1. 1 Category Theory in a ( E,M ) -category Some aspects of category theory, in particular related to universality, can be developed in any finitely complete category with a factorization system on it. Claudio Pisani

2. 2 Our abstraction is modeled on the comprehensive factorization system on Cat [Street and Walters, 1973], where E and M are the classes of final functors and discrete fibrations. Discrete fibrations can be defined as the functors orthogonal to the codomain functor t : 1 → 2

3. 3 The bifibration associated to the comprehensive factorization system was presented by Lawvere [1969, 1970] as an instance of hyperdoctrine: The relevance of the same reflection in the calculus of (co)limits was shown by Paré [1973]. as an instance of the comprehension scheme [1970,1973]. He also considered the adjunction

4. 4 We here develop further this thesis: Part of the logic of category theory rests ultimately on the logic of factorization systems (that is, on their own universal properties).

5. 5 Let C be an arbitrary finitely complete category with a factorization system on it. We refer to the arrows in E as final maps, and to the arrows in M with codomain X as discrete objects over X: Discrete objects over 1 are sets. Arrows x : 1→ X with a terminal domain are objects (in an internal sense) of the codomain X.

6. 6 The basic fact about factorization systems is that any factorization p = me in C gives rise to the following universal property: That is, m is the reflection ↓ p of p in discrete objects over X, with reflection map e. In particular, a map is final iff its reflection is an isomorphism, i.e. is terminal in M /X: ↓ e = 1

7. 7 If X is terminal, we get the set S = Γ ! P of components of P as the reflection of the terminal map P → 1, giving the usual universal property. In particular an object P of C is connected ( that is Γ ! P = 1, or equivalently, the terminal map is final ) iff any map P → S to a set is constant.

8. 8 If P is terminal, we get the principal object m = ↓ x over X, as the reflection of an object x of X. In Cat, we get the discrete fibration corresponding to the presheaf represented by x, with its universal element e. The universal property now expresses exactly the (discrete fibration version of ) the Yoneda Lemma.

9. 9 Summarizing, the reflection of categories over a base in discrete fibrations is a generalization of both components and representability. Thus, these concepts can be defined in any ( E, M )-category C, and their decisive universal properties are those of the reflection

10. 10 While in Cat representability is usually defined in terms of arrows, we here define arrows of X in terms of representability (principal objects over X). (We assume that a factorization has been chosen for any object x : 1 → X ) We say that an arrow in X from x to y is a map λ : x → ↓ y with a principal codomain over X :

11. 11 Composition of arrows is defined via the Kleisli construction. We thus get is the principal category of X, i.e. the full subcategory of discrete objects over X generated by the reflections ↓ x, with inclusions : If C = Cat, we get the base category X itself.

12. 12 Arrows are a particular case of cones, that is maps over X with a principal codomain λ : p → ↓ x So a universal cone is simply a reflection in principal objects. We in fact have two colimit functors, that is the partially defined left adjoints to the above inclusions of the principal category:

13. 13 Then Colim exists iff colim exists, and Colim p = colim ↓ p, for any p : P → X Furthermore, each (universal) cone λ : p → ↓ x has a (universal) “kernel cone” λ’ : ↓ p → ↓ x

14. 14 Pullbacks preserve discrete objects: The subfibration of the codomain fibration is itself a bifibration (or, in terms of doctrines, an eed ): and in Cat give the discrete fibration version of the substitution (or composition) of a functor f in the presheaf m. In Cat, it gives the (discrete fibration version of the) left Kan extension of a presheaf along f.

15. 15 given a map f : X → Y in C and an object y : 1 → Y, a universal arrow from f to y is a final object of the pullback of ↓ y along f : Thus, such a universal arrow exists, iff the pullback f* ↓ y = Δ f ↓ y is itself principal over X. Since f* ↓ y in Cat is the usual “comma” category, sometimes denoted by ( f ↓ y ), we define:

16. 16 If a universal arrow from f to y exists for any y, that is, if Δ f preserves principal objects, we say that f may have a right adjoint. A genuine right adjoint of f : X → Y should be defined as a map g : Y → X in C such that If this is the case, f may have a right adjoint, since then and the latter preserves principal objects:

17. 17 Proposition: Existential quantification preserves principal objects: So, we have a functor C → Cat

18. 18 Thus, any cone λ : m → ↓ x with a discrete domain m : M → X has a direct image along f : X → Y : If λ : p → ↓ x is a universal cone (a colimit), we say that f : X → Y preserves the colimit if the direct image of its kernel (universal) cone λ’ : ↓ p → ↓ x is itself universal. If C = Cat, the above condition is equivalent to the standard one.

19. 19 Some theorems Theorem 1 (Paré1973, for Cat) If two maps p : P → X and q : Q→ X in C have isomorphic reflections ↓ p = ↓ q then Colim f p = Colim f q for any f : X → Y (either existing if the other one does). In particular, if e : P → X is final then Colim e p = Colim p for any f : X → Y

20. 20 If the discrete reflection of p : P → X is already principal, it is an absolute colimit of p: Theorem 2: If x is the absolute colimit of p : P → X, then fx is the absolute colimit of fp : P → X, for any f. In particular, absolute colimits are preserved by any map.

21. 21 Theorem 3: If f may have a right adjoint, then it preserves all colimits. Proof. Using the natural isomorphisms Unicity: Existence:

22. 22 Theorem 5: There is a universal arrow from f to y if and only if the pullback Δ f ↓ y has a colimit in X, preserved by f itself. Theorem 4: If a final map has a colimit, then it is absolute, and it is (the reflection of) a final object. Both of them rest on the following general fact: If an object X of a category C has a reflection X’ in a full subcategory C’, and if the reflection map e : X → X’ has a retraction r, then it is in fact the inverse of e :

23. 23 Proof of Theorem 4. If the identity map ( i.e. the terminal object 1 in M /X ) has a reflection λ : 1 → ↓ x in principal objects over X, then the terminal map is a retraction of λ. So ↓ x = 1 = ↓ 1 that is, x is a final object and an absolute colimit of the identity. The same is true for any final map e, since ↓ e = 1 Proof of Theorem 5. (Non-trivial direction.) In this case, the retraction-inverse of the universal cone λ : Δf ↓ y → colim(Δf ↓ y) is given by the adjunct u*: colim(Δf ↓ y) → Δf ↓ y of the universally induced

24. 24 So far, we have used only the ( E, M ) -structure of Cat. Now we briefly hint to how one can exploit other aspects of the rich structure of Cat: Power objects Duality and exponentials Arrow object Adding more structure

25. 25 We say that the map y : X → P X in C presents P X as a power object of X, or that y is a Yoneda map if the following composites constitute an (adjoint) equivalence between discrete objects over X and principal objects over P X : Power objects play the role of the presheaf categories in Cat. Power objects

26. 26 In Cat, the composite takes an object A : X op →Set of the presheaf category on X, in the corresponding discrete fibration. The fiber over x is given by the natural transformations P X ( X ( -, x ), A ). Thus the fact that it is an equivalence may be seen as a strong form of the Yoneda Lemma. The other composite takes a discrete fibration on X to the corresponding presheaf, expressed in the well-known form of a colimit of representable presheaves.

27. 27 Theorem : If X has a power object in C, then the discrete reflection ↓ p of a map p : P → X can be expressed as: Proof:

28. 28 Corollary (Paré1973, for Cat) Two maps p : P → X and q : Q→ X in C have isomorphic reflections ↓ p = ↓ q if and only if Colim f p = Colim f q for any f : X → Y (either existing if the other one does). In particular, we obtain the classical characterizations of final maps and absolute colimits which are usually taken as definitions in Cat : A map e : P → X is final if and only if Colim e f = Colim f for any f : X → Y x is the absolute colimit of p : P → X if and only if Colim f p = f x for any f : X → Y

29. 29 So far we have done “one-sided category theory” (say, “left-sided” ), modelling our theory on one possible choice of the comprhensive factorization system. Duality and exponentials In fact, one can here define not only left-handed concepts, such as colimits, but also right-handed ones. E.g., given two objects x and y of X, we can define their product, as the following universal arrow: More generally, if C is cartesian closed, we can similarly define the limit of any map X → Y.

30. 30 However, in order to have a balanced “two-sided” theory, we assume a “duality functor”, that is an equivalence ( - )’ : C →C, playing the role of ( - ) op : Cat → Cat Then we have another factorization system, and all the corresponding definitions and properties. For instance, a map f : X → Y is initial iff f ’ is final. The reflection ↑ p in right discrete maps can be obtained as (↓ p’ )’. And a limit is a reflection in right principal objects, which can be obtained as (Colim p’ )’. If the power object Ω = P 1 exists, we call it the object of internal sets. Assuming that C is cartesian closed, we say that X is locally small if there is a hom map h : X’ x X → Ω such that both the transposes X → Ω X’ and X’ → Ω X are Yoneda maps.

31. 31 Which axioms are appropriate for the duality functor? First, the duality functor should fix sets in a strong sense. For example, in Cat, 2 is fixed by duality, but not the functor s : 1 → 2. On the contrary any functor between sets is fixed. In particular, since sets are both left and right discrete, the same is true for the constant objects over X, obtained by pulling back a set along the terminal map.

32. 32 We can require that, as in the case C = Cat, the left discrete objects m in M /X and the right discrete objects m’ in M ’/X, are both exponentiable in C/X, and that the exponentials m m’ and m’ m are right and left discrete respectively. So, we have a complement operator, parametrized by sets, which transforms right discrete objects in left ones, and conversely. The following law can then be proved [Pisani, 2007]: Γ ! ( q* ↓ p ) = Γ ! ( p* ↑ q ) In particular, the value of the reflection ↓ p at x is given by the reflection formula: (↓ p )x = Γ ! ( p* ↑ x )

33. 33 Arrow object We say that a bipointed object s, t : 1 → 2 in C is an arrow object if t generates the factorization system. That is, an object is discrete iff it is orthogonal to t, as in Cat ( in particular, t is final ). A map λ : 2 → X is an arrow of X (in an internal sense). If there is a duality functor, we also require that it “exchange” the source and target maps. Since t and the composite ! t : 1 → 2 → 1 are both final, also ! : 2 → 1 is final, that is 2 is connected.

34. 34 Then it is easy to see that our sets coincide with those defined in [Lawvere,1965] : all the arrows 2 → S are identities (constant maps). Furthermore, any object X of C “is” a graph, and any discrete object over X “is” a graph fibration. As shown in [Lawvere, 1965], by posing appropriate conditions on the pushouts 3 and 4 of 2, one gets an associative composition of internal arrows. In this case, any object X of C “is” a category X*, and any discrete object m over X “is” a discrete fibration m*.

35. 35 Then one can prove that we so obtain a functor from X* to the principal category on X and so also a functor F : X* → M /X and that m* corresponds to the presheaf : M /X ( F -, m ) Given an arrow λ : 2 → X, one gets an arrow in the old sense by taking the source s λ’ of the arrow λ’ obtained by lifting the target of λ to the universal element of the corresponding principal object. There is a coherence between the category of internal arrows and the principal category:

36. 36 Further examples While our abtraction enlightens some aspects of category theory, one may wonder if there are other relevant instances of the theory. Indeed, if the objects 1 → X are in M, then the principal category of X is discrete and a map P → X has a colimit iff it is constant in a strong sense. In fact, most of the “classical” factorization systems give rise to rather uninteresting “category theories”. For instance, in Set with the ( iso, all ) or the (epi, mono) factorization system, a mapping may have (and has) a right adjoint iff it is a bijection.

37. 37 Power objects are given by P X = ↓X, the poset of lower sets, and in particular Ω = 2. Also interesting is the case of reflexive graphs, with graph fibrations as discrete objects over X. In this case, while an arrow object does exist, there is no internal composition. On the other hand, the principal category is the free category on the graph. The case of irreflexive graphs is very different: sets are endomappings, while e.g. the dot graph and the arrow graph are not connected (in fact, the latter is not an arrow object, since the dot is not terminal). Doing category theory in Pos, with cofinal mappings as final maps, and the inclusions A → X of lower sets as discrete objects over X, may give new perspectives on the two-valued nature of posets.

38. 38