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Geometria Superiore Reference Cards The composition of mono (resp. epi, bijections) is again a mono (resp. epi, bijection); if the product gf is mono (resp. epi) then Categories and f is mono (resp. g is epi). c 2000 M. Cailotto, Permissions on last. v0.0 Send comments and corrections to [email protected] In the U −, mono (resp. epi, bijection) correspond to injective (resp. surjective, bijective) maps. Categories. A family of f : X →X is epimorphic if the Q i i C(X,W )→ C(Xi,W ) is injective for any W ∈ obW . Dually, a Universes. A universe U is a set, whose elements are called i family f : X →X is monomorphic if the map C(W,X)→Q C(W,X ) U-sets, subject to the following axioms: i i i i is injective for any W ∈ obW . (U1) if x ∈ U and y ∈ x then y ∈ U; Left and Right Invertible. Let f : X →Y be a of C. (U2) if x,y ∈ U then {x,y} ∈ U; It is left invertible (or a split mono) if there exists g ∈ C(Y,X) (U3) if x ∈ U then P(x) ∈ U; such that gf = idX . Equivalently the map of set C(f,Z) = ◦f : C(Y,Z)→C(X,Z) is surjective for any Z ∈ obC. It is right in- (U4) if (x ) is a famly of elements of U and I ∈ U, then α α∈I vertible (or a split epi) if there exists g ∈ C(Y,X) such that fg = the union ∪α∈I xα is an element of U. idY . Equivalently the map of set C(Z,f) = f◦ : C(Z,X)→C(Z,Y ) The axiom of universes says that for any set x there is a univers is surjective for any Z ∈ obC. It is an if it is left U such that x ∈ U. In particular for any univers U there is a and right invertible. univers V such that U ∈ V (remark that U/∈ U). The composition of split mono (resp. split epi, ) Therefore in the theory of category one has to decide the univers is again a split mono (resp. split epi, isomorphism); if the in which we may take the elements, but we can increase the product gf is split mono (resp. split epi) then f is split mono univers if it is needed. (resp. g is split epi). Categories. An U-category C is a collection obC of objects A split mono (resp. split epi, isomorphism) is mono (resp. epi, and for any pairs X,Y ∈ obC a set (i.e. an element of U) of bijection). morphisms Hom (X,Y ) (indicated also as C(X,Y ) or Y (X)) C The morphism f is an isomorphism iff it is mono and split epi with the notion of composition, i.e. for any X,Y,Z ∈ obC of a (i.e. C(Z,f) is bijective for any Z ∈ obC), iff it is split mono map C(X,Y )×C(Y,Z)→C(X,Z) indicated as a product (f,g) 7→ and epi (i.e. C(f,Z) is bijective for any Z ∈ obC). gf = g ◦f, subject to the following conditions: Exact diagrams. A diagram in USet as (C1) for any object X there exists idX ∈ C(X,X) (the identity f g1 morphism of X) such that idX f = f for any f ∈ C(Z,X), X −−→Y −−→Z Z ∈ obC and fidX = f for any f ∈ C(X,Z), Z ∈ obC; g2 (C2) the product of morphisms is associative, i.e. h(gf) = (hg)f is exact if it is commutative (g1f = g2f), f is injective and in C(X,W ) for any f ∈ C(X,Y ), g ∈ C(Y,Z) and h ∈ C(Z,W ). g1(y) = g2(y) implies there exists x such that y = f(x); we say We will indicate with morC the class of morphisms of C. also that (X,f) is a kernal of the pair (g1,g2). An U-category C is U-small if the collection of objects is an A similar diagram in C is exact if for any A ∈ obC the diagram g element of U; it is finite if the class obC is a finite set; it is f 1 C(A,X)−−→C(A,Y )−−→C(A,Z) discrete if the only morphisms are the identities. g2 . Let C be a category; the opposite category is exact in USet; in other words if it is commutative (g1f = o o o C is defined by obC = obC and C (X,Y ) = C(Y,X) for any g2f), f is a and for any u : A→Y such that o X,Y ∈ C . g1u = g2u there exists v : A→X with u = fv. For any categorical property of objects or morphisms, we de- A diagram in C of the form fine the property by requiring this property (of objects or f1 g morphisms) on the dual category. X −−→Y −−→Z f Product of Categories. C ×C0 is the category whose object 2 (resp. morphisms) are the pairs of objects (resp. morphisms). is exact if the diagram f Category of Morphisms. F(C) is the category whose objects g 1 C(Z,A)−−→C(Y,A)−−→C(X,A) are the morphisms of C and morphisms between are the com- f2 mutative squares. is exact in USet for any A ∈ obC; i.e. if it is commutative Localized Categories. Let X ∈ obC; the category C/X (resp. (gf1 = gf2), g is an epimorphism and for any u : Y →A such X/C) has objects the morphisms in C of target X (resp. of that uf1 = hf2 there exists v : Z→A with u = vg. source X) and morphism the commutative triangles. In particular a similar diagram in USet is exact iff it is com- Names? USet, UGrp, UAb, UT op mutative (gf1 = gf2), g is surjectiive and the equality g(y1) = Properties of morphisms. g(y2) implies that y1 ∼ y2 where ∼ is the in Y generated by (f1(x),f2(x)) and (f2(x),f1(x)) for all x ∈ X. , Epimorphisms, Bimorphisms. Let f : X →Y This is true iff Z=∼Y/ ∼; we say also that (Z,g) is a of be a morphism of C. It is a monomorphism (mono for short) the pair (f1,f2). if it is left eraseable, i.e. if fu = fv implies u = v for any u,v ∈ −→ C(Z,X) and Z ∈ obC. Equivalently the map of set C(Z,f) = In any category C the diagram X −→Y −→Y tX Y is exact iff −→ f◦ : C(Z,X)→C(Z,Y ) is injective for any Z ∈ obC. It is an epi- X −→Y is a split mono; and X ×Y X −→X −→Y is exact iff morphism (epi for short) if it is right eraseable, i.e. if uf = vf X −→Y is a split epi. implies u = v for any u,v ∈ C(Y,Z) and Z ∈ obC. Equivalently Effective Morphisms. A morphism f : X →Y is left effective if the map of set C(f,Z) = ◦f : C(Y,Z)→C(X,Z) is injective for it factorizes any morphism u : A→Y having a bigger left equal- any Z ∈ obC. It is a bijection (or a bimorphism) if it is mono ization set: i.e. if for any g1,g2 we have that g1f = g2f im- and epi. plies g1u = g2u then u = fv. In case we have in C the push-out

1 GeoSupRC - Categories and Functors c 2000 by MC −→ i1,i2 : Y −→Y tX Y the condition is equivalent to: if i1u = i2u i×id id ×i O × M −−−−→M M × M ←−−−−M M × O then u = fv. O O O ∼= c ∼= A morphism f : X →Y is right effective if it factorizes any mor- y y y M −−−−−→ M ←−−−−− M. phism u : A→Y having a bigger right equalization set: i.e. if = = for any g1,g2 we have that fg1 = fg2 implies ug1 = ug2 then −→ u = vf. In case we have in C the pull-back p1,p2 : X ×Y X −→X the condition is equivalent to: if up1 = up2 then u = vf. Functors. Remark that a morphism is an isomorphism iff it is a (left or Let C and C0 two categories; a (covariant) F : C→C0 is right) effective bijection. the data of a map obF : obC→obC0 and for any X,Y ∈ obC of a 0 Effective Monomorphisms and Epimorphisms. A morphism f map F : C(X,Y )→C (FX,FY ) respecting identities (F (idX ) = is an effective monomorphism if it is left effective and mono. In idFX for any X) and compositions (F (gf) = F (g)F (f) for any −→ composable f,g). case we have in C the push-out i1,i2 : Y −→Y tX Y the condition −→ o 0 is equivalent to: the diagram X −→Y −→Y tX Y is exact, i.e. f A contravariant functor is just a functor C →C . is the of (i1,i2); or again f is the kernel of some pair A functor is faithful (resp. full, fully faithful) if for any X,Y ∈ −→ j1,j2 : Y −→W . Effective mono is equivalent to C(A,f) = f◦ to obC the map C(X,Y )→C0(FX,FY ) is injective (resp. surjec- be a kernel. tive, bijective). Remark that f is iso iff it is epi and effective mono. The esssential is the collection of X0 in C which are 0 A morphism f is an effective epimorphism if it is right effective isomorphic in C to an object of the form FX for X in C.A −→ functor is essentially surjective if the essential image is the all and epi. In case we have in C the pull-back p1,p2 : X ×Y X −→X −→ of C0. the condition is equivalent to: the diagram X ×Y X −→X −→Y is exact, i.e. f is the cokernel of (p1,p2); or again f is the The collection of U-categories has a structure of category using kernel of some pair q1,q2 : W →X. Effective epi is equivalent the functors as morphisms. Call this UCat. to C(f,A) = ◦f to be a kernel. Morphisms of Functors. Let F,G : C→C0 be two functors; Remark that f is iso iff it is mono and effective epi. a morphism ϕ : F →G is just the data for any X ∈ obC of a morphism ϕ(X): FX →GX in C0 subject to the compatibility: Strict Monomorphisms and Epimorphisms. A morphism f is for any morphism f : X →Y the diagram a strict monomorphism if for any epimorphism f0 : X0 →Y 0 the 0 0 0 natural map C(Y ,X)→C(f ,f) given by h 7→ (hf ,fh) is a bi- ϕX jection. Remark that a strict mono is mono, and that an effec- FX −−→ GX tive mono is strict mono. F f y y Gf FY −−→ GY A morphism f is a strict epimorphism if for any monomorphism ϕY f0 : X0 →Y 0 the natural map C(Y,X0)→C(f,f0) given by h 7→ (hf,f0h) is an bijection. Remark that a strict epi is epi, and commute, i.e. G(f)ϕ(X) = ϕ(Y )F (f). that an effective epi is strict epi. The collection of functors Cat(C,C0) has a structure of category The composition of strict mono (resp. strict epi) is again a under the obvious composition of morphisms of functors. strict mono (resp. strict epi); if the product gf is strict mono The collection of U-categories has a structure of 2-category us- (resp. strict epi) then f is strict mono (resp. g is strict epi). ing the functors as (1-)morphisms, and the morphism of func- tors as 2-morphism. Remark that f is iso iff it is mono and strict epi or iff it is epi and strict mono. In order to distinguish the two types of compositions for mor- phisms of functors, we write ψ ◦ϕ = ψϕ : F →H for the “vertical Summary. We have the following implications: composition” of ϕ : F →G and ψ : G→H; and ϕ0 •ϕ : F 0F →G0G iso ⇒ split mono ⇒ effective mono ⇒ srict mono ⇒ mono; for the “horizontal composition” of ϕ : F →G and ϕ0 : F 0 →G0. and Let C,C0,C00 three categories; the composition of functors gives iso ⇒ split epi ⇒ effective epi ⇒ strict epi ⇒ epi. rise to the functor: Cat(C,C0)×Cat(C0,C00)→Cat(C,C00) send- 0 0 0 0 In the category USet a monomorphism (resp. epimorphism) is ing (F,F ) to F F , and (ϕ,ϕ ) to ϕ •ϕ, having the following a aplit mono (resp. split epi). properties: id 0 •id = id 0 , Abstract Categories. An abstract category in C is the data F F F F of two objects O and M, and of morphisms s,t : M →O (source (ψ ◦ϕ)•(ψ0 ◦ϕ0) = (ψ •ψ0)◦(ϕ•ϕ0), and target), i : O→M (identity), c : M ×O M →M (composi- F 00(F 0F ) = (F 00F 0)F and ϕ00 •(ϕ0 •ϕ) = (ϕ00 •ϕ0)•ϕ (associa- tion) subject to the following conditions: si = idO = ti (source tivity), and target of the identities), c(c×idM ) = c(idM ×c) (associa- 0 0 ϕ •idF = ϕ and idF 0 •ϕ = ϕ (identity). tivity), c(i×idM ) = idM = c(idM ×i) (compositions with iden- tities). The properties correspond to the commutativity of the Equivalence of Categories. A functor F : C→C0 is an equiva- following diagrams: lence if there exists a functor G : C0 →C and two isomorphisms ∼ ∼ i s of functors FG=idC0 and GF =idC (G is called a quasi inverse, M −−→O−−→M it is defined up to isomorphism); equivalently if it is fully faith- t ful and essentially surjective (this result needs for some version of the choice axiom). c×id M × M × M −−−−→M M × M O O O Remark that the data of G with an isomorphism ϕ : FG→idC0   0 0 idM ×cy y c is equivalent to the data for any X ∈ obC of a pair (X,u) 0 M ×O M −−−−−−−→ M with X ∈ obC and u : F (X)→X an isomorphism; in fact we c associate to such a G the pairs (G(X0),ϕ(X0)).

2 GeoSupRC - Categories and Functors c 2000 by MC Categories of Diagrams. A type of diagram is a small category Kernel and Cokernel. Let f,g : X →Y two morphisms in C; if D; a diagram of type D in a category C is a functor D→C; the the functor in C∨ sending W ∈ obC in ker(f,g : C(W,X)→C(W,Y )) diagrams of type D in C form a category CD = Cat(D,C), and (the equalizator of two maps of sets, i.e. the subset of the we have a canonical functor C→CD sending any object in the domain where the functions coincide) is representable, we call constant functor having that object as value. This functor is ker(f,g) (kernel or equalizator) the representative, and α : ker(f,g)→X faithful, and fully faithful if the category D is connexe (i.e. any the canonical morphisms given by the Yoneda yoga (necesser- two objects of D can be connected by a sequence of morphisms, ily mono with fα = gα), determining the indipendently of the directions). C(W,ker(f,g))=∼ker(f,g : C(W,X)→C(W,Y )) for any W ∈ obC. Representability. If the functor in C∧ sending W ∈ obC in ker(f,g : C(Y,W )→C(X,W )) Put C∧ = Cat(C,USet) and C∨ = Cat(Co,USet). Then we have (the equalizator of two maps of sets, i.e. the subset of the do- canonical functors h∧ : Co →C∧ and h∨ : C→C∨ given resp. by main where the functions coincide) is representable, we call h∧(X)(Y ) = C(X,Y ) and h∨(X)(Y ) = C(Y,X), and the natural coker(f,g) (cokernel or coequalizator) the representative, and action on morphisms. γ : Y →coker(f,g) the canonical morphisms given by the Yoneda yoga (necesserily epi with γf = γg), determining the universal ∧ ∨ A functor F ∈ C (resp. F ∈ C ) is representable if it is isomor- property C(coker(f,g),W )=∼ker(f,g : C(Y,W )→C(Y,W )) for any ∧ ∨ ∧ phic in C (resp. in C ) to a functor of the form h (X) (resp. W ∈ obC. h∨(X)), i.e. if it is in the essential image of h∧ (resp. h∨). In the category USet: the kernel is just the usual equalizator, ∧ . For any F ∈ obC and X ∈ obC, the map the cokernel is just the quotient of Y by the equivalent relation ∧ ∧ Φ: C (h (X),F )→F (X) given by ϕ 7→ Φ(ϕ) = ϕ(X)(idX ) is a generated by the pairs (f(x),g(x)) varying x ∈ X. bijection with inverse the map Ψ sending ξ ∈ F (X) to the mor- phism Ψ(ξ) defined as Ψ(ξ)(Y )(f) = F (f)(ξ). In particular for Pull-back (fibred product) and Push-out (amalgamed sum). Let ∧ ∧ ∧ ∧ ∼ f g F = h (Y ) we have C (h (X),h (Y ))=C(Y,X), so that the X →Z←Y two morphisms in C; if the functor in C∨ send- functor h∧ is fully faithful. ing W ∈ obC to C(W,X)×C(W,Z) C(W,Y ) (the fibred product Dually for F ∈ obC∨ and X ∈ obC; with the same definition we of sets, i.e. the subset of the cartesian product equalizing ∨ ∨ have a canonical bijection C (h (X),F )→F (X). the given maps) is representable, we call X ×Z Y (pull-back or fibred product) the representative, and p : X × Y →X, In particular if F is a representable functor in C∧, the iso- X Z p : X × Y →Y the canonical morphisms given by the Yoneda morphism F =∼h∧(X) is determined by an element ξ ∈ F (X); Y Z yoga (necessarily fp = gp ), determining the universal prop- the pair (X,ξ) is the representant of F , which is characterized X Y erty C(W,X × Y )∼C(W,X)× C(W,Y ) for any W ∈ obC. by the following universal property: for any Y ∈ obC and any Z = C(W,Z) η ∈ F (Y ) there exists a unique f : X →Y such that F (f)(ξ) = η. f g Dually, given the diagram X ←Z→Y if the functor in C∧ send- ∨ Dually: if F is a representable functor in C , the isomorphism ing W ∈ obC in C(X,W )× C(Y,W ) is representable, we F =∼h∨(X) is determined by an element ξ ∈ F (X); the pair C(Z,W ) call X t Y (push-out or amalgamed sum) the representative, (X,ξ) is the representant of F , which is characterized by the Z and i : X →X t Y , i : Y →X t Y the canonical morphisms following universal property: for any Y ∈ obC and any η ∈ F (Y ) X Z Y Z given by the Yoneda yoga (necessarily iX f = iY g), determining there exists a unique f : Y →X such that F (f)(ξ) = η. ∼ the universal property C(X tZ Y,W )=C(X,W )×C(Z,W ) C(Y,W ) Final, Initial and Zero Objects. Any set P with one element is for any W ∈ obC. a final object in the category USet, i.e. for any S ∈ obUSet we ∼ have only one element in USet(U,P ). The functor sending any Remark that X ×Z Y =ker(fpX ,gpY : X ×Y →Z) canonically, ∼ X ∈ obC in a fixed set with one element is either a covariant and X tZ Y =coker(iX f,iY g : Z→X tY ), also canonically. More- ∼ and contavariant functor; if it is representable as an element of over we have canonical identifications ker(f,g : X →Y )=(X ×Y C∧ the representative I ∈ obC is a initial object of C (character- X)×X×X X (where we use the diagonal immersion of X in ∼ ized by C(I,X) having exactly one element for any X); if it is X ×X) and coker(f,g : X →Y )=(Y tX Y )tY tY Y (where we representable as an element of C∨ the representative F ∈ obC is use the canonical morphism of Y tY in Y ). In particular the a final object of C (characterized by C(X,F ) having exactly one existence of kernel and finite products is equivalent to the ex- element for any X). An final and initial object is called a zero istence of fiber product and finite products, and also to the object; if the category has a zero object, for any X,Y ∈ obC we existence of fiber product and final object (remark that the call zero morphism from X to Y the unique morphism which product is a fiber product over the final object). Dually the ex- factorizes through the zero object. istence of cokernel and finite sums is equivalent to the existence of amalgamed sum and finite sums and also to the existence of In the category USet: P is a final object, and the empty set ∅ amalgamed sum and initial object (the sum is the amalgamed is the initial object. sum over the initial object). ∨ Product and Sum. Let X,Y ∈ obC; if the functor in C send- A morphism f : X →Y is mono iff the morphisms X × X −→X ing W ∈ C in C(W,X)×C(W,Y ) (cartesian product in USet) Y −→ coincide, iff the diagonal morphism X →X ×Y X (defined by is representable, we call X ×Y (product) the representative, the universal property asking for the compositions with the and pX : X ×Y →X, pY : X ×Y →Y the canonical morphisms projections to be the identity of X) is an isomorphism. given by the Yoneda yoga, determining the universal property C(W,X ×Y )=∼C(W,X)×C(W,Y ) for any W ∈ obC. Dually, a morphism f : X →Y is epi iff the canonical morphisms Y −→Y t Y coincide, iff the codiagonal morphism Y t Y →Y Dually, if the functor in C∧ sending W ∈ C in C(X,W )×C(Y,W ) −→ X X (defined by the universal property asking for the compositions (cartesian product in USet) is representable, we call X tY with the injections to be the identity of Y ) is an isomorphism. (sum) the representative, and iX : X →X tY , iY : Y →X tY the canonical morphisms given by the Yoneda yoga, determin- In the category USet: the fibred product is just the usual fiber ing the universal property C(X tY,W )=∼C(X,W )×C(Y,W ) for product, the sum is just the quotient of the disjoint union of any W ∈ obC. sets with the equivalent relation generated by f(z) ∼ g(z). In the category USet: the product is just the cartesian product, Projective and Injective Limits. Let I a small U-category, and the sum is just the disjoint union of sets. F : I→C a functor; define the functor in C∨ sending W ∈ obC

3 GeoSupRC - Categories and Functors c 2000 by MC to←− limI C(W,F ) = lim←−i∈I C(W,F i) (projective of the pro- is an effective epimorphism. The equivalence relation is uni- jective system I→USet given by the composition of F with versal effective if it is effective and π is a universal effective h∧(W ); the projective limit is just the subset of the prod- epimorphism. Q uct i∈obI C(W,F i) given by the elements compatible with General Yoneda Lemmas. Let F ∈ C∧; define the category the maps of the system). If the functor is representable, we C/F having objects the morphisms X = h∧(X)→F in C∧ with call lim←−I F = lim←−i∈I F i the representative, pi : lim←−I F →F (i) for X ∈ C (that is the pairs (X,ξ) with ξ ∈ F (X)), and mophisms any i ∈ obI the canonical morphisms given by the Yoneda yoga the morphisms h∧(X)→h∧(Y ) compatible over F (i.e. the (necessarily we have the compatibilities F (ϕ)pi = pj if ϕ : i→j morphisms Y →X such that F (ϕ)(η) = ξ). Then we have an is any morphism of I) determining the universal property given inductive system h∧/F : C/F −→C∧ and a canonical morphism by C(W,lim F )=∼lim C(W,F ) for any W ∈ obC. ∧ ∧ ←−I ←−I −→lim h /F = lim−→ h (X)−→F Let I a small U-category, and F : Io →C a functor; define the C/F X∈C/F ∧ functor in C sending W ∈ obC to lim←−I C(F,W ) = lim←−i∈I C(F i,W ) in C∧ given for any Y ∈ obC by (projective limit of the projective system I→USet given by lim C∧(h∧(Y ),h∧(X))=∼ lim C(X,Y )−→F (Y ) the composition of F with h∨(W ); the projective limit is just −→ −→ X∈C/F X∈C/F the subset of the product Q C(F i,W ) given by the ele- i∈obI ∧ ∼ ∧ ∼ ments compatible with the maps of the system). If the func- sending h (f)=f to (h (f)α)(Y )(idY )=α(Y )(f) = F (f)(ξ) (where ∧ tor is representable, we call lim F = lim F i the representa- α : h (X)→F ). It is an isomorphism with inverse sending η ∈ −→I −→i∈I F (Y ) to the morphism id where Y is considered in C/F by tive, i : F (i)→lim F for any i ∈ obI the canonical morphisms Y i −→I means of η. given by the Yoneda yoga (necessarily we have the compati- bilities iiF (ϕ) = ij if ϕ : i→j is any morphism of I) determin- We may deduce the general Yoneda lemma: ing the universal property C(lim F,W )=∼lim C(F,W ) for any ∧ ∼ ∼ −→I ←−I C (F,G)= ←−lim G(X)= ←−lim −→lim C(Y,X) W ∈ obC. X∈C/F X∈C/F Y ∈C/G In the category USet the projective limit is just the usual pro- and the reverse Yoneda lemma jective limit already described, and the inductive limit is given C∧(F,h∧(X))=∼ lim C(Y,X) . by the quotient of the disjoint union by the equivalence relation ←− X∈C/F generated by the maps of the system. Remark that product, fibred product, kernel (resp. sum, amal- Dually: let F ∈ C∨; define the category C/F having objects the gamed sum, cokernel) are just projective (resp. injective) lim- morphisms X = h∨(X)→F in C∨ with X ∈ C (that is the pairs (X,ξ) with ξ ∈ F (X)), and mophisms the morphisms h∨(X)→h∨(Y ) its on the index categories given by {i1,i2}, {i1 →i3 ←i2} and −→ compatible over F (i.e. the morphisms X →Y such that F (ϕ)(η) = {i1 −→i2} (only the non obvious morphisms are indicated). ξ). Then we have an inductive system h∨/F : C/F −→C∨ and Moreover we have canonical isomorphisms a canonical morphism ∼ Q −→Q ←−limF =ker( i∈obI F (i)−→ ϕ∈morI F (tϕ)) lim h∨/F = lim h∨(X)−→F I −→ −→ C/F X∈C/F where the two morphisms are defined by composition with the in C∧ given for any Y ∈ obC by canonical projections pϕ to be the ptϕ and the com- ∧ ∧ ∧ ∼ position F (ϕ)psϕ; and −→lim C (h (Y ),h (X))= −→lim C(Y,X)−→F (Y ) ∼ ` −→` X∈C/F X∈C/F −→limF =coker( ϕ∈morI F (tϕ)−→ i∈obI F (i)) ∨ ∼ ∨ ∼ I sending h (f)=f to (h (f)α)(Y )(idY )=α(Y )(f) = F (f)(ξ) (where where the two morphisms are defined by composition with the α : h∨(X)→F ). It is an isomorphism with inverse sending η ∈ canonical injections iϕ to be the injection itϕ and the compo- F (Y ) to the morphism idY where Y is considered in C/F by sition isϕF (ϕ). means of η. In particular the existence of projective (resp. injective) lim- We may deduce the general Yoneda lemma: ∨ ∼ ∼ its is equivalent to the existence of (arbitrary) products and C (F,G)= ←−lim G(X)= ←−lim −→lim C(X,Y ) kernels (resp. of (arbitrary) sums and cokernel); while the ex- X∈C/F X∈C/F Y ∈C/G istence of finite projective (resp. injective) limits is equivalent and the reverse Yoneda lemma to the existence of (finite) products and kernels (resp. of (fi- ∧ ∨ ∼ nite) sums and cokernel), i.e. to the existence of final object C (F,h (X))= ←−lim C(X,Y ) . and fibred products (resp. initial object and amalgamed sums). X∈C/F Equivalence Relations. Let F,G ∈ C∨; a pair p,q : F −→G is Universality, Couniversality. A property of morphism f : an equivalence relation on G if for any W ∈ obC the induced X →Y is said to be universal (resp. couniversal) if it is sta- morphism F (W )→G(W )×G(W ) gives a bijection of F (W ) ble for cartesian (resp. cocartesian) diagram, i.e. if for any with an equivalence relation on G(W ). If X,Y ∈ obC, a pair g : W →Y (resp. g : X →W ) the morphism pW : W ×Y X →W −→ f,g : Y −→X is an equivalence relation if the induced diagram (resp. iW : W →Y tX W ) has the same property. of representable functors is. Mono, split mono, effective mono, strict mono, to be isomor- Remark that if π : X →Y is a morphism in C for which the phism are universal properties; dually epi, split epi, effective fiber product X ×Y X is representable, then the canonical pro- epi, strict epi, to be iso are couniversal properties. −→ jections X ×Y X −→X gives an equivalence relation. We say that a morphism f : X →Y has universal(ly) a prop- 0 An equivalence relation p,qR−→X on X ∈ obC is effective if erty if it has the property and for any base change Y →Y the p 0 0 0 0 0 R→X canonical morphism f : X →Y (where X = X ×Y Y ) has the there exists a morphism π : X →Y such that the square q↓ ↓π property. Dually, a morphism f : X →Y has couniversal(ly) a X→Y 0 π property if it has the property and for any fibre change X →X 0 0 0 0 0 is cartesian and cocartesian. In particular π is the cokernel of the canonical morphism f : X →Y (where Y = Y tX X ) has (p,q), so it is determined up to a unique isomorphism, and it the property.

4 GeoSupRC - Categories and Functors c 2000 by MC Remark that universal effective epi and couniversal effective (i) if kernel or cokernel are representable in C and the functors mono are stable by composition. Fi commute with, then conservative implies faithful; Adjonction. Two functors F : C→C0 and G : C0 →C are adjoint (ii) if the fiber product (resp. amalgamed sum) exists in C, and one of the other (F left adjoint and G right adjoint) if one of the functors Fi commute with, then the family Fi faithful the following equivalent conditions holds: or conservative implies that a morphism u is mono (resp. epi) iff F (u) is for all i; (Ad1) the two (bi)functors Co ×C0 →USet given by sending (X,X0) i to C0(FX,X0) and C(X,FX0) are isomorphic; (iii) if the fiber product (resp. amalgamed sum) exists in C, the functors Fi commute with, and every bimorphism is (Ad2) there exist two morphisms α : FG→idC0 and β : idC →GF 0 0 0 0 an isomorphism, then faithful implies conservative; such that for any X ∈ C we have G(αX )β(GX ) = idX0 (iv) if the fiber product (resp. amalgamed sum) exists in C, and and for any X ∈ C we have α(FX)F (βX) = idX ; the functors F commute with, then conservative for mono 0 0 ∨ 0 0 i (Ad3) for any X ∈ obC the functor in C sending X to C (FX,X ) (resp. epi) implies conservative; is representable and for any X ∈ obC the functor in C0∧ sending X0 to C0(X,GX0) is representable. (v) if kernel (or cokernel) and fiber product (resp. amalgamed sum) exists in C, and the functors Fi commute with, then Remark that if F admits a right adjoint G this is unique up to the notions of faithful, conservative, conservative for mono isomorphism, and dually. (resp. for epi) are equivalent. A left adjoint functor commute with arbitrary inductive limits, Generation and Cogeneration. Let C be a category and D and a right adjoint functor commute with arbitrary projective be a full . We say that D is: limits. (Ge) generator by epi (resp. effective epi) if for any X ∈ obC the Remark that F is faithfull (resp. fully faithful) iff β is a family of morphism D/X is epi (resp. effective epi); monomorphism (resp. an isomorphism); dually G is faithfull (Gm) generator (resp. for the mono, for the effective mono) if (resp. fully faithful) iff α is an epimorphism (resp. an isomor- for any u : Y →X (resp. mono, effective mono) such that phism); C(W,Y )→C(W,X) is a bijection for any W ∈ obD, then u Morphisms for . If F,G and F 0,G0 are two is an isomorphism. pairs of adjoint functors, then there is a canonical (anti)bijection Consider the family H of functors {h∧(X) ∈ C∧|X ∈ obD} and 0 0 between morphisms of functors Cat(F,F )=∼Cat(G ,G). the canonical functor h∧ : C→C∧ →D∧. Then we have the |D Existence of Universal Objects. In a category C, the existence following fact: of products (resp. sums) for any pair of objects is equivalent (Ge) D is generator by epi iff H is faithful iff h∧ is faithful; to the existence of a right (resp. left) adjoint to the canonical |D (G) D is generator iff H is conservative iff h∧ is conservative; diagonal immersion C→C ×C. |D The existence of kernels (resp. ) for any pair of mor- (Gm) D is generator for the mono iff H is conservative for mono phism is equivalent to the existence of a right (resp. left) ad- iff h∧ is conservative for mono; |D joint to the canonical inclusion C→F(C)×C×C F(C) with the obvious category structure on the rhs. (Gem) D is generator for the effective mono iff H is conservative for effective mono iff h∧ is conservative for effective mono; The existence of fibred products (resp. amalgamed sums) is |D equivalent to the existence of a right (resp. left) adjoint to (Gee) D is generator by effective epi iff the canonical morphism o 0 ∧ the canonical functor C→F(C)×C F(C) (resp. C→F(C) ×C −→lim X →X is an isomorphism for all X ∈ obC, iff h o D/X |D F(C) ). is fully faithful. Properties of Functors. In particular we have the following implications: Faithfull. Let F : C→C be a family of functors; the family if (Ge) ⇐ (Gee) ⇒ (G) ⇒ (Gm) ⇒ (Gem) i i Q faithfull if the map C(X,Y )→ iCi(FiX,FiY ) is injective for and the following conditioned implications: any X,Y ∈ obC, i.e. if two morphisms of C are equal when they (Ge) ⇒ (Gee) if in C the epi families are effective; became equal under all the Fi. Q (Gem) ⇒ (Gm) if in C the monomorphisms are effective; The family Fi is faithful iff the product functor F = Fi : Q i (G) ⇒ (Ge) if in C the kernels are representable; C→ Ci is faithful. i (Gm) ⇒ (G) if in C the fibred products are representable; Conservative. A family of functors F : C→C is conservative i i (Gm) ⇒ (Gee) if every family of morphism X →X factorizes (resp. conservative for mono, epi,...) if for any morphism u of i by effective epimorphisms followed by a mono Xi →Y →X; C, we have that Fi(u) isomorphism (resp. mono, epi,...) for all i implies that u is iso (resp. mono, epi,...). (Gem) ⇒ (Ge) if every family of morphism Xi →X factorizes Q by epimorphisms followed by an effective mono Xi →Y →X. The family Fi is conservative iff the product functor F = iFi : C→Q C is. In particular all the notions of generations coincide in the fol- i i lowing two cases: Exactness. A functor F : C→C is left (resp. right) exact if it i (i) kernel and fibred product are representable, all monomor- commutes with finite projective (resp. injective) limits (i.e. the phisms are effective and all epi families are effective; canonical morphisms F (lim←−Xi)→←−limFXi (resp.−→ limFXi →F (lim−→Xi)) are isomorphisms). This is equivalent to: F sends final object (ii) every family of morphism Xi →X factorizes by epimor- in final object and commutes with product and kernel (resp. phisms followed by an mono Xi →Y →X, all monomor- initial object in initial object and commutes with sum and cok- phisms are effective and all epi families are effective. ernel); or also to: F sends final object in final object and com- Dually: say that D is: mutes with fibred product (resp. initial object in initial object (cGe) cogenerator by mono (resp. effective mono) if for any X ∈ and commutes with amalgamed sum). obC the family of morphism X/D is mono (resp. effective Relations. mono);

5 GeoSupRC - Categories and Functors c 2000 by MC (cGm) cogenerator (resp. for the epi, for the effective epi) if for any In particular if the left exact functors over the category D are u : Y →X (resp. epi, effective epi) such that C(X,W )→C(Y,W ) determined by a single object, then a D-algebraic structure on is a bijection for any W ∈ obD, then u is an isomorphism. C is just an object X of C such that for any S ∈ obC the set Consider the family H of functors {h∨(X) ∈ C∨|X ∈ obD} and X(S) have a functorial (in S) structure of type D. the canonical functor h∨ : C→C∨ →D∨. Then we have the Structure. Let Mon be the smallest category with fi- |D following fact: nite products (let 1 be the terminal object), an object G, and morphisms p : G×G→G and 1 : 1→G subject to the follow- (cGm) D is cogenerator by mono iff H is faithful iff h∨ is faithful; |D ing conditions: g(idG ×g) = g(g ×idG) (associativity), g(1× ∼ ∼ (cG) D is cogenerator iff H is conservative iff h∨ is conservative; idG)=idG=g(idG ×1) (identity) and eventually gs = g (com- |D mutativity) where sc : G×G→G×G is the canonical morphism (cGe) D is cogenerator for the epi iff H is conservative for epi iff of exchange defined by p{1,2}sc = p{2,1}. h∨ is conservative for epi; |D Then MonC is the category of C-; a C-monoid is an (cGee) D is cogenerator for the effective epi iff H is conservative object X ∈ obC with a functorial monoid structure on X(S) for for effective epi iff h∨ is conservative for effective epi; any S ∈ obC. |D structure. Let Grp be the smallest category with fi- (cGem) D is cogenerator by effective mono iff the canonical mor- nite products (let 1 be the terminal object), an object G, phism lim X →X0 is an isomorphism for all X ∈ obC, −→X/D and morphisms p : G×G→G, i : G→G and 1 : 1→G subject iff h∨ is fully faithful. |D to the following conditions: g(idG ×g) = g(g ×idG) (associativ- ∼ ∼ ∼ ∼ In particular we have the following implications: ity), g(1×idG)=idG=g(idG ×1) (identity), g(i×idG)=1=g(idG × (cGm) ⇐ (cGem) ⇒ (cG) ⇒ (cGe) ⇒ (cGee) i) (inverses) and eventually gsc = g (commutativity). and the following conditioned implications: Then GrpC is the category of C-groups; a C-group is an object X ∈ obC with a functorial group structure on X(S) for any (cGm) ⇒ (cGem) if in C the mono families are effective; S ∈ obC. (cGee) ⇒ (cGe) if in C the epimorphisms are effective; Structure. Let An be the smallest category with finite (cG) ⇒ (cGm) if in C the cokernels are representable; products (let 1 be the terminal object), an object G, and mor- (cGe) ⇒ (cG) if in C the amalgamed sums are representable; phisms s : G×G→G, p : G×G→G, o : G→G, 0 : 1→G and 1 : 1→G subject to the following conditions: s,o,0 gives a commu- (cGe) ⇒ (cGem) if every family of morphisms X →Xi factorizes tative group structure, p,1 gives a (commutative) monoid struc- by an epi followed by effective monomorphisms X →Y →Xi; ture, and p(idG ×s) = s(p×p)(idG ×sc×idG)(∆×idG×G), where (cGee) ⇒ (cGm) if every family of morphism X →Xi factorizes ∆ is the diagonal immersion of G. by an effective epi followed by monomorphisms X →Y →Xi. Then AnC is the category of C-rings; a C-ring is an object X ∈ In particular all the notions of cogenerations coincide in the obC with a functorial ring structure on X(S) for any S ∈ obC. following two cases: Left Structure. Let Mod the smallest category with fi- (i) cokernel and amalgamed sums are representable, all epi- nite products containing obAnt1 obGrp, and morphisms morAnt morphisms are effective and all mono families are effective; morGrpt{a : A×G→G}, subject to the following conditions: a(1×id )=∼id =∼a(id ×1), a(id ×a) = a(p×id ), a(id ×s) = (ii) every family of morphism X →Xi factorizes by an epi fol- G G G A G A s(a×a)(id ×sc×id )(∆ ×id ). lowed by an monomorphisms X →Y →Xi, all epimorphisms A G A G×G are effective and all mono families are effective. Then ModC is the category of C-modules over a C-rings; a C- Injective and Projective Objects. An object I of C is module is the data of a C-ring A and of an object X ∈ obC with injective if for any monomorphism X →Y the induced map a functorial module structure on X(S) over the ring A(S) for C(Y,I)→C(X,I) is surjective. any S ∈ obC. Remark that arbitrary product of injective objects is injective, and the converse holds if the category admits a zero object. Functoriality of Functors. An object P of C is projective if for any epimorphism X →Y Let C, C0 and D be categories and u : C→C0 a functor. Then the induced map C(P,X)→C(P,Y ) is surjective. Remark that by composition we have a natural functor for any X ∈ obC the functor h∨(X) is a of C∨. u∗ = Cat(u,D): Cat(C0o,D)−→Cat(Co,D); Remark that arbitrary sums of projective objects is projective, remark that u∗ commutes with projective and inductive limits and the converse holds if if the category admits a zero object. (in particular it is exact). Algebraic Structures on Categories. A (finite) algebraic If C is U-small and D admits inductive limits, then u∗ ad- structure is a small reduced category D with (finite) projective o mits a right adjoint u! i.e. for any F ∈ obCat(C ,D) and G ∈ limits. 0o ∗ obCat(C ,D) we have an isomorphism Cat(u!F,G)←Cat(F,u G); An algebraic structure (of type D) on C (category with projec- the functor is defined by the position (u!F )(Y ) = lim−→Y/u(C) F pY tive limits) is a left D→C. We define the category where Y/u(C) is the category given by the pairs (X,m) with of D-algebraic objects of C as DC = LEx(D,C) full subcategory X ∈ obC and m : Y →u(X), and the functor pY : Y/u(C)→C of Cat(D,C) of left exact functors. sends (X,m) to X. For any S ∈ C the composition with h∧(S) gives a D-algebraic Dually, if C is U-small and D admits projective limits, then u∗ o o structure with value in USet, i.e. to a bifunctor D ×C →USet. admits a left adjoint u∗, i.e. for any F ∈ obCat(C ,D) and G ∈ 0o ∗ The data of an object D→C in DC is equivalent to the data of a obCat(C ,D) we have an isomorphism Cat(u F,G)←Cat(F,u∗G); map obD→obC (the support of the D-algebraic structure) and the functor is defined by the position (u∗F )(Y ) = lim←−u(C)/Y F pY of a functor D→C∨ extending the map obD→obC→obC∧ in where u(C)/Y is the category given by the pairs (X,m) with the obvious sense (i.e. the data for any S ∈ obC of a D-algebraic X ∈ obC and m : u(X)→Y , and the functor pY : Y/u(C)→C structure in USet functorially in S). sends (X,m) to X.

6 GeoSupRC - Categories and Functors c 2000 by MC ∗ In particular we have the sequence of adjoint functors u!,u ,u∗ In the category USet, the filtrant inductive limits (i.e. induc- (every functor is left adjoint of the functor in its right). tive limits indexed on filtrant categories) are exact, i.e. com- mutes with finite projective limits (in particular with fibred Suppose: C admits finite projective limits, D admits finite pro- product). jective and arbitrary inductive limits and filtred inductive lim- its are exact (i.e. commute with finite projective limits), u is A filtrant category I is essentially small if it admits a small 0 left exact. Then finite projective limits are representable in full subcategory I which is cofinal, i.e. such that for any func- 0 Cat(Co,D) and Cat(C0o,D), and the functor u is left exact. tor F : I→C the inclusion i : I →I induced an isomorphism ! ∨ −→limI0 F ◦i←−→limI F in C ; or equivalently such that for any G : Let now D = USet. then we have a sequence of three adjoint o 0 ∨ ∨ 0∨ ∨ ∨ 0∨ I →C the inclusion i : I →I induced an isomorphism lim←−I0 G◦ functors ∨u,u ,u∨ where u : C →C and ∨u,u∨ : C →C . ∧ i←←−limI G in C . Suppose C a small U category; then we have the following prop- Remark that for any essentially small filtrant category I there erties: exists a small filtrant ordered set E with a cofinal functor (i) u∨ : C0∨ →C∨ commutes with inductive and projective lim- (E,6)→I. its; Reverse Inductive Limits. Let F : I→C with I a small fil- −→ trant category; we define the functor lim F : Co →Set (i.e. in (ii) u : C∨ →C0∨ commutes with projective limits; I ∨ ∨ −→ ∨ ∨ C ) by limI F = lim−→I h (F ) = lim−→i∈I h (F i) (inductive limit in (iii) u : C∨ →C0∨ commutes with injective limits; it can be de- −→ ∨ the category C∨), i.e. (lim F )(X) = lim C(X,F ) = lim h∧(X)F . fined in such a way that the diagram I −→I −→I ∨ ∼−→ It is representable if there exists L ∈ obC such that h (L)=limI F , u 0 ∼ C −−→ C i.e. for any W ∈ obC we have C(W,L)=−→limI C(W,F ), bijec- ∨   0∨ h y y h tion realized by the universal property of (the class of) a mor- phism f : L→F (i ): for any u : W →F (i) there exists a unique C∨ −−→ C0∨ 0 ∨u ϕ : W →L such that the classes of fϕ and u coincide, i.e. such s0 s that there exists i0 →k←i with F (s0)fϕ = F (s)u. 0∨ ∨ ∨ commutes, i.e. h u = ∨uh ; for any F ∈ C we have that ∼ The representative L is characterized by the following proper- = 0∨ ∨u(F )→−→limC/F h u; if the finite projective limits exist in ties: C and u is left exact, then ∨u is left exact; (1) for any i ∈ obI there exists ρi : F (i)→L; ∨ (2) there exists i ∈ obI and a morphism f : L→F (i ); (iv) u is fully faithful iff ∨u is fully faithful (iff idC∨ →u ∨u is 0 0 ∨ iso) iff u∨ is fully faithful (iff u u∨ →idC∨ is iso). such that s0 s Let C and C0 be two small U-categories, and v : C0 →C, u : C→C0 (a) for any i there exists i0 →k←i such that F (s0)fρi = F (s); ∨ adjoint functors; then we have canonical isomorphism v →∨u (b) ρi0 f = idL; ∨ and v∨ →u compatible with the adjonction; therefore if u ad- (c) for any s : i→j we have ρi = ρjF (s). mits a left adjoint, then ∨u commutes with projective limits; we In fact the bijections C(W,L)→lim C(W,F ) are realized by ∨∼ ∼ ∨ −→I have a sequence of four adjoint functors ∨v,v =∨u,v∨=u ,u∨. sending ϕ to the class of fϕ with inverse sending the class of fi to ρifi. −→ Ind-Categories. Any functor T : C→C0 preserve the representative L of lim F , −→ I i.e. T (L) is a representative of limI T ◦F . −→ (Pseudo)-Filtrant Categories. A category I is pseudo- If lim F is represented by L, then also lim F is represented filtrant if the following conditions hold: I −→I by L, using the bijection C(L,W )→←−lim C(F,W ) sending ϕ to 0 0 I (PF 1) any diagram j←i→j can be completed with j→k←j to the sequence (ϕρi), and inverse sending (fi) to fi0 f. form a commutative square; ∨ −→ ∼ Remark that C (limI F,H)=←−limI H ◦F and that for F : I→C (PF 2) any diagram i−→j can be completed to i−→j−→k commu- and G : J →C we have −→ −→ −→ −→ tative (any two parallel morphisms can be equalized). C∨(limF,limG)=∼limlimC(F,G) . I J ←−−→ A non empty, pseudo-filtrant category is filtrant if it is con- I J nected (i.e. any two objects can be connected by a sequence of Ind-Objects. We define the category of Ind-object of C equiv- morphisms, independently of the directions). alently as: (i) the full subcategory IndC of C∨ whose objects are the func- Remarks. tors isomorphic to filtrant inductive limits of representable (i) under the condition (PF 1), a category is connected iff the functors; 0 following condition (C ) holds: for any i,j ∈ obI there ex- (ii) the category Ind(C) whose objects are the filtrant induc- ists k ∈ obI and a diagram i→k←j; tive systems, i.e. the functors F : I→C from a small fil- (ii)(PF 2) and (C0) imply (PF 1); trant category, and morphisms defined by Ind(C)(F,G) = lim lim C(F i,Gj). (iii) in particular, I is filtrant iff it is non empty, (PF 2) and ←−i∈I −→j∈J 0 The equivalence Ind(C)→IndC ⊆ C∨ between the two categories (C ) hold. −→ is defined by sending a functor F : I→C to lim F . Remark that a category with amalgamed sums and cokernels is I Then we have that the functor h∨ : C→C∨ extends to a fully pseudo-filtrant, and a category with finite sums and cokernels faithful functor h∨ : Ind(C)→C∨ making commutative the dia- is filtrant. gram Let X : I→USet an inductive system in USet indexed by a i filtrant set (category?); then the inductive limit lim−→i∈I X(i) is C −−→ Ind(C) ` ∨   ∨ defined by i∈obI X(i)/ ∼ where ∼ is the equivalent relation h y y h ∨ given by: ai ∼ aj for ai ∈ X(i) and aj ∈ X(j) iff there exists IndC −−→ C i k > i,j such that X(i,k)(ai) = X(j,k)(aj) in X(k).

7 GeoSupRC - Categories and Functors c 2000 by MC Remark that in general the canonical functors Cat(I,C)→Ind(C) (b0) for any X0 ∈ obC0 the functor in C∨ sending X to C(fX,X0) are neither full nor faithful. is ind-representable; Suppose that C admits the filtrant inductive limits; then the (c) there exists a functor G : IndC0 →IndC such that we have a ∼ bifunctorial isomorphism IndC(X,GZ0)∼IndC0(FX,Z0) for canonical bijection C(lim−→I F,W )=Ind(C)(F,W ) show that−→ limI = is the left adjoint of the canonical inclusion C→Ind(C). More- any X ∈ obC and Z0 ∈ obIndC0; over the following conditions are equivalent: 0 0 (c ) there exists a functor G0 : C →IndC such that we have a 0 ∼ 0 0 (a) the canonical functor C→Ind(C) commutes with the filtrant bifunctorial isomorphism IndC(X,G0X )=C (FX,X ) for inductive limits; any X ∈ obC and X0 ∈ obC0; (b) for any X ∈ obC the functor h∧(X) ∈ C∧ commutes with (d) the functor Ind(F ) : Ind(C)→Ind(C0) admits a right ad- the filtrant inductive limits; joint; (e) if C is equivalent to a small category: F is right exact. (c) the functor lim−→ : Ind(C)→C is fully faithful (and so an equiv- alence of categories). Remark that if F admits a right adjoint G0 : C0 →C, then it ad- 0 In particular we have Ind(Ind(C))=∼Ind(C). mits an ind-adjoint which is canonically isomorphic to Ind(G ). Strict Ind-objects. An ind-object ϕ : I→C is strict if I is (the Presentation of Ind-objects. category associated to) a small ordred set, and one of the fol- lowing equivalent condition holds: Pro-Categories. −→ (i) the canonical morphisms ϕ(i)→limI ϕ are monomorphisms Reverse Projective Limits. Let F : Io →C with I a small in C∨; ←− filtrant category; we define the functor limI F : C→Set (i.e. in (ii) for any i j the transition morphism ϕ(i)→ϕ(j) is a monomor- ∧ ←− ∧ ∧ 6 C ) by limI F = lim−→I h (F ) = lim−→i∈I h (F i) (inductive limit in phism in C. ∧ ←− ∨ the category C ), i.e. (limI F )(X) = lim−→I C(F,X) = lim−→I h (X)F . It is essentially strict if it is isomorphic in Ind(C) to a strict ∧ ∼←− It is representable if there exists M ∈ obC such that h (M)=limI F , one. ∼ i.e. for any W ∈ obC we have C(M,W )=−→limI C(F,W ), bijec- Ind-Representability. A functor F ∈ C∨ is ind-representable tion realized by the universal property of (the class of) a mor- ∨ if it is in the essential image of the inclusion IndC→C , i.e. phism f : F (i0)→M: for any u : F (i)→W there exists a unique if it is isomorphic to an inductive limit in C∨ of representable ϕ : M →W such that the classes of ϕf and u coincide, i.e. such functors. s0 s that there exists i0 →k←i with ϕfF (s0) = uF (s). An ind-representable functor F is left exact, i.e. the canonical The representative M is characterized by the following proper- morphism F (lim−→I ϕ)→←−limI F ϕ is an isomorphism for all finite ties: projective system ϕ : I→C. (1) for any i ∈ obI there exists σi : M →F (i); Criterion of ind-represantability. The following conditions are equivalent: (2) there exists i0 ∈ obI and a morphism f : F (i0)→M; (a) F is ind-representable; such that s0 s (b) the category C/F is essentially small and filtrant; (a) for any i there exists i0 →k←i such that σifF (s0) = F (s); (b) fσ = id ; (b0) if the category C is equivalent to a small category: C/F is i0 M filtrant; (c) for any s : i→j we have σi = F (s)σj.

(c) if in C the finite inductive limits are representable: F is a In fact the bijections C(M,W )→−→limI C(F,W ) are realized by left exact functor and C/F is essentially small; sending ϕ to the class of ϕf with inverse sending the class of f to f σ . (c0) if the category C is equivalent to a small category and in i i i ←− C the finite inductive limits are representable: F is a left Any functor T : C→C0 preserve the representative M of lim F , ←− I exact functor. i.e. T (L) is a representative of limI T ◦F . Remark that if C has finite inductive limits, then F left exact ←− If limI F is represented by M, then also←− limI F is represented implies that C/F has finite inductive limits also, in particular by M, using the bijection C(W,M)→←−limI C(W,F ) sending ϕ to it is filtrant. the sequence (σiϕ), and inverse sending (fi) to ffi0 . For F ∈ C∨, let Sub(F ) be the full subcategory of C/F given by ∧ ←− ∼ o Remark that C (limI F,H)=←−limI H ◦F and that for F : I →C the injective morphisms (i.e. the representable sub-functors of and G : J o →C we have F ). Then the following are equivalent: ∧ ←− ←− ∼ C (limF,limG)=←−lim−→limC(G,F ) . (i) F is strictly ind-representable (i.e. ind-representable by a I J I J strict ind-object); Pro-Objects. We define the category of Pro-object of C equiv- (ii) the category Sub(F ) is filtrant, essentially small and cofi- alently as: nale in C/F . (i) the full subcategory ProC of C∧ whose objects are the func- 0 ∨ 0∨ ∨ Ind-Adjoints. Consider F : C→C a functor, and F : C →C tors isomorphic to filtrant inductive limits of representable the canonical inverse image; we say that F admits an ind- functors; adjoint if one of the following equivalent conditions are sat- isfied: (ii) the category Pro(C) whose objects are the filtrant projec- tive systems, i.e. the functors F : Io →C from a small fil- ∨ 0 (a) F sends IndC in IndC; trant category, and morphisms defined by Pro(C)(F,G) = 0 ∨ 0 (a ) F sends C in IndC; ←−limi∈I −→limj∈J C(Gj,F i). (b) for any Z0 ∈ obInd(C0) the functor in C∨ sending X to The equivalence Pro(C)o →ProC ⊆ C∧ between the two cate- 0 0 0 o ←− Ind(C )(FX,Z ) = lim−→C(fX,Z ) is ind-representable; gories is defined by sending a functor F : I →C to limI F .

8 GeoSupRC - Categories and Functors c 2000 by MC Then we have that the functor h∧ : Co →C∧ extends to a fully Pro-Adjoints. Consider F : C→C0 a functor, and F ∧ : C0∧ →C∧ faithful functor h∧ : Pro(C)o →C∧ making commutative the di- the canonical inverse image; we say that F admits a pro-adjoint agram if one of the following equivalent conditions are satisfied: ∧ 0 io (a) F sends ProC in ProC; Co −−→ Pro(C)o 0 ∧ 0 ∧   ∧ (a ) F sends C in ProC; h y y h 0 0 ∧ ProC −−→ C∧ (b) for any Z ∈ obPro(C ) the functor in C sending X to i 0 0 0 Ind(C )(Z ,FX) = lim−→C(Z ,fX) is pro-representable; 0 0 0 ∧ 0 Remark that in general the canonical functors Cat(Io,C)→Pro(C) (b ) for any X ∈ obC the functor in C sending X to C(X ,fX) are neither full nor faithful. is pro-representable; 0 Suppose that C admets the filtrant projective limits; then the (c) there exists a functor G : ProC →ProC such that we have 0 ∼ 0 0 canonical bijection C(W,lim F )=∼Pro(C)(W,F ) show that lim a bifunctorial isomorphism ProC(GZ ,X)=ProC (Z ,FX) ←−I ←−I 0 0 is the right adjoint of the canonical inclusion C→Pro(C). More- for any X ∈ obC and Z ∈ obIndC ; over the following conditions are equivalent: 0 0 (c ) there exists a functor G0 : C →ProC such that we have a 0 ∼ 0 0 (a) the canonical functor C→Pro(C) commutes with the fil- bifunctorial isomorphism ProC(G0X ,X)=C (X ,FX) for trant inductive limits; any X ∈ obC and X0 ∈ obC0; (b) for any X ∈ obC the functor h∨(X) ∈ C∨ commutes with (d) the functor Pro(F ) : Pro(C)→Pro(C0) admits a left adjoint; the filtrant inductive limits; (e) if C is equivalent to a small category: F is left exact. (c) the functor lim←− : Pro(C)→C is fully faithful (and so an equiv- Remark that if F admits a left adjoint G0 : C0 →C, then it admits alence of categories). a pro-adjoint which is canonically isomorphic to Pro(G0). ∼ In particular we have Pro(Pro(C))=Pro(C). Presentation of Pro-objects. o Strict Pro-objects. A pro-object ϕ : I →C is strict if I is (the Generalized Adjonctions. We say that F : C→IndC0 and category associated to) a small ordered set, and one of the G : C0 →ProC are adjoint if there is a bifunctorial isomorphism following equivalent condition holds: IndC0(X0,FX)=∼ProC(GX,X0) for any X ∈ obC and X0 ∈ obC. ←− (i) the canonical morphisms limI ϕ→ϕ(i) are epimorphisms Intersection of Ind and Pro. The canonical square in C∧; C −−→ Pro(C) (ii) for any i 6 j the transition morphism ϕ(j)→ϕ(i) is an epi-   morphism in C. y y It is essentially strict if it is isomorphic in Pro(C) to a strict Ind(C) −−→ Pro(Ind(C)) one. is cartesian in the following sense, that an object of Pro(Ind(C)) Constant Pro-objects. which is in the image either of Pro(C) and of Ind(C) is really in Pro-Representability. A functor F ∈ C∧ is pro-representable C; i.e. Pro(C)∩Ind(C) in Pro(Ind(C)) is just C. if it is in the essential image of the inclusion ProC→C∧, i.e. if it is isomorphic to an inductive limit in C∧ of representable 2-Categories and 2-Functors. functors. A pro-representable functor F is left exact, i.e. the canonical 2-Categories. morphism F (lim←−I ϕ)→←−limI F ϕ is an isomorphism for all finite 2-Functors. projective system ϕ : I→C. Criterion of pro-represantability. The following conditions are Fibred and Cofibred Categories. equivalent: Relative Categories. Let E be a category; we define the (a) F is pro-representable; category Cat/E whose objects are the functors p : F →E, and (b) the category C/F is essentially small and filtrant; morphisms the commutative triangles of functors. More gener- ally we have a natural structure of 2-category in Cat/E, using (b0) if the category C is equivalent to a small category: C/F is the 2-commutative triangles as morphisms, and the commuta- filtrant; tive triangles of morphisms of functors as 2-morphisms. (c) if in C the finite projective limits are representable: F is a Let F ∈ ob(Cat/E)(p : F →E understood), α : T →S a morphisms left exact functor and C/F is essentially small; in E; we say that a morphism v : X →Y in F is above α if 0 (c ) if the category C is equivalent to a small category and in p(v) = α. C the finite projective limits are representable: F is a left Let F,G : F →G be functors of E-categories (they form commu- exact functor. tative triangles with the canonical morphisms); a morphism Remark that if C has finite projective limits, then F left exact u : F →G is an E-morphism of E-functors if for any X ∈ obF, implies that C/F has finite inductive limits also, in particular u(X) is an idp(X)-morphism (i.e. q(u(X)) = idp(X), being it is filtrant. p(X) = qF (X) = qG(X)). ∧ For F ∈ C , let Sub(F ) be the full subcategory of C/F given by We define the category E(F,G) having objects the E-functors the injective morphisms (i.e. the representable sub-functors of from F to G, and morphisms the E-morphisms of E-functors. It F ). Then the following are equivalent: is given by ker(Cat(F,G)−→Cat(F,E)) where the two functors (i) F is strictly pro-representable (i.e. pro-representable by a send F to p (constant) and qF . strict pro-object); The usual composition functor induces the composition E(F,G)× (ii) the category Sub(F ) is filtrant, essentially small and cofi- E(G,H)→E(F,H) giving a functor (Cat/E)o ×Cat/E →Cat send- nale in C/F . ing (F,G) to E(F,G) ⊆ (Cat/E)(F,G).

9 GeoSupRC - Categories and Functors c 2000 by MC Products of Relative Categories. Given two categories F, G Remark that: any E-equivalence is cartesian, the cartesian- in Cat/E, we define the fiber product F ×E G having objects ity of functor is invariant under isomorphism, the composi- obF ×obE obG and morphisms morF ×morE morG. Then we tion of cartesian functors is cartesian. Moreover, if F →G is have for any H ∈ Cat/E a canonical isomorphism of categories an E-equivalence, then the induced functors Ecart(G,H)→E- ∼ E(H,F ×E G)=E(H,F)×E(H,G). cart(F,H) and Ecart(H,F)→Ecart(H,G) are equivalences for If λ : E0 →E is a functor, then we have the base change functor any H ∈ ob(Cat/E). ∗ 0 0 0 λ : Cat/E →Cat/E sending F to F = F ×E E ; it is the adjoint We define the 2-category Cart/E of cartesian categories over E of the obvious restriction functor sending q : G→E0 to λq : G→E. having objects the objects of Cat/E, and morphisms the cate- The base change functor commutes with projective limits and gories Ecart(F,G); it is a sub-2-category of Cat/E? induces an isomorphism E0(F0,G0)→E(F0,G). The cartesian sections of F over E form a category←−− LimF/E = In particular, for F = E we put Γ (G/E) = E(E,G) (sections of G Ecart(E,F); it gives a functor Cart/E →Cat. over E). Prefibred and Fibred Categories. A category F over E Remark that for any category H we have that the canonical is prefibred over E if for any f : T →S in E the inverse image ∗ functor Cat(H,E(F,G))→E(F ×H,G) is an isomorphism. functor f : FS →FT exists; it is fibred over E if moreover the Fiber Categories. Let F ∈ ob(Cat/E), and S ∈ obE; we define class of cartesian morphisms is stable under composition (the composition of any two cartesian morphisms is cartesian). the fiber category of F over S as FS = F ×E {S,idS}, i.e. the objects are the objects of F over S, and the morphisms the A subcategory of a (pre)fibred category is say to be a sub- morphisms of F over the identity of S. (pre)fibred category if it is (pre)fibred category and the inclu- We say that an E-functor F : F →G is an E-equivalence if one sion functor is cartesian. of the following equivalent conditions is satisfied: A prefibred category F is fibred iff the following condition hold: ∼ ∼ for any α : η→ξ in F cartesian over f : T →S, any g : U →T and (i) there exists an E-functor G : G→F with FG=idG and GF =idF which are E-(iso)morphisms; any ζ ∈ obFU the map Fg(ζ,η)→Ffg(ζ,ξ) sending u to αu is 0 0 0 0 0 an isomorphism. (ii) for every E →E, the functor F = F ×E E : F →G is an equivalence of categories; In particular, if α is an isomorphism, then f = p(α) is an iso- morphism and α is cartesian; the converse is true if F is fibred (iii) F is an equivalence and for any S ∈ obE the induced func- over E. tors FS : FS →GS are equivalences of categories; Moreover, if α : η→ξ and β : ζ→η are morphisms of F fibred (iii0) F is fully faithful and for any S ∈ obE and any η ∈ obG S over E with α cartesian, then β is cartesian iff αβ is. there exists ξ ∈ obFS with an S-isomorphism u : F (ξ)→η. Remark that if F : F →G is an E-equivalence then for any H ∈ Let F a fibred category over E; define Fe the subcategory of Cat/E, the canonical functors E(G,H)→E(F,H) and E(H,F)→E(H,G) F having the same objects, and as morphisms the cartesian are E-equivalences. morphisms of F (in particular the morphisms of FeS are the The functor p : F →E is transportable if for any isomorphism isomorphisms of FS). Then Fe is a fibred category over E, since in the bijection F (η0,η)→F (η0,ξ), for an f-morphism α : T →S in E, and any ξ ∈ obFT there exists an isomorphism T f u : ξ→ξ0 with p(u) = α. In that case, every E-functor F →G α in F, the T -isomorphisms correspond to the cartesian mor- which is an equivalence of categories is an E-equivalence. phisms). In particular the cartesian sections E →F correspond to arbitrary functors E →F, but the canonical faithful func- Cartesian Morphisms and Functors. Let p : F →E, α : e η→ξ be a morphism of F, S = pξ, T = pη, f = pα. We say tor E(E,Fe)→←−−LimF/E = Cart/E(E,F) is not full, so it is not an that α is a cartesian morphism (of F w.r.t. E) if one of the isomorphism. following equivalent conditions holds: For a category F over E the following facts are equivalent: 0 0 (i) for any η ∈ obFT and any f-morphism u : η →ξ there ex- (i) every morphism of F is cartesian; 0 ists a unique T -morphismu ¯ : η →η such that u = αu¯; (ii) F is a category fibred in gruppoids: F is fibred and every 0 (ii) for any η ∈ obFT the composition v to αv gives a bijection fiber FS is a gruppoid (i.e. every morphism in FS is an 0 0 FT (η ,η)→Ff (η ,ξ); isomorphism); o 0 0 (iii) if E is a gruppoid: the category F is a gruppoid and the (iii)(η,α) represents the functor FT →Set sending η to Ff (η ,ξ). functor F →E is transportable. If for f : T →S in E and ξ ∈ obFS there exists a cartesian mor- phism α : η→ξ, then η is determined in obFT up to a unique Remark that if F →G is an E-equivalence, then F is (pre-)fibred isomorphism; we say that η is the inverse image of ξ by f, and iff G is. ∗ ∗ write η = fF (ξ) = f (ξ) = ξ ×S T . Moreover, if F and G are prefibred categories over E and F : If for all ξ ∈ obFS there exists the inverse image by f, then we F →G is a cartesian E-functor, then F is faithful (resp. fully ∗ call inverse image by f the functor f : FS →FT defined up to faithful, E-equivalence) iff for any S ∈ obE the induced functor isomorphisms. FS : FS →GS is faithful (resp. fully faithful, E-equivalence). α ξ ← η Let F = F1 ×E F2 the product of two E categories; then a mor- λ↓ ↓µ 0 phism α = (α ,α ) of F is cartesian iff the two morphisms α , If we have a commutative square 0 0 in F, with α, α 1 2 1 ξ ←η α are cartesian. So an E-functor F = (F ,F ): G→F is carte- α0 2 1 2 f-morphisms, λ an isomorphism, µ a T -morphism, then α is sian iff the two E-functors F1, F2 are. Therefore we have canon- cartesian iff α0 is. ical isomorphisms Cart/E(G,F)→Cart/E(G,F1)×Cart/E(G,F2) and LimF/E →LimF /E ×LimF /E. An E-functor f : F →G is cartesian if it sends cartesian mor- ←−− ←−− 1 ←−− 2 phisms of F to cartesian morphisms of G; we indicate with Let F in Cat/E, and λ : E0 →E be a functor. We define F0 = 0 0 0 0 Ecart(F,G) the full subcategory of E(F,G) whose objects are F ×E E ∈ Cat/E . Then a morphism α of F is cartesian iff its the cartesian functors. Remark that any morphism of E is first projection α is a cartesiam morphism of F. So if a func- cartesian for the E-structure of E given by the identity. tor F : F →G is cartesian over E then the functor F 0 : F0 →G0

10 GeoSupRC - Categories and Functors c 2000 by MC is cartesian over E0. In particualr we have a canonical mor- (a) ϕ = id for any S; idS FS phism Cart/E(F,G)→Cart/E0(F0,G0) and in the isomorphism ∗ (b) ϕ ◦(F •c ) = ϕ ◦(ϕg •f ) for any g : U →T in E; E0(F0,G0)→E(F0,G) the cartesian functors over E0 correspond fg U f,g f to the functors which sends morphism whose first projection the morphisms are the data for any S ∈ obE of a morphism is cartesian to cartesian morphisms. Moreover we have that uS : FS →GS subject to 0 0 0 ∗ ←−−LimF /E is equivalent to the full subcategory of E(E ,F) of (c) uS ◦ϕf = ψf ◦(uT •f ) for any f ∈ morE. the E-functors sending all morphisms to cartesian morphisms. Then the canonical functor Cat(F,C)→H(F,C) sending a func- 0 0∼ 0 In particular, if F is fibred, we have obLim←−−F /E =E(E ,Fe) (re- tor F to the data F = F i and ϕ = F •α , and a morphism 0 0 o S S f f mark that Lim←−−(F ×E E )/E is a functor (Cat/E) ×(Cart/E)→Cat u : F →G to the data uS = u•iS, is an isomorphism of cate- on E0 and F). gories. If F is (pre-)fibred over E, then F0 is (pre-)fibred over E0. If G is another category over E with a normalized clivage, then Clivages. A clivage of a category F over E is the choice for the functors E(F,G) corresponds to the data for any S ∈ obE any f : T →S morphism of E of an inverse image functor f∗ : of a functor FS : FS →GS and for any f ∈ morE, f : T →S, of a morphism ϕ : F f∗ →f∗F , subject to the following condi- F →F ; the clivage is normalized if id∗ = id for any S ∈ f T F G S S T S FS obE. tions: (a) ϕ = id for any S; Remark that F admits a clivage iff it is prefibred, and in that idS FS case it admits a normalized clivage. (b) ϕ ◦(F •cF ) = (cG •F )◦(g∗ •ϕ )◦(ϕ •f∗ ) for any fg U f,g f,g S G f g F The class of (resp. normalized) clivages is in bijection with the g : U →T in E; class of K ⊆ morF such that: every α ∈ K is cartesian, for any moreover the cartesian functors corresponds to the data with morphism f : T →S in E and any objects ξ ∈ obF there exists S ϕ isomorphisms. a unique f-morphism of target ξ in K (resp. and for any object S Scindages. If we have a (1-)functor F : Eo →Cat, then we have ξ ∈ obF, idξ ∈ K). The elements in K are called the transport morphisms for the given clivage. also a 2-functor with cf,g = id(fg)∗ , hence a (fibred) category F over E with a clivage which is called a scindage. A clivage The functorial morphisms α (ξ): f∗(ξ)→ξ, i.e. α : i f∗ →i f f T S of a category over E is a scindage iff it is defined by a functor, if iS : FS →F, and the universal properties gives rise for any ∗ ∗ ∗ iff (fg) = g f , iff cf,g = id ∗ , iff the composition of two g f ∗ ∗ ∗ (fg) composition U →T →S to a canonical morphism cf,g : g f →(fg) transport morphism is again a transport morphism. such that for any ξ ∈ obF we have α (ξ)c (ξ) = α (ξ)α (f∗(ξ)). S fg f,g f g We have a canonical equivalence between categories over E with Remark that a category over E with a clivage is a fibred cat- a scindage and the category Cat(Eo,Cat). egory iff for any f,g ∈ morE the morphisms c are isomor- f,g Remark that a fibred category over E does not admits neces- phisms. sarily a scindage; but if F is fibred and has rigid fibers (i.e. for The data of a prefibred category F over E with a normalized any S we have FS(ξ,ξ) reduced to the identity) then F admits clivage (resp. a clivage) is equivalent to the data of a normalized a scindage (every isomorphism between functors from A to C o 2-functor (resp. a 2-functor) E →Cat, i.e. the data of: is the identity if C is rigid and reduced). (i) a map obE →obCat (given by S to FS); Limits of Relative Categories. Let F be a category over E, ∗ C be the class of cartesian morphisms of F; then the functors (ii) a map morE →morCat (given by f : T →S to f : FS →FT ); Cat→USet sending C to obCat (F,C) (functors F →C which (iii) a map morE × morE →2morCat (given by (f,g) to c : C obE f,g sends C to the isomorphisms of C) and Cart/E(F,C ×E) are g∗f∗ →(fg)∗); canonically isomorphic (the cartesian morphisms of C ×E are (iv) (resp. a map obE →2morCat sending S to α : id∗ →id ); S S FS precisely the morphisms (m,f) with m an isomorphism of C) subject to the following conditions: and representable. If F is fibred over E, define Lim F = F = F(C−1) be the h g f ∗ ∗ −−→Eo C (i) for V →U →T →S we have cf,gh ◦(cg,h •f ) = cfg,h ◦(h • representative. It is in fact the inductive limit in the sense ∗ cf,g), i.e. for any ξ ∈ obFS we have cf,gh(ξ)cg,h(f ξ) = of 2-functors of the representation of F as Eo →Cat; for any ∗ cfg,h(ξ)h (cf,g(ξ)); S ∈ obE we have a canonical uS : FS →F →−−→LimEo F and for ∗ (ii) for f we have c = id ∗ and c = id ∗ (resp. c = any f : T →S we have a commutative triangle u f = u , with f,idT f idS ,f f f,idT T S α •f∗ and c = f∗ •α , i.e. for any ξ ∈ obF c (ξ) = the universal property. T idS ,f S S f,idT α (f∗(ξ)) and c (ξ) = f∗(α (ξ))). o T idS ,f S Remark that even if we have a scindage F : E →Cat, the canon- o Given a normalized 2-functor F : E →Cat we can reconstruct ical morphism Lim−−→Eo F →−→limEo F is not an equivalence; it is an ` equivalence if Eo is pseudofiltrant. the prefibred category F: the objects are obF = S∈obE obF (S); for ¯ξ = (S,ξ),η ¯ = (T,η) in obF the morphisms are defined by If E has the following properties: F(¯η,¯ξ) = ` h (¯η,¯ξ); where h (¯η,¯ξ) = F (T )(η,f∗(ξ)); f∈E(T,S) f f (L1) any X →Z←Y can be completed in a commutative square ¯ ¯ ¯ ¯ for g : U →T , ζ ∈ obF (U), the composition hf (¯η,ξ)×hg(ζ,η¯)→hfg(ζ,ξ) by X ←W →Y ; is defined by sending (u,v) to uv = c (ξ)g∗(u)v; the canonical f,g (L2) for any commutative X −→Y −→Z there exists W −→X −→Y forget gives a functor p : F →E for which we have isomorphisms −→ −→ commutative; of categories F (S)→FS for any S ∈ obE. then the class C of cartesian morphism has the following prop- Functors of normalized clivage. Let F be a category over E erties with a normalized clivage; for S ∈ obE we have iS : FS →F (F 0) the isomorphisms belongs to C; the inclusion of each fiber, and for any f ∈ morE we have αf : ∗ iT f →iS. Define H(F,C) the following category: the objects (F 1) C is stable by composition; are the data for any S ∈ obE of a functor FS : FS →C and for c any f ∈ morE, f : T →S, of a morphism ϕ : F f∗ →F , sub- (F 2) any diagram X →Z←Y can be completed to a commuta- f T S c ject to the following conditions: tive square by X ←W →Y ;

11 GeoSupRC - Categories and Functors c 2000 by MC c c (F 3) for any commutative X −→Y −→Z there exists W −→X −→Y then a fibred category over E is the data of two categories, commutative; two bifunctors of morphism, and the composition... ? −1 and the category FC = F(c ) can be represented as obFC = A category over E with a normalized clivage is the data 0 obF and FC (X,Y ) = lim−→X0∈S/X F(X ,Y ), with the usual com- of two categories Fa, Fb and two adjoint functors G : position of localized categories. Fa →Fb, F : Fb →Fa which are equivalence of categories one quasi-inverse of the other (we identify F = f∗ and The canonical functor Q : F →F has the inverse image Q∨ : C G = g∗, the isomorphisms being u = c : FG→id and F ∨ →F ∨ which is fully faithful and injective on the objects, g,f Fa C v = c : GF →id ). ∨ ∨ f,g Fb with the left adjoint Q! : F →FC left exact (if the finite pro- jective limits are representable in F, then also Q is left exact). (iv) the trivial fibration over E for a category C is just the projection F = C ×E →E; it is a fibred and cofibred cat- Let E be a cofiltrant category, and F be a fibred category over egory having canonical scindage and coscindage given by E. Then for any S ∈ obE and F/S = F × (E/S) we have a E Eo →Cat and E →Cat sending any S to C. We have Γ (F/E)=∼Cat(E,C). canonical equivalence−−→ Lim o (F/S)→−−→Lim o F. Moreover (E/S) E (v) the category of morphisms of E is F(E) = Cat(∆1,E) where if E0 →E is cofinal, then the canonical Lim (F × E0)→Lim F −−→E0o E −−→Eo ∆1 = {s→t} (category with two objects and one morphism). is an equivalence. It is a category over E by t : F(E)→E sending a morphism Let F →E be a fibred category. Then the functor Cat→U- to its target. We have F(E)S = E/S. Set sending C to obCart/E(C ×E,F) is rapresentable by the If f : T →S is a morphism of E, then we have a canonical category Lim o F = Cart/E(E,F). ←−−E direct image f∗ : FT →FS by composition; moreover we If we choose a clivage F : Eo →Cat, we have the canonical mor- have a canonical coscindage of F(E) over E, so that a phisms vS : Lim←−−Eo F →FS (evaluation in S) for any S ∈ obE, fortiori it is cofibred over E. ∗ and for any f : T →S a commutative triangle f vS = vT , mak- Remark that the cartesian morphisms in F(E) corresponds ing Lim←−−Eo F the projective limit in the sense of 2-categories. to the cartesian square of E, so that the inverse image Remark that even if we have a scindage, the canonical mor- functor for f exists in F(E) iff E admits the fiber product phism lim←−Eo F →←−−LimEo F is not an equivalence. over f. In particular F(E) is fibred (iff it is prefibred) iff Cofibred and Bifibred Categories. A morphism α : η→ξ the fiber product exists in E. of a category F over E (f = p(α): T →S) is cocartesian iff it Problems. Let F be a category over E with a cli- is cartesian in the opposite category Fo over Eo. This means g1 f vage. A diagram S00 −→S0 −→S in E with fg = fg = h is F- equivalently that: for any ξ0 ∈ F the map F (ξ,ξ0)→F (ηξ0) −→ 1 2 S S f g2 given by the composition v to vα is a bijection; or that the pair exact if for any ξ,η ∈ F the diagram 0 0 S (ξ,α) represents the functor F →USet sending ξ to F (ηξ ). ∗ S f ∗ g We say that (ξ,α) is a direct image of η by f (in F over E). f ∗ ∗ −→1 ∗ ∗ FS(ξ,η)−→FS0 (f ξ,f η)−→FS00 (h ξ,h η) g∗ If (ξ,α) exists for any η ∈ obFT , we say that the direct image 2 F ∗ ∗ ∗ functor for f exists, and we write f∗ (η) = f∗(η) = ξ. In that is exact in USet (remark that g f is identified to h by the case we have an isomorphism of (bi)functors FS(f∗(η),ξ)→Ff (η,ξ) clivage cf,g). o from F ×FS. 0 T Gluing data. For an object ξ ∈ obFS0 a gluing data relative If for a morphism f of E we have inverse and direct images in 0 ∗ 0 ∗ 0 0 0 to (g1,g2) is an isomorphism r(ξ ): g1(ξ )→g2(ξ ). If ξ ,η ∈ F, then they are adjoint functors: 0 0 obFS0 are endowed with gluing data r(ξ ),r(η ), a morphism ∗ 0 0 ∗ 0 FS(f∗(η),ξ)→Ff (η,ξ)←FT (η,f (ξ)) u : ξ →η in FS is compatible with that data if g2(u)r(ξ ) = ∗ 0 ∗ so that if f∗ exists, then f exists iff f∗ admits a right adjoint. r(η )g1(u) If f and g admits direct and inverse images, then the mor- We have then the category of objects of FS0 endowed with ∗ ∗ ∗ phisms cf,g : g f →(fg) and cf,g :(fg)∗ →g∗f∗ are in corre- gluing data rel. to (g1,g2). spondence for the general properties of adjoint functors, and The inverse image by f of an object ξ of FS is canonically one is an isomorphism iff the other is. endowed with a gluing data rel. to (g1,g2), and the inverse o We say that F if (pre-)cofibred over E if F is (pre-)fibred image of a morphism of FS is compatible with the canonical o ∗ over E ; a category is (pre-)bifibred over E if it is (pre-)fibred gluing data. Therefore f gives a canonical functor from FS to and (pre-)cofibred over E. In particular: if F if prefibred and the category of FS0 endowed with gluing data rel. to (g1,g2). precofibred over E then it if fibred iff it is cofibred. The diagram is F-exact iff this functor is fully faithful. Examples. The gluing data r(ξ0) is effective rel. to f if there exists ξ ∈ obF such that ξ0∼f∗ξ. If the diagram is F-exact, then ξ (i) if E is a discrete category, and E = obE, then Cat/E has S = is determined up to a unique isomorphism; in that case, if objects the collections indexed by E of categories; every any gluing data is effective, the inverse image by f gives an category over E is fibred and cofibred, every E-functor equivalence of categories of F with the category of gluing (i.e. collections indexed by E of functors) is cartesian; S ∼Q ∼Q data rel. to (g1,g2). E(F,G)= Cat(Fi,Gi) and Γ (F/E)= Fi. i∈E i∈E 0 0 −→ 0 Descent data. In the case of the diagram S ×S S −→S →S (ii) if E is the category {a→b}, then a category F over E 0 0 0 we have the diagonal morphism ∆ : S →S ×S S , and for ϕ : is the data of two categories Fa, Fb and a bifunctor H : ∗ 0 ∗ 0 o p1ξ →p2ξ to be an effective gluing data the following (cocycle) Fa ×Fb →USet (we identify H(η,ξ) = Ff (η,ξ) = F(η,ξ)). conditions are necessary: It is fibred (iff it is pre-fibred) iff H is representable w.r.t. ∗ ∗ ∗ 0∼ ∗ 0 0 ξ; it is cofibred (iff it is pre-cofibred) iff H is representable (i) ∆ (ϕ) = idξ0 , because ∆ pi ξ =(pi∆) ξ = ξ ; w.r.t. η. ∗ ∗ ∗ 0 0 0 (ii) p1,3(ϕ) = p2,3(ϕ)p1,2(ϕ), where pi,j : S ×S S ×S S →S ×S 0 (iii) if E is a connected rigid gruppoid with two objects (i.e. S given by p1pi,j = pi and p2pi,j = pj. f 0 0 −→ An F-descent data over ξ ∈ obF 0 rel. to f : S →S is a gluing {a←−b} with the two morphisms one inverse of the other), S g data rel. to (p1,p2) satisfying the cocycle conditions.

12 GeoSupRC - Categories and Functors c 2000 by MC A morphism f : S0 →S is an F-descent morphism if the natural 0 0 −→ 0 diagram S ×S S −→S →S is F-exact; it is a strict F-descent 0 morphism if every F-descent data over ξ ∈ obFS0 rel. to f is effective, i.e. if FS is equivalent to the descent data on FS0 . Remark that if f is an F-descent morphism and admits a sec- 0 tion s : S→S with fs = idS, then it is a strict F-descent mor- 0 phism, because for any ξ ∈ obFS0 whit descent data decsends to ξ = s∗(ξ0). If the category E has finite products and fiber products, then a morphism f : S0 →S is a strict descent morphism (i.e. for the category E/S with the natural clivage) iff it is a universal effective epimorphism. Intuitive presentation. For T ∈ obE/S we write t,t0,t00 ∈ S0(T ) = 0 0 0 ∗ 0 FS(T,S ); if ξ ∈ FT write ξt = t (ξ ) ∈ FT . Then the gluing 0 data for ξ rel. to (p1,p2) is the data for any T ∈ obE/S and any 0 0 0 0 t,t ∈ S (T ) of an isomorphism ϕt,t0 : ξt →ξt0 , and the cocycles conditions are 0 (i) ϕt,t = id 0 for any T ∈ obE/S and any t ∈ S (T ); ξt 0 00 0 (ii) ϕt,t00 = ϕt,t0 ϕt0,t00 for any T ∈ obE/S and any t,t ,t ∈ S (T ). 2 In particular we see that (ii) implies (i), for ϕt,t = ϕt,t and ϕt,t is an isomorphism; in fact the condition (ii) makes equivalent the condition (i) and the condition that ϕt,t is an iso.

Copyright c 2000 M. Cailotto, December 2000 v0.0 Dip. di Matematica Pura ed Applicata, Univ. Padova (Italy) Thanks to TEX, a trademark of the American Mathematical Society, and DEK. Permission is granted to make and distribute copies of this card provided the copyright notice and this permission notice are preserved on all copies.

13 GeoSupRC - Categories and Functors c 2000 by MC