Balanced Category Theory

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 projectionsmx :X/x  X and coslice projectionsnx : x\X  X(to be thought of as a sort of neighborhoods of the category X at x).

4. A functor f : X  Yis a discrete fibrationiff the obvious slice functorsef,x :X/x  Y/fx areisomorphisms 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 projectionmx (in Cat/X) iff its domain has a terminal object iff it corresponds to a representable presheaf.

5. x\p P Q p q nx x\X f ix 1 X Y fx\Y A functor p : P  Xis 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: nfx

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. 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 . Indeed, concepts such as terminal object, representability, universal arrow and colimits, with their reciprocal interplay, are enlightened in this generalized context.

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

10. efx Y/fx 1 ex mfx mx f X/x X Y We also get slice mapsef,x :X/x  Y/fx for any map f : X  Y in C : ef,x These are final maps, that is are in E.

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

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

13. A cone in X is simply a map over Xto a sliceg : p  X/x . A colimiting cone is universal among cones from p : g X/x X/y P p X

14. Y X f Theorem: Adjunctible maps preserve colimits. Proof : g P Y/y X/x’ l ! ? p ef,x Y/fx X/x my

15. f/y Y/y Y X f Theorem : A map f is adjunctible at y iff f/y has a colimit preserved by f. Proof : ef,x Y/fx X/x l 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 Tis itself final: exT XxT YxT T p n e X Y 1

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

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

19. 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 complementA  D is a lower set (in particular, the complement of an upper set is a lower set ). 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).

20. Now we come to the last axiom. S := M/1 = M’/1 is the full reflective subcategory of C of “internal sets”. The reflectionpo : C Sis the “components functor”. The internal set of componentspo(X) is obtained by factorizing the terminal map: X  po (X) 1 by either one of the two factorization systems. 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. A (mx) [mx] (mx) A m m ix nx x x\X X X 1 1 The internal (enriched) value of a m at x is the internal set (mx) given by the following pullback: i By factorizing the point x through the coslicex\X , we get a factorization of the above pullback. Thus, po[mx] = (mx) .

23. Thus, po[mx] = po(x\X x m) = (mx) . This is the “co-Yoneda Lemma”, since in Cat the functor po ( - x - ) : M’/X xM/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 functorC (1,-) , we have |(mx)| = C/X(x,m) , the set of points overx .

24. The elements 1 ex X/x [x,x] ix mx nx X x\X give biuniversal elements of the profunctor | po ( - x - ) | : M’/X xM/X  Set Therefore, the two underlying categories are dual.

25. i [x,y] (x,y) (x,y) e The internalhom-set X(x,y) = po[x,y] enriches bothC/X (X/x,X/y)andC/X (y\X,x\X): (x,y) 1 ey e i [x,y] (x,y) X/y my ix nx x\X X 1 Thus we have a cylinder: where the common retraction is given by the components map.

26. (x,y) 1 X [s] X/y [x,y] [a] s po(X) 1 x\X a (x,y) 1 To any element s of po(X)there corresponds a connected subobject[s] which is included in X as a discrete bifibration : b e i i e n b m

27. i [a] 1 e Thus, by the characterization of the slice projections, we have the isomorphisms [a] = a\(X/y) = (x\X)/a In Cat , the interval object is the category of factorizations of the arrow a , with the two factorizations through the identities.

28. 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 SetX = SetX* as internal sets. Thus we can develop (balanced) category theory relative to a groupoid.

29. 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.

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