well as exact sequences of proper morphisnis.


4.4. Therefore, most of “diagrammatic lemmas” can be checked in Mlc, a concrete category, and extended to every exact category E via the transfer functor SubE. I t would be too long, here, to formalize this fact in a precise metatheorem; we just remark that the thestk of the lemma should not contain “commutativity conditions” (SubE need not be faithful) and must avoid “existence statements” (SubE need not be full).

4.6. For example, the “snake lemma” in [16, p. 2021 should be rewritten in the following (more precise) form : in the given hypotheses the relation

(1)

is a proper morphism, which supplies the usual “Ker-Cok” exact sequence. As regards the proof of the lemma in Mlc, we show for example that 6 = (d., 6’) is everywhere defined, i.e. 6’(1) = 1:

(2)

6 = (cok /) +i‘gi?(ker h) : Ker h -+ Cok f

6*( 1) = (ker h)’ e.g’m! (cok f ) ’ (1) = (ker h)’ e.g’m!(l) = (ker h)’ e.g.e”(O)

= (ker h)’e.e’h‘(O) = (ker h)’ (h‘(0) A e.(l))

= (ker h)‘ (ker h). (1) = 1.

However, one should notice that this argument is practically the same that one could give in any exact category E, by writing @(I) instaed of &(l), and so on.

Thus, the above mentioned metatheorem has probably more a conceptual than a practical interest.

4.6. Remark. As a consequence of 4.1, it is easy to see that Mlc is not abelian: otherwise, the (exact) transfer functor Sub : K-Vct + Mlc of the (abelian) category of vector spaces on A’ would preserve finite products: since Sub ( K ) = {0, 1) is always a two- point lattice, while the cardinality of Sub (K2) depends on K , there is a contradiction. Analogously, one could use the transfer functor Sub : G --f Mlc of the category of abelian groups: Sub (G) is a two-point lattice for any simple group G, but Sub (Z, @ Z,) and Sub (Z, 0 Z,) g Sub (Z,) have respectively 5 and 4 elements.

4.7. Last we remark that, for any exact functor P : E -+ E’, there is a canonical “liorizontal transformation of vertical (exact) functors” (or, according to [ 1, p. 2511, an Mhc-wise transformation) :

(1) Sub1 : SUbE -+ SUbp * F : E -+ Mhc

associating to every E-object A the Mlh-morphism

(2) SuhF ( A ) : SUbE ( A ) --f SUbp ( F A ) , 2 I+ im F(2)

am! to every E-niorphism u : A -+ B the Mhc-square

-t Subp (FA) SubE ( A ) - SubdA)

( 3 )

SubE ( B ) Bub,o -b SUbEt ( F B )

The functor F is locally faithful (resp. full) iff all the mappings (2) are injective (resp. surjective).


6. Transfer categories and projective spaces

The transfer functor associates, to any exact category E, a “transfer” exact cate-

6.1. For any exact category E, the EX-factorization (1.10) of the functor SUbE : E --t Mlc

gory Trn (E) which is a sort of “category of projectivities” of E.

will be written :

(1)

Notice that, as SUbE reflects the isomorphisms (4.2), Trn E is the (ordinary) quotient of E modulo the transfer congruence -.\ associated to SubE: if u, v are parallel maps in E

(2)

E % Trn ( E ) % Mlc.

u w6 v iff u* = v* iff u* = v*.

6.2. We say that E is a transfer exact category if SubE is faithfull (iff E = Trn E). For any exact category E, 8, : Trn E --f Mlc is isomorphic to Sub,,, E via

(1) 1~ : SUbE ( A ) + S U b T r n E ( A ) , x H f.

This proves that Trn (E) is transfer: i t will be called the tramJer (exact) d e g o r y associated to E; the exact quotient E --f Trn ( E ) is a universal arrow, in an obvious sense.

6.3. Vector spaces. Let E = K-Vct be the (abelian) category of vector spaces on the commutative field K ; the transfer congruence 6.1.2 is characterized by:

(1) u wS v iff u = 1v for some non null scalar 1.

Actuallp), the sufficiency of the right hand condition being trivial, let us suppose that u* = v* (and u* = v*): if

p = coim u = cok (u*(O)) = cok (v*(O)) = coim v

m = im u = u*(l) = v,(l) = im v

there are unique isomorphisms a, b such that u = map, v = mbp, and:

a, = (m*m*) a,(p,p*) = m*u,p* = m*v*p* = b*

thus we only need to prove our property for a, b : A --f B (isomorphisms). For every x E A , x + 0, a* and b, coincide on the subspace (x) of A spanned by x , hence there is a unique (and non zero) scalar 1, such that a(x) = I , - b(z). Now, let y E A , y + 0; if y = Ax, for some non zero scalar 1, then I,b(y) = a(y ) = 1a(x) = 1 - 1,b(x) = A&) and 1, = I , ; otherwise, x and y are linearly independent, and:

I Z W 4 + 1,WY) = 44 + 4 Y ) = 4% + Y) = 1,,b(X + Y) = %+,b(4 + &+“b(Y)

since b(x) and b(y) too are linearly independent, 1, = I,+, = A,. It follows, that SubE i.s not faithful (and E = K-Vct is not transfer) whenever R has more than two elements ( K + Z,), and only then.

4, The proof is analogous to that of Thm. 2 in [19].


5.4. Projective spaces. Thus Trn E is the quotient of E = K-Vct modulo 5.3.1; we say that Trn (K-Vct) is the (exact) category of projective spaces and projectivities on K, K-Prs for short; it admits the following concrete representations in terms of (faithful) functors, allowing to recognize “usual” projective spaces. a) pmjective spaces as lattices of subspaces of vector spaces

( 1 ) F , = ( K - P ~ S % MIC % Set)

the functor P I being defined in 2.3, and S, in 5.1.

b) projective spaces as pointed sets, quotients of vector spaces

(2) where F, (A) is the quotient of the set underlying the vector space A , modulo the usual equivalence relation ( x - y if there exists 1 E K, 1 + 0, such that x = Ay), pointed a t the class U = {0} ; besides, for a linear mapping u : A + B, F2(E) : F2(A) 3 E;(B) is the pointed mapping Z I+ u(z). F, is faithful because every subspace of a vector space is the union of its 1-dimensional subspaces, and images preserve unions.

c) Projective spaces as (non pointed) sets [2]

F, : K-Prs -+ Set,

(3) F~ = (K-Prs F,, Set, J, sfn)

where J is the obvious equivalence between Set,, and the oategory Sfn of sets and func- tions (i.e. partially defined mappings):

J (S , 2 0 ) = s - {xo). It will be noticed that various authors (not BOURBAKI [2]) only consider as projectivities the isomorphisms of K-Prs (in some concrete description), which we call Cso- projectivities.

6.6. Finite dimensional iso-projectivities. Let A and B be vector spaces on K , of finite dimension m > 1, and xl, . . ., x , + ~ a projective frame of A , i.e. all xi are lines of A (1-dimensional subspaces) and the union of any m of them is lA. It is well known [2, p. 2881 that the mapping

(1) H (u*(x~), - - - 3 u*(Zm+l))

is a biunivocal correspondence between the iso-projectivities ;ii : A --f B and the projective frames of B.

Z2, K-Prs is not provided with equalizers (hence i t is not abelian). Actually, if e l , ..., em is the canonical base of Km (m > l), the projective frames

5.6. It follows easily that, for K

(1)

(2)

are different; thus iii = id (Km) is different from the iso-projectivity B : Km --f Km such that v*(xi) = x: (i = 1, . . ., m + 1); however, they coincide on xl, .. ., x, and x1 v . . . v x, = 1.

z1 = ( e l ) , . . . x i = (e l ) , . . .

xrn = (ern),

x; = (e , ) ,

Xm+1 = (e l + e2 + *.* + e m )

x : + ~ = (el - e2 - . -. - e m )


5.7. It follows also that SubE : E + M k .iS not full, for K * Z,. Actually, i t is easy to see that the Mlc-isomorphisms (v, v-l) : Sub ( K 2 ) -+ Sub (K2) are in biunivocal correspondence with the permutations v’ of the set L of lines of K 2 (take v‘ to be the restric- tion of v to L); since K2 has more than three lines (for K * Z2), the conclusions follows from 5.5.

Instead, if m > 2, the fundamental theorem of projective geometry [2, p. 2901 shows that any Mlc-isomorphism (21, v-l) : Sub (Km) -+ Sub (Km) comes from some Vct- isomorphism u : Kffl + Km, hence from one iso-projectivity ?i : K m -+ Km.

6.8. The abelian category E = K-Vct is provided with a monoidal closed structure [15], given by the adjunctions -0, B 4 HornK (B, -). It is easy to see,from charac- terization 5.3.1, that the functors (3, : E x E -+ E and Hom, : E* x E --f E agree with wS, and induce functors

(1)

(2)

Similarly, the adjunctions “on E”

(3)

are preserved by quotientation, defining a monodal closed structure on K-Prs, whose identity is K (the one-point projective space in description 5.4.3).

@K : K-Prs x K-Prs + K-Prs;

HornK : K-Prs* x K-Prs -+ K-Prs;

Z 8, B = u 0, c

Hom, (a, e ) = V - - a ?i.

Q$,c : Hom, ( A @ B, C) -+ Hom, (A, Horn, (B, C))

5.9. Romark. The following is an alternative categorical construction leading to projective spaces.

On a category E, any subgroup x of the abelian group no = Aut (1E) of the auto- morphisms of the identical functor, defines a congruence mn: two parallel morphisms u, v E E(A, B) are n-equivalent if there is some ;I E x such that u = v - 1” (= i l B . v). When E is exact, also the quotient category En = El-,, and the quotient functor E -+ En are so; moreover, trivially, u -n v implies Sub (u) = Sub (v), so that there is a factorization of quotient functors:

(1) E --f En --f En. +Trn E. I t

We just saw that, for E = K-Vct, the quotient En, + Trn E is an isomorphism (5.3.1 and,)). Instead, in the category of abelian groups, the morphisms Z -+ Z, give three wnp-classes and two -s-classes.

6. Distributive and boolean exact categories

Distributive exact categories arose in categorical homological algebra, in connection with canonical isomorphisms between subquotients [ 113, [ 121 ; boolean exact categories were first considered by SCEIWAB [22], as “exact inverse” categories. The transfer functor of an exact category E allows to build a d.iStrz%utive expansion Dst, (E) and a boolean

6) For E = R - Mod, no is the group of invertible8 in the centre of the ring R.

Grandia, Transfer Functors and Projective Spacee 159

expansion Bln, ( E ) of every exact category E, via "horizontal commas of vertical functors" into the double category Mhc.

6.1. Definition. We say that the exact category E is distributive (resp. boolean) iff all its lattices of subobjects are distributive (resp. boolean algebras); by 1.4, one can equivalently use the lattices of quotients.

Say D l c (resp. Blc) the full subcategory of M l c defined by the distributive lattices (resp. boolean algebras); by 2.5 it is also locally and colocally full in Mlc, and therefore it is an exact subcategory; by 2.5.2 i t is also distributive (resp. boolean).

Any distributive (resp. boolean) exact category E has a restricted transfer functor SubE : E --f Dlc (resp. E --f Blc).

6.2. Theorem [ 111, [ 121. The following conditions on the exuct category E are equiralent :

a ) E is distributive b) for any morphism u, the mapping us preserves binary intersections (hence it is a homo-

c) for any m o r p h h u, the mapping us preserves binary unions a) for any morphism u, the mapping uo preserves binary unions e ) for any morphism u, the mapping uQ preserves binary intersections f ) the category of relations Re1 (E) is orthodox (that is, idempotent endomorphism.s are

g ) the &gory Re1 (E) is quasi-inverse [lo] h) i f the diagram (1) is commutative in E and its upper square is a pullback, 80 is the lower

morphism of lattices)

stuble for composition)

one

i) i f the diagram (2) is commutative in E and its upper square is a pushout, 80 i s the lower one.

6.3. Remark. As a consequence, any modular connection (u., u.): X + Y between distrdutive lattices is composed of lattice homomorphisms ; this can be directly checked as follows (for x,, x2 E X ) :

(1) U. (21 A 2 2 ) = U.(Z1 A x2) V U.U.0 = U . ( ( 2 , A Z2) V U ' o )

= U . ( ( Z 1 V U ' o ) ) A ((2, V U ' o ) ) = 2L.(U'U.(Z1) A U'U.(Z,))

= U.U*U.(X,) A U.(Z2) = U.(Z1) A U.(x2).

Actually, via the transfer functor in D l c (6.1), this argument is a proof of a) + b) in 6.2.


6.4. Theorem. Let E be a %-category. The following are equivalent: a ) E is a boolean exact category b) E is exact and inversea) c ) E is distrz’butive exact and has a regular involution d ) E is inverse and factorizing, has a zero object and jor each object A the semilattice Prj ( A )

i s a small boolean algebra. Proof. a) + b). As the distributive exact category E can be imbedded into its

canonical inverse symmetrization O(E) = Re1 (E)/@ [ll], we need only to prove that O(E) = E; actually if m, m’ E SUbE ( A ) are complementary subobjects and p = cok m‘, the morphism i = pm

(1) . Z - b A -%.

is an isomorphism as:

(2) (3)

ker i = is(0) = mSpS(0) = ms(m‘) = 0 cok i = io(0) = p4mo(0) = po(cok m) = 0 .

m(i-’p) m = mi-% = m Thus in Re1 (E), f i and i-1p are @-equivalent [8] :

(4) (5 ) (i- lp) m(i-1p) = i-lii-lp = i - lp

and, in OE = (Re1 E)/@, f i = i - lp belongs to E ; by duality, all @-involute of epimorphisms of E belong to E, and the conclusions follows.

b) + c). To prove that E is distributive, we verify the condition 6.2h). The upper square of 6.2.1 is a Cartesian square of monics in the inverse category E : by [ 10, 4.241, i t is bicommutative ; analogously, the four vertical “mixed” squares are commutative in E, hence bicommutative. It follows that the lower square of monics is bicommutative too, hence a pullback.

c) + a). We have to prove that every lattice SubE ( A ) is complemented. Define the mapping : (6) Since the involution of E is an anti-automorphism of exact categories, this mapping is order reversing (if m < n in SUbE ( A ) then G<fi among the epis of domain A and m l = ker f i > ker 6 = n l ) and involutory (m* 1 = ker ((ker %)*) - (cok ker f i ) - (6)- = m ) ; thus i t is an anti-isomorphism of ordered sets, and therefore of lattices with extremes. Finally m A m1 = 0, since in the Cartesian square

( )I : SubE ( A ) --f SubE ( A ) ; m1 = ker 6.

(7) 4 t-

O) A category A is inueree if each morphism a E A(A’, A”) has a unique generalized inuerae d E A(A“, A’) veryfying: ada = a and iiad = 6; then a I+ d is a regular involution for A and clearly the only one; moreover the projections coincide with the idempotante and commute: the ordered sets Prj (A) are 1-similattices. Inverse categories, generalizing inverse semigroups, are studied in [8], [lo], [14], [21]. Exaot inverse categories were introduced and characterized in [22], where it is proved that they are boolean.

Grandis, Transfer Functors end Projective Spaces 161

a) & b) 3 d). Use the order isoniorphisnis 1.10.4 with A = E. d) 3 b). If m E SubE ( A ) , it is easy to check that m is a kernel for the complemen-

tary of mi5 in Prj, ( A ) ; therefore E is normal (and conormal), factorizing and well- powered (by the above isomorphisms), i.e. it is exact.

6.5. The category of small cyclic groups is exact, distributive, non boolean. The category of small sets and partial bijections is boolean exact.

However, exact categories “in nature” are rarely distributive ; a non-trivial abelian category is never so, since for any non-zero object A the subobjects of A @ A given by t,he first and second injection of A and the diagonal do not behave distributively.

6.6. To build a distrzhtive expansion Dst, (E) of the exact category E, consider the diagram

DSti E - D l c t;

where LY : TP1 -+ SubE P, : Dst, (E) 3 Mhc is a horizontal comma square of vertival exact functors (1’ is the vertical inclusion): the objects of Dst, (E) are the triples ( X , A ; h : X -+ SubE (A) ) , where X is a distributive lattice, A an object of E and h a homo- morphism of lattices (preserving 0 and 1); a morphism (u, v ) : ( X , A ; h) -+ ( Y , B ; k) is given by morphisms u = (u., u‘) : X 3 Y and P, : A + B respectively in Dlc and E, 80 that ku. = v,h and hu’ = v*k; the composition and the functors PI, P, as well as the universal property of (1) are clear.

6.7. Dst, (E) is an exact category. The functor PI : Dst, (E) -+ Dlc is isomorphic to the transfer functor of Dst, (E), via

(1)

(2)

,l,x,A;h) : X 3 Sub ( X , A ; h )

m o ) = (XO, A,; ho)

where X , = ( x E X I z 5 zo), A , = Cod h(z,) and for any z E X o , ho(z) = h(z,)* (h(s)) . Thus Dst, (E) is a distributive exact category, to be called the dislra7mlive ( c m m a )

expansion oj E. In [ 121 the author used a smaller distributive expansion E# isomorphic to the full subcategory Dst (E) of Dst, (E) whose objects are the triples ( X , A ; h) where h is an inclusion.

6.8. Any exact functor F : D -+ E, where D is a distrz3utive exact category, defines a horizontal transformation (notations as in 6.6)

(1) /!? = SubF : T SUbD +Sub, * F : D +Mhc

therefore there is exactly one functor F# : D -tDst, E such that P,F* = P, PIP*

(2)

= SubD, otF* = p : F*(B) = (Sub (D) , P(D); Sub (B) S u b F ‘ D ) ~ Sub F ( D ) ) .

6.9. Analogously to 6.6, the vertical inclusion Blc --f Mhc determines the boolean expansion Bln, (E) of the exact category E : i t is the full subcategory of Dst, (E) deter-

11 Math. Nachr. Bd. 118


mined by those objects ( X , A ; h) where X is a boolean algebra. It is a boolean exact category, which solves a universal problem analogous to 6.8.

6.10. Last, we remark that a boolean exact category E has transfer mappings both for subobjects and for projections (1.10). The covariant parts agree (via the order isomorphisms 1.10.4): if u E E(A‘, A’’) and 2 E SubE (A’ ) , it is easy to check that:

(1) UP(2j’) = u*(4 (u*(4)-.

Instead, as regards the contravariant parts, u* obviously agrees with the right adjoint of up: Pri (A’) -+ Prj (A”) , that is

(2)

(3)

which coincides with up iff up(l) = 1 (iff u is mono). For further properties of u n (and uA = C A ) see [22], from where we borrowed the notation.

U A : Prj (A”) -+ Prj (A’ )

u A ( f ) = max {e E Prj (A‘) I up(e) < f ) = uP(f) v (uP(1))~

7. Other examples of transfer functors

As we saw, E X has a nice behaviour w.r.t. the transfer of subobjects: any exact category E is provided with an exact transfer functor SubE : E + Mlc towards a fixed exact category; the factorization structure of E X allows to define (in E X ) a ‘%ransfer category” Trn (E) of E which can be considered as a sort of “category of projectivities” of E; exact subcategories of Mlc (e.g. Dlc or Blc) determine, via horizontal commas, “expansions” of exact categories.

We consider here, very shortly, some other categories of structured categories which present analogous behaviours. The cases 7.3 and 7.4 concern the transfer of projections instead of subobjects.

7.1. Let C be a category provided with a factorization structure (Cl, C,), where C, c Mon C, having counterimages of C,-subobjects; we also suppose that C is a &-category, well-powered w.r.t. C,-subobjects. Say MFC the category of such categories and functors which preserve the factorization structure and counterimages.

Any category C in MFC is provided with an obvious transfer functor

(1) Sub= : C --+SIC

where SIC is the category of small 1-semilattices and Galois connections (u., u’) : X -+ Y

(2’) u. : X -+ Y and u’ : Y --t X are increasing mappings

(2“) u’u. 2 lx; u.u‘ 5 1, provided with the factorization structure (P, M), where the P-morphisms and M-morphisms are respectively characterized by

(3) u.(l) = 1

(4) u‘u. = lx.

Thus SIC and the transfer functor (1) are in MFC.

Grandis, Transfer Funotors snd Projeotive Spaces 163

7.2. Consider the category MIN of well-powered %-categories C provided with the following limits:

- counterimages of subobjects (i.e. pullbacks of - 3 - t-< -) - (infinite) intersection of subobjects (i.e. generalized pullbacks of small indexed

The functors in MIN have to preserve monics, their counterimages and their intersections.

As the counterimages of subobjects necessarily preserve intersections, any u E C(A, B ) defines an intersection-preserving mapping between complete lattices, u* : Sub (B) + Sub ( A ) which has left adjoint:

(1)

families of monics with the same codomain).

u.,, : Sub ( A ) 3 Sub (B) ; u*(z) = min {g 6 Sub (B) I u*y 2 2). Thus we have a functor

(2) Sub= : C -+ Clg

towards the category of complete (small) lattices and Galois connections, studied in [13] under the name of “category of sup-lattices” (as a morphism (u., u‘) : X -+ Y is determined by its “covariant part” u., which is a sup-preserving mapping): i t is a bi- complete selfdual category. The subobjects of X can be chosen to be of the following form [13, Ch. I, $41:

i (3) X , c p t X ; i(.) = 2 ; i d p

where X , is any sup-closed subset of X ; thus the complete lattice Sub (X) is contained in BX E 9: Clg is in MIN, and analogously the functor (2).

7.3. Inverse categories. Let IN be the category of inverses) %-categories with small

Any IN-category A is provided with a transler functor: projection sets, and of functors between them.

(1)

where Slt is the IN-category of small 1-semilattices and transfer pairs (a., a’): E 3 F satisfying :

(2) a. : E -+ F and a’ : F -+ E are homomorphisms of semi lattice^^)

(3) u’a.(e) = ea.(l) , for every e E E

(4) a.a‘(j) = /a‘(l), for every/ E F. Here, for any object A of A, Prj, ( A ) is the (small) set of projections of A , and for any A-morphism a : A‘ 3 A”, PrjA (a) = (up, up) is defined in 1.10.2-3. A functor P : A 4 B between inverse categories determines an obvious horizontal transformation of vertical functors:

(7) Prj, : Prj, -+ Prj, P : A -+ Sht; e I+ P(e)

towards the double category of 1-semilattices, their 1-preserving homomorphisms and their transfer pairs, with “bicommutative” cells.

Prj, : A -+ S l t

’) These homomorphisms, generally, do not preserve 1.

11*

164 Grsndis, Transfer Functors and Projective Spaces

The CAT-factorization of the transfer functor PrjA : A --f Slt yields an inverse category Trn (A). It is easy to define locally booleuns) inverse categories (all 1-semilattices of projections have to be boolean algebras), and locally boolean expansions of inverse categories.

The transfer functor of a locally boolean inverse category takes its values in Blt, the full subcategory of Slt determined by boolean algebras; it is isomorphic to Blc (6.1) via functors identical on the objects and transforming the transfer pair (u., a') : X +- Y nto the modular connection (u., u') : X -+ Y so that:

(8) U. = U.; U'(p) = a'(?/) V ( U * ( l ) ) ' ; a ' ( y ) = U ' ( y ) A (U. (O)) ' .

Last, we remark that any inverse category A canonically embeds in a factorizing inverse category Fct (A) whose objects are the projections of (all the objects of) A, and whose morphisms ( a ; e, f ) : e --f f are determined by A-morphisms a such that a = fae. The 1-semilattices Prj, ( A ) and SubFct A ( lA) are canonically isomorphic.

7.4. Regular involution categories. These facts extend to categories provided with a regular involution.

Let RI be the category of %-categories, provided with a regular involution a I+ 6, with small projection sets, and of involution-preserving functors. Por any object A of the RI-category A, the (small) set PrjA (A) is provided with the (generally non associative) operation 0 [lo]:

(1)

with identity 1 A and associated order 3:

(2) e <f if e = f 0 e (iff e = ef, iff e = fe).

Besides, every A-morphism a : A' +A" defines mappings a,, up (1.10.2-3) so that:

(3) a, : Prj (A') -+ Prj (A") and UP : Prj (A") -+ Prj (A') are increasing

e f = efe = e p ( f ) = e P ( f )

mappings (with regard to 4)

(4)

(5 )

U P ( / (ape)) = ( a P f ) e

a,(e 0 W f ) ) = (ape) 0 f - Thus we have a transfer functor in RI:

(6) PrjA : A + A,

where A, is the category of small (possibly non associative) monoids with identity S = (S , a), such that e = f 0 e is an order relation, with mappings (u., a ' ) : S 3 1' verifying conditions analogous to (3)-(5); the involution in A, is (a,, a')* = (u*, u.), regular because :

(7) a.a'a.(e) = u.(1 u'a.(e)) = a . ( ~ ) 0 a.(e) = a.(e).

Since an inverse category has a canonical order [8], a stronger notion of boolean inverse category is possible (in partioular all horn-sets have to be boolean algebras).

Grandis, Transfer Functors and Projective Spaces

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