The category of categories with pullbacks is cartesian closed

The category of categories with pullbac ks is cartesian closed John Bourk e email: johnb@mat hs.usyd.edu .a u 10/04/20 09 Abstract W e show that the category of categories with pullbac ks and pullback preser ving functor s is cartesian closed. Consider the category , Cat pb , whose ob jects are categories with pu llb ac ks and w hose morp hisms are pu llbac k p r eserving fun ctors. Our ai m is to sho w this ca tegory is cartesian closed. Firstly ob s erv e that Cat pb has pro ducts. Giv en A,B of Cat pb , th e cartesian pro d u ct in Cat, A × B, has pullbac ks, constructed p oint wise. As p ullbac ks in A × B are p oin twise, th e p r o j ections from A × B preserve them. It is straightfo rw ard to c hec k the universal prop erty , and so th e cartesia n pro duct in Cat pb is just the ordinary cartesian pro duct of categories. Up on observing th at Cat pb has pro du cts, we p ro ceed to s h o w that it is cartesian closed. T o do so is to pro vid e a righ t adjoin t to the fu nctor Cat pb Cat pb - × A / / for ea ch ob ject A of Cat pb . In k eeping with conv en tion the righ t adjoin t is d enoted by [A,-] pb and r eferred to as the in ternal hom. Giv en B of Cat pb , w e d efine the in tern al hom [A,B] pb to b e the category whose ob j ects are pull- bac k preserving functors from A to B and whose morph isms are cartesian n atural transformations b et ween suc h fun ctors. (Recall that a natural transformation is carte sian if its naturalit y s quares are pu llbac k squ ares). The first thing w e sh ould v erify is that [A,B] pb is actually an ob ject of Cat pb , which is to sa y th at it has pullbac ks. Lemma 1. [A,B] pb has pu llbac ks. Pr o of. Given pullbac k pr eserving f u nctors F,G and H from A to B, and a pair of cartesian natural transformations F H t + 3 and G H u + 3 w e m u st construct the pu llbac k in [A,B] pb . As the catego r y B has p u llbac ks, pullbac ks in the ordinary functor catego r y [A,B] exist and are constructed p oin t wise. Consider the pullbac k in [A,B], P F G H r + 3 s   t   u + 3 . W e w ill sho w that this is the pullb ac k in [A,B] pb . T o do so we must firstly sho w that this square liv es in [A,B] pb , wh ic h is to say that P preserv es p u llbac ks and that r and s are cartesian. 1 • P pr eserves pullbacks: (This is just a sp ecial case of the general fact that the limit of a d iagram of Φ-con tinuous functors is alw a ys Φ-con tinuous, for a t yp e of li mit Φ. W e giv e an explicit p ro of b elo w ). Giv en a pullback square in A, sa y a c b d α   β / / θ / / φ   , w e m ust sho w that the square P a P b P c P d P α   P β / / P θ / / P φ   is a pullbac k too. In the composite square P a F a P b F b r a / / P α   F α   r b / / F c F d F β / / F θ / / F φ   b oth sm aller squares are pullbac k s , the left square as r is cartesian, the r igh t han d square as F preserve s pullbac k s . Thus the comp osite square is a pullback. The equations F β ◦ r a = r c ◦ P β and F θ ◦ r b = r d ◦ P θ hold by naturalit y of r and so we ma y rewrite th e ab o ve square as P a P c P b P d P β / / P α   P φ   P θ / / F c F d r c / / r d / / F φ   . The righ t hand square in the comp osite is a pullbac k as r is cartesian. Therefore as the comp osite square is a p ullbac k, the left hand square must b e a pullbac k to o. • r , s are cartesia n: W e sh all consider the ca se of r. Giv en a morphism a b α / / of A w e must sho w that the squ are P a P b F a F b P α   r a / / F α   r b / / is a pullbac k. Both squares in the comp osite P a Ga F a H a s a / / r a   u a   t a / / Gb H b Gα / / H α / / u b   2 are p ullbac ks: the left hand squ are b ecause pullbac ks in the functor category are constructed p oin t wise; the right hand square b ecause u is cartesian. Therefore the comp osite squ are is a pullbac k. By naturalit y of t and s w e ma y rewrite this comp osite as P a P b F a F b P α / / r a   r b   F α / / Gb H b s b / / t b / / u b   . The right hand squ are is a pullbac k, and so as the comp osite is, it follo ws that the left h and square is a pullback. Th us the square P F G H r + 3 s   t   u + 3 . lies in [A,B ] pb . Its universal prop erty is easily c hec ke d up on n oting that if r 1 and r 2 are vertic ally comp osable natural transformations suc h that r 2 is cartesian and r 2 ◦ r 1 is cartesian, then r 1 is cartesian. Giv en a morphism B C F / / of Cat pb , there is an ind uced pullbac k preserving fu nctor [A,B] pb [A,C] pb [A,F] pb / / giv en b y composition with F, and so w e obtain an end ofunctor Cat pb Cat pb [A,-] pb / / . W e shall show that [A,-] pb is right adjoin t to - × A b y pro viding a unit and counit satisfying the triangle equations. These ma y b e lifted to Cat pb directly from the case of C at. In the ca se of the cartesian closedness of Cat, th e unit and counit are giv en b y ev aluation and co ev aluatio n , B [A , B × A] coev B / / and [A,B] × A B ev B / / . That these lift directly to the case of Cat pb is the con ten t of the foll owing lemma. Lemma 2. 1. Th e restriction of [A,B] × A B ev B / / to [A,B] pb × A preserve s pullbac ks (lies in Cat pb ). W e define the counit comp onen ts via the restriction as [A,B] pb × A B ev B / / and these are natural in B. 2. B [A , B × A] coev B / / preserve s p ullbac ks, and its image lies in [A , B × A] pb . W e defin e the unit comp onents via this factorization as B [A , B × A ] pb coev B / / and these are natural in B. 3 Pr o of. 1. Let ( P , a ) ( F , b ) ( G, c ) ( H, d ) ( r , α ) + 3 ( s,β )   ( t,θ )   ( u,φ ) + 3 b e a pullb ack diagram in [A,B] pb × A (co r resp ondin g to a pullback square in eac h of [A,B] pb and A). The image of this p ullbac k square under ev B is the outer square of P a F a F b Ga H a H b Gc H c H d r a / / F α / / u a / / H α / / u c / / H φ / / s a   Gβ   t a   H β   t b   H θ   . The top left square is a p ullbac k as it is a comp onen t of the pullbac k square in [A,B] pb . The b ottom righ t square is a pullbac k as it is the image of the pullbac k square in A, under the p ullbac k p reserving fun ctor H. The top right and b ottom left squares are pu llbac ks as b oth t and u are cartesian. Consequentl y the outer square is a pullbac k square as desired. It is straigh tforw ard to v erify that the counit components s o defined constitute a natural transformation [A,-] pb × A 1 Cat pb ev / / . 2. T o see that B [A , B × A] coev B / / preserve s pu llbac ks, n ote that we ha ve the p r o duct in Cat [A , B × A] ∼ = [A,B] × [A,A], and that the follo wing diagram comm utes: B [A , B × A] [A,B] [A,A] coev B / / ? ?        ? ? ? ? ? c 1 A + + V V V V V V V V V V V V V V V V V V V V V ∆ 3 3 h h h h h h h h h h h h h h h h h h h h h where ∆( b ) is th e constant fun ctor at b f or eac h ob j ect b of B, c 1 A ( b ) = 1 A is the ident it y functor on A, and the unlab elled arro ws are the pro jections from the pro duct. Both ∆ and c 1 A clearly p reserv e p ullbac ks, so coev B = (∆ , c 1 A ) p r eserv es p ullbac ks. T o see that the image of coev B lies in [A , B × A] pb , w e must show firstly that giv en an ob j ect b of B, the fu nctor coev B ( b ) = (∆ , c 1 A )( b ) = (∆( b ) , 1 A ) preserves pu llbac ks. Certainly the constan t functor ∆( b ) at b preserve s pullbac ks, as d o es 1 A , so that coev B ( b ) = (∆( b ) , 1 A ) preserve s p ullbac ks. Giv en a morphism a b α / / of B, w e m u st v erify that the natural trans- formation coev B ( α ) is cartesian. No w coev B ( α ) = (∆( α ) , 1 1 A ), and as b oth ∆( α ) and 1 1 A are cartesian, it follo ws that coev B ( α ) is. 4 It is straigh tforward to v erify that the unit comp onents so defin ed constitute a natural trans- formation 1 Cat pb [A , - × A] pb coev / / . Theorem 3. The in ternal hom, unit and coun it defined thus far giv e Cat pb the structure of a cartesian closed cate gory . Pr o of. It r emains to ve rify the triangle equations for the unit and counit. Being defined exactly as in the case of Cat (where the triangle equ ations hold) they certainly hold in Cat pb . Th er efore Cat pb is cartesian close d . As C at pb is cartesian closed we ma y enric h ov er it. Cat pb obtains th e stru cture of a Cat pb - catego r y , Cat pb , by defining Ca t pb (A,B) = [A,B] pb for A and B catego r ies with pullb ac ks. I n deed Cat pb is a ca r tesian c losed Cat pb -categ ory . No w eve r y Cat pb -categ ory in particular has an under lyin g Cat-categ ory (2-catego r y). T o b e precise (follo wing [1]) the f orgetful functor Cat pb Cat U / / is finite pro duct preserving and th us induces a 2-functor Cat pb -Cat Cat-Cat U* / / . Giv en a Cat pb -categ ory A, U*A h as ob jects as A, and for ob jects a , b of A, U*A( a , b ) = U(A( a , b )). In particular U* Cat pb is the 2-category consisting of categories with pullbac ks, pullbac k preservin g functors and cartesian natural tr ansformations. W e ha ve seen that the underlying cat egory , Cat pb , of U* Cat pb is cartesian closed. W e conclud e by sho wing that U* Cat pb is a cartesian closed 2-cate gory . Corollary 4. Th e 2-catego r y of categories with pullb acks, p u llbac k preserving fu nctors and carte- sian natural transformations, U* Cat pb , is a cartesian closed 2- category . Pr o of. F or A,B,C ob jects of Cat pb w e ha v e U* Cat pb (A × B,C) = U( C at pb (A × B,C)) = U([A × B,C] pb ) ∼ = U([A,[B,C] pb ] pb ) = U* Cat pb (A,[B,C] pb ) naturally in A and C. I would lik e to thank m y su p ervisor Dr. S tephen Lac k for askin g me whether Cat pb is cartesian closed. References [1] S. Eilen b erg and G. M. Kelly . Closed Categorie s. Pr o c e e ding of the Confer enc e on Cate goric al Algebr a (La Jolla, 1965), S pringer-V erlag, Berlin-Heidelb erg-New Y ork, 196 6; 421-56 2. 5