1. Balanced Category Theory Claudio Pisani Calais, June 2008

2. Balanced category theory is an abstraction of category theory based on the following classes of functors: E : final functors; M : discrete fibrations; E ’ : initial functors; M ’ : discrete opfibrations. Since [Street and Walters, 1973], we know that these form two “comprehensive” factorization systems on Cat.

3. Less known is the following “reciprocal stability” property: the pullback of a final (resp. initial) functor along a discrete opfibration (resp. fibration) is itself final (initial). Recall that discrete fibrations and final functors can be characterized in terms of slice projections m x : X/x  X and coslice projections n x : x\X  X (to be thought of as a sort of neighborhoods of the category X at x).

4. A functor f : X  Y is a discrete fibration iff the obvious slice functors e f,x : X/x  Y/fx are isomorphisms for any object x of X (“local homeomorphism” property). Discrete fibrations over X “are” the presheaves over X: Cat/X (m, m ’ ) = Nat (m, m ’ ) A discrete fibration is isomorphic to a slice projection m x (in Cat/X) iff its domain has a terminal object iff it corresponds to a representable presheaf.

5. A functor p : P  X is final iff it is “locally connected”: the pullback x\p along any coslice projection x\X  X has a connected domain. (E.g., X  1 is final iff X is connected, and x : 1  X is final iff it is a terminal object.) Now, the “reciprocal stability” property is easily proved: QP Y X q p f x\p x\X nxnx 1 ixix fx\Y n fx

6. Thus, Cat is a “balanced factorization category” (bfc): a finitely complete category with two reciprocally stable factorization systems which generate the same discrete objects: M /1 = M ’ /1 ( = Set ). Every left exact category with a factorization system satisfying the Frobenius law is a bfc. We will soon present other interesting instances of bfc’s.

7. We see an object of a bfc C as a “category” (of type C ), in the same sense for which an object of a topos T is (variable) set (of type T ). Thus, balanced category theory is a 1-categorial abstraction of category theory, based on an axiomatization of Cat (with the comprehensive f.s.)

8. Indeed, concepts such as terminal object, representability, universal arrow and colimits, with their reciprocal interplay, are enlightened in this generalized context. We now show that it deserves to be called “category theory”, by illustrating how several classical universal concepts can be treated in any bfc C.

9. We get the “Yoneda Lemma” for free: First, we define slices and coslices by factorizing a point (in Cat, an object) with the two factorization systems: 1 X/x A X exex m mxmx x X 1 exex mxmx 1 X x\X ixix nxnx Thus, as in Cat, the slice projections are (up to isomorphisms in C /X) those maps in M whose domain has a final point. 1 X x

10. We also get slice maps e f,x : X/x  Y/fx for any map f : X  Y in C : 1 X/x Y/fx exex X Y f e fx mxmx m fx e f,x These are final maps, that is are in E.

11. For any object X of C, we have an underlying category X, obtained by restricting C /X to slices: We so get a underlying functor C  Cat (and another one using coslices instead). Y f Y/fx X X/x e f,x X/x’ Y/fx’ e f,x’ and for any map f : X  Y in C, we have a functor f : X  Y : X/x’ X X/x m fx’

12. X Y f X/gyY/y The underlying functor preserves adjunctible maps. A map is adjunctible if the pullback of a slice over Y is a slice over X. 1 exex X/x Y/fx e f,x mymy All adjunctions in Cat are instances of the adjunction of the bifibration of the f.s. gy

13. A cone in X is simply a map over X to a slice  : p  X/x. X P X/x X/y p A colimiting cone is universal among cones from p : 

14. Theorem: Adjunctible maps preserve colimits. Proof : X Y f P X/x Y/fx Y/y X/x’ ? ! e f,x  p mymy

15. Theorem : A map f is adjunctible at y iff f/y has a colimit preserved by f. Proof : X Y f f/yY/y X/x Y/fx e f,x q

16. We have not yet exploited the reciprocal stability law. As a first consequence, if e : X  Y is a final map (is in E ) and T  1 is a discrete opfibration (is in M ’ ) then e x T : X x T  Y x T is itself final: TYxTYxT 1 Y n p XxTXxT X e exTexT

17. Theorem : If S  1 is in M, T  1 is in M ’ and the exponential S T exists in C, then S T  1 is itself in M. Proof : Let  be the reflection map of S T in M /1. We want to show that it is an iso. Since  is final, also  x T is final. Thus  x T  be is orthogonal to S and, by adjunction,  is orthogonal to S T. So, having a retraction,  is an iso.

18. Corollary If X is a category and m and n are a d.f. and a d.o.f. over X respectively, then the exponential m n in Cat/X is itself a d.f. Since the slices of a wbfc’s are wbfc’s, we have a categorical explanation of the following fact about Cat : We say that C is a weak bfc, if we do not assume the last axiom to hold.

19. Indeed, P (L) is a wbfc : an inclusion of the subset X in Y is in M ( M ’ ) iff X is a lower (upper) set of Y, and it is in E ( E ’ ) iff Y generates (or cover) X from above (below). P (L) is a bfc iff the order is groupoidal (an equivalence relation). Another instance of the same theorem is the elementary fact that if L is a poset, A a lower subset and D an upper subset, then the relative complement A  D is a lower set (in particular, the complement of an upper set is a lower set ).

20. Now we come to the last axiom. S := M /1 = M ’ /1 is the full reflective subcategory of C of “internal sets”. The internal set of components   (X) is obtained by factorizing the terminal map: X    (X)  1 by either one of the two factorization systems. The reflection   : C  S is the “components functor”. Two objects linked by a final or initial map have the same set of components.

21. Balanced category theory is largely enriched over internal sets. If m : A  X is a map in M, and x : 1  X a point of X, the external (not enriched) value of m at x is the set of points of A over X : mx = Cat/X (x, m) = M /X (X/x, m)

22. The internal (enriched) value of a m at x is the internal set (mx) given by the following pullback: 1 x (mx) A X m By factorizing the point x through the coslice x\X, we get a factorization of the above pullback. 1 ixix (mx) [mx] x\X nxnx A X m Thus,  o [mx] = (mx). i

23. By factorizing the point x through the coslice x\X, we get a factorization of the above pullback: 1 ixix (mx) [mx] x\X nxnx A X m i Thus, [mx] and (mx) have the same components set:  o [mx] =  o (mx) = (mx).

24. This is the “co-Yoneda Lemma”, since in Cat the functor  o ( - x - ) : M ’ / X x M /X  S gives the usual tensor product of a covariant and a contravariant presheaf. Note that (mx) really enriches mx: if |-|: C  Set is the points functor C (1,-), we have |(mx)| = C /X(x,m), the set of points over x. Thus,  o [mx] =  o (x\X x m) = (mx).

25. give biuniversal elements of the profunctor |  o ( - x - ) | : M ’ /X x M /X  Set The elements x\X nxnx [x,x] X/x X mxmx 1 Therefore, the two underlying categories are dual. exex ixix

26. 1 ixix (x,y) [x,y] x\X nxnx X/y X mymy (x,y) 1 eyey i e Thus we have a cylinder: (x,y) [x,y] i (x,y) e where the common retraction is given by the components map. The internal hom-set X(x,y) =  o [x,y] enriches both C /X (X/x,X/y) and C /X (y\X,x\X) :

27. To any element s of   (X) there corresponds a connected subobject [s] which is included in X as a discrete bifibration : (x,y) X/y x\X   (X) 1 s [s] X 1   [x,y] (x,y)  b b i ei e n m

28. In Cat, the interval object 1  i e is the category of factorizations of the arrow  with the two factorizations through the identities.  =  \(X/y) = (x\X)/  Thus, by the characterization of the slice projections, we have the isomorphisms

29. Note that a slice C /X of a wbfc is a bfc iff X is groupoidal, meaning that M /X = M ’ /X. For example, if X is a groupoid, Cat/X is a bfc, with presheaves Set X = Set X* as internal sets. Thus we can develop (balanced) category theory relative to a groupoid.

30. The category of posets is a bfc with the following factorization systems: a morphism f : X  Y is in M ( M ’ ) iff it is (isomorphic to) a lower (upper) set inclusion, and it is in E ( E ’ ) iff it is cofinal (coinitial) in the classical sense. Its category of internal sets is S = 2, and internal hom-sets give the usual interpretation of posets as 2-enriched categories.

31. Balanced category theory seems to offer an alternative to the commonly accepted view that category theory is intrinsically 2- (or bi- or higher-) categorical. Conclusions We have seen that category theory largely depends on the universal properties that Cat inherits from being a bfc.