Categories as models on a suitable algebraic theory
aa r X i v : . [ m a t h . C T ] S e p CATEGORIES AS MODELSON A SUITABLE ALGEBRAIC THEORY
KUERAK CHUNG AND GIOVANNI MARELLI
Abstract.
We explain how categories, and groupoids, can beseen as models for a Lawvere G r -theory, where G r is the cate-gory of graphs, and show that for Lawvere G r -theories finitelypresentable models are finitely presentable objects. Introduction
Lawvere theories were introduced by Bill Lawvere in his doctoralthesis [L64] in 1963 as a categorical formulation of universal algebra.The correspondence between Lawvere theories and finitary monads on
Set is one of the deepest relationships in category theory. In [P99]Lawvere theories were generalized to enriched Lawvere theories, sub-stituting
Set with an arbitrary base category V satisfying axioms thatmake V an appropriate base category for enrichment in the sense of[K82], and a correspondence between V -enriched Lawvere theories and V -enriched monads on V was achieved. A further step was taken in[NP09] and [LP11] with the notion of Lawvere A -theories: first a cate-gory V in which to enrich and then a base V -category A were chosen.The correspondence above was extended to one between Lawvere A -theories and finitary V -enriched monads on the V -cateogry A . Thisallowed to view as models for Lawvere A -theories structures for whichthis interpretation was not possible with A = V .In this paper we first show, as an application of what explainedabove, that categories and groupoids can be seen as models for certainLawvere G r -theories, where A = G r is the category of graphs and V = S et .Another property of Lawvere theories on Set is that a model M for agiven theory is finitely presentable exactly when M od ( M, − ) : M od → S et preserves filtered colimits, where M od denotes the category ofmodels for the given theory. This provides an equivalence betweenan extrinsic (the former) and an intrinsic (the latter) characterization
Key words and phrases.
Lawvere theories, finitely presentability, MSC201018C10. of finitely presentability. We show that this still holds for categories,seen, as said, as models for a Lawvere G r -theory, where the fact that A = G r is decisive. We do not know if this equivalence holds for genericLawvere A -theories and at the moment we have not counterexamples.The paper is organized as follows: in the second chapter we remindthe notion of graph and resume their basic properties; in the thirdwe remember Lawvere A -theories, for a locally finitely presentable V -category A , where V is a locally finitely presentable symmetric monoidalclosed category, and their V -category of models, particularly we showhow categories and groupoids can be seen each one as models for asuitable Lawvere G r -theory, where G r denotes the category of graphs;finally, in the fourth, we show that finitely presentable categories arejust finitely presentable models, establishing an equivalence betweenan intrinsic and extrinsic characterization.We would like to thank Bernhard Keller, who gave us a motivationfor studying this kind of problems, and for useful discussions. We wishto thank also Ross Street, Stephen Lack and John Power for usefulexplanations and suggestions.2. Graphs
We introduce here the notion of graph, explaining some of theirproperties, and the category of graphs and graphs morphisms.
Definition 2.1.
A (directed) graph G consists of (1) a class G , whose elements are called vertices (or 0-cells); (2) for each pair ( A, B ) ∈ G × G a set G ( A, B ) , whose elementsare called the arrows (or 1-cells or edges) from A to B . Equivalently, we can assign a graph G by giving a class G of verticesand a class G of arrows, together with two maps of classes s, t : G →G , called source and target, such that the arrows with given sourceand target form a set. Definition 2.2.
A morphism of graphs α : G → H between two graphs G and H consists of (1) a map α : G → H (2) for each ( A, B ) ∈ G × G a map α A,B G ( A, B ) → H ( αA, αB )Equivalently, a morphism of graphs α is assigned by giving maps α : G → H and α : G → H commuting with s and t . Proposition 2.3.
Small graphs and morphisms of graphs form a cat-egory, which we denote by G r . ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 3
Another useful characterization of graphs is that of presheaves over asuitable category. Let S et be the category of sets and D is the subcat-egory of S et , whose objects are the sets ¯0 := { } and ¯1 := { , } , andwhose non-trivial morphisms are the obvious inclusions i , i : { } →{ , } to 0 , • ¯0 i * * i • ¯1 Proposition 2.4. G r is isomorphic to S et D op .Proof. Given a graph G = ( G , G , s, t ) we define a presheaf Φ on D by setting Φ(¯0) = G , Φ(¯1) = G , Φ( i ) = s , Φ( i ) = t ; conversely,the same definitions assign to a given presheaf Φ a graph G . Givena morphism α : G → H , clearly from the equality above, it defines amorhism between presheaves Φ and Ψ defined by G and H respectively,and the converse holds too. (cid:3) As examples we compute the graphs associated to the representablefunctors h ¯0 ( − ) = Hom D ( − , ¯0) and h ¯1 ( − ) = Hom D ( − , ¯1) in S et D op . Example 2.5.
From the definition of D , we have that h ¯0 (¯0) = { id ¯0 } and h ¯0 (¯1) = ∅ , so that h ¯0 is the graph with one vertex and no arrows; • id ¯0 . Instead h ¯1 (¯0) = { i , i } and h ¯1 (¯1) = { id ¯1 } , so that h ¯1 is a graph withtwo vertexes and one arrow id ¯1 from i to i ; • i id ¯1 / / • i . Corollary 2.6. G r is locally finitely presentable.Proof. It follows from the fact that G r is a category of presheaves byproposition 2.4. (cid:3) In particular, G r is complete and cocomplete such that limits andcolimits can be computed pointwisely, or, equivalently, according todefinition 2.1, cellwisely.The following proposition establishes a relation between the category C at of small categories and the category Gr of graphs: Proposition 2.7.
As a functor between S et -categories, the forgetfulfunctor U : C at → G r has a left adjoint F .Proof. See [Bo94]. (cid:3)
Remark 2.8. G r is a symmetric monoidal closed category. G r and C at are enriched over G r , however proposition 2.7 does not extend to G r -adjunction. KUERAK CHUNG AND GIOVANNI MARELLI Lawvere A -theories As explained in remark 2.8 we will be concerned with Lawvere A -theories when A = Gr and V = S et , however, following [NP09], weintroduce them in generality. Suppose that V is locally finitely pre-sentable as a symmetric monoidal closed category and that A is alocally finitely presentable V -category. Denote by A fp a skeleton ofthe full sub- V -category of A given by finitely presentable objects of A . Let i : A fp → A be the inclusion V -functor and ˜ i the followingcomposition: A Y / / [ A op , V ] [ i op , V ] / / [ A opfp , V ]where Y is the enriched Yoneda embedding. As to G r , note that finitelypresentable objects are just finite graphs; we will denote G r fp simplyby G r f . Definition 3.1.
A Lawvere A -theory is a small V -category L togetherwith an identity-on-objects strict finite V -limit- preserving V -functor J : A opfp → L . Definition 3.2.
Given a Lawvere A -theory ( L , J ) , its V -category ofmodels is defined by the following pull-back in the V −
Cat of locallysmall V -categories: M od ( L ) P L / / U L (cid:15) (cid:15) [ L , V ] [ J, V ] (cid:15) (cid:15) A ˜ i / / [ A opfp , V ]We quote the following result from [NP09]: Proposition 3.3. U L is finitary monadic, particularly it has a left V -adjoint F L For simplicity, when the theory L is fixed, we will use the notation U and F for the forgetful functor and its left adjoint.As said, we want to show that categories can be seen as models foran A -Lawvere theory with V = S et and A = G r .Let −→ h ¯0 in S et D op −→ • a ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 5 and −→ h ¯1 in S et D op −→ • a / / • b . By abuse of notations, s and t denote the two morphisms of graphsfrom −→ −→ −→ a and b respectively • a t ( ( s • a / / • b Note that the graph −→ a , b and c and two arrows from a to b and from b to c −→ • a / / • b / / • c is the push-out of s and t in Gr ~ t / / s (cid:15) (cid:15) ~ s ′ (cid:15) (cid:15) ~ t ′ / / ~ −→ ∼ = −→ −→ −→ • a / / • b / / • c / / • d is isomorphic to −→ −→ −→ −→ −→ Gr .In general, −→ n ; = • a → • a · · · → • a n ∼ = −→ −→ · · · + −→ −→
1. We may consider that above graphs and morphisms are in Gr f andabove finite colimits are those in Gr f since i : Gr f → Gr preservesfinite colimits.Note that for any graph G Gr ( −→ , G ) ∼ = G , Gr ( −→ , G ) ∼ = G , Gr ( −→ n , G ) ∼ = G × G G × G · · ·× G G . In particular, we have the following cartesian (pullback) diagram Gr ( −→ , G ) in Set corresponding to the pushout diagram −→ −→ −→ KUERAK CHUNG AND GIOVANNI MARELLI Gr ; G × G G t ′ / / s ′ (cid:15) (cid:15) G s (cid:15) (cid:15) G t / / G . Denote the obvious inclusions in Gr by l j : −→ → −→ , j = 1 , , l jk : −→ → −→ , ( j, k ) = (1 , , (2 , Definition 3.4. L C is the Lawvere G r -theory having the following pre-sentation;generators: m : −→ → −→ , e : −→ → −→ axioms(relations): ~ m / / s ′ op (cid:15) (cid:15) ~ s op (cid:15) (cid:15) ~ m / / t ′ op (cid:15) (cid:15) ~ t op (cid:15) (cid:15) ~ ψ / / φ (cid:15) (cid:15) ~ m (cid:15) (cid:15) ~ s op / / ~ , ~ t op / / ~ ~ m / / ~ ,~ e / / id (cid:29) (cid:29) ;;;;;;;; ~ s op (cid:15) (cid:15) ~ e / / id (cid:29) (cid:29) ;;;;;;;; ~ t op (cid:15) (cid:15) ~ δ / / id (cid:29) (cid:29) ;;;;;;;; ~ m (cid:15) (cid:15) ~ ρ o o id (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ~ , ~ , ~ . where ψ, φ, δ, ρ are the unique morphisms in L C making the followingdiagrams in L C commute ~ l op x x qqqqqqqqqqqqqq ψ (cid:15) (cid:15) l op & & MMMMMMMMMMMMMM ~ l op x x qqqqqqqqqqqqqq φ (cid:15) (cid:15) l op & & MMMMMMMMMMMMMM ~ m (cid:29) (cid:29) ;;;;;;;; ~ s ′ op (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) t ′ op (cid:29) (cid:29) ;;;;;;;; ~ id (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ~ id (cid:29) (cid:29) ;;;;;;;; ~ s ′ op (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) t ′ op (cid:29) (cid:29) ;;;;;;;; ~ m (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ~ t op (cid:29) (cid:29) ;;;;;;;; ~ s op (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ~ t op (cid:29) (cid:29) ;;;;;;;; ~ s op (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ~ , ~ ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 7 ~ ∼ = ~ × ~ ~ s op (cid:15) (cid:15) δ id ( ( QQQQQQQQQQQQQQQ ~ ∼ = ~ × ~ ~ id (cid:26) (cid:26) t op / / ρ ~ e (cid:29) (cid:29) ;;;;;;;; ~ e $ $ IIIIIIIIIIII ~ t ′ op / / s ′ op (cid:15) (cid:15) ~ s op (cid:15) (cid:15) ~ t ′ op / / s ′ op (cid:15) (cid:15) ~ s op (cid:15) (cid:15) ~ t op / / ~ , ~ t op / / ~ . Note that such unique morphisms ψ, φ, δ, ρ exist in L C , since thebottom diagrams are cartesian in L C and the outer diagrams commute(by axioms).The next theorem says that categories are the models for this theory. Theorem 3.5.
The category
M od ( L C ) of L C -models is equivalent tothe category C at .Proof. From definition 3.2 we have that for any model M there existsa graph G ∈ G r such that M ◦ J = G r ( i − , G ).The first two diagrams yield the following commutative diagrams in Set G × G G M ( m ) / / s ′ (cid:15) (cid:15) G s (cid:15) (cid:15) G × G G M ( m ) / / t ′ (cid:15) (cid:15) G t (cid:15) (cid:15) G s / / G , G t / / G which says that when applying ”the composition” M ( m ) to a pair ofarrows ( f, g ) such that t ( f ) = s ( g ), we get an arrow g ◦ f := M ( m )( f, g )such that s ( g ◦ f ) = s ( f ) , t ( g ◦ f ) = t ( g ).Apply M to the commutative diagram which was used to define ψ ,we have the commutative diagram G × G G × G G p s s hhhhhhhhhhhhhhhhhhh M ( ψ )=( M ( m ) ,id ) (cid:15) (cid:15) p * * UUUUUUUUUUUUUUUUUUUU G × G G M ( m ) % % KKKKKKKKKK G × G G s ′ w w ooooooooooooo t ′ ' ' OOOOOOOOOOOOO G id ~ ~ }}}}}}}} G t ' ' PPPPPPPPPPPPPPP G s w w nnnnnnnnnnnnnnn G KUERAK CHUNG AND GIOVANNI MARELLI where p , p are the obvious projections. Indeed, M ( ψ ) is the obviousprojection ( M ( m ) , id ), since the bottom diagram is cartesian in Set and the outer diagram commutes (by the second axiom). By analogousconsideration, we have that M ( φ ) = ( id, M ( m )).Thus, the third diagram yields the commutative diagram G × G G × G G M ( m ) ,id ) / / ( id,M ( m )) (cid:15) (cid:15) G × G G M ( m ) (cid:15) (cid:15) G × G G M ( m ) / / G which expresses the associativity of the composition M ( m ), i.e., h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f for any triple ( f, g, h ) of arrows with t ( f ) = s ( g ) , s ( h ) = t ( g ).The 4-th, 5-th diagrams yield the commutative diagrams G M ( e ) / / id BBBBBBBB G s (cid:15) (cid:15) G M ( e ) / / id BBBBBBBB G t (cid:15) (cid:15) G , G which say that ”the unit map” M ( e ) assigns an arrow id a := M ( e )( a ) ∈ G with S ( id a ) = a = t ( id a ) to each vertex a ∈ G .Similar arguments for showing M ( ψ ) = ( M ( m ) , id ) show that M ( δ ) = ( M ( e ) , id ) : G ∼ = G × G G → G × G G , M ( ρ ) = ( id, M ( e )) : G ∼ = G × G G → G × G G . Thus, the last diagram yields the commutative diagram G ∼ = G × G G M ( e ) ,id ) / / id ( ( RRRRRRRRRRRRRRR G × G G m (cid:15) (cid:15) G × G G ∼ = G id,M ( e )) o o id v v lllllllllllllll G which says that f ◦ id a = f for any ( a, f ) ∈ G × G with s ( f ) = a and g = id b ◦ g for any ( g, b ) ∈ G × G with t ( g ) = b .All of these say that ( G, M ( m ) , M ( e )) is a category.For the converse, given a category C , define the functor M : L C → Set by the following; M ( G ) = Gr ( G, U ( C )) for G ∈ ob ( L C ) = ob ( Gr opf ), ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 9 M ( α ) = Gr ( α, U ( C )) for morphisms α in Gr f , M ( m ) : U ( C ) × U ( C ) U ( C ) → U ( C ) , ( f, g ) g ◦ f , M ( e ) : U ( C ) → U ( C ) , a id a .Then, all diagrams commute obviously. Finally, one can easily checkthat two constructions are mutually inverse. (cid:3) Remark 3.6.
For the Lawvere theory L C we have defined, the functors U L C and F L C coincide with forgetful functor and free construction ofproposition 2.7.In a similar way we can show that groupoids are models for a Lawvere G r -theory. Definition 3.7. L G is the Lawvere G r -theory having the following pre-sentation:generators: m : −→ → −→ , e : −→ → −→ , ι : −→ → −→ } axioms(relations): all those appearing in definition 3.4 plus ~ ι / / t op (cid:29) (cid:29) ;;;;;;;; ~ s op (cid:15) (cid:15) ~ ι / / s op (cid:29) (cid:29) ;;;;;;;; ~ t op (cid:15) (cid:15) ~ ξ / / t op (cid:15) (cid:15) ~ m (cid:15) (cid:15) ~ ζ o o s op (cid:15) (cid:15) ~ , ~ , ~ e / / ~ ~ e o o where ξ and ζ are the unique morphisms in L G making the followingdiagrams in L G commute ~ ι (cid:26) (cid:26) id ξ (cid:29) (cid:29) ~ id (cid:26) (cid:26) ι ζ (cid:29) (cid:29) ~ s ′ op (cid:15) (cid:15) t ′ op / / ~ s op (cid:15) (cid:15) ~ s ′ op (cid:15) (cid:15) t ′ op / / ~ s op (cid:15) (cid:15) ~ t op / / ~ , ~ t op / / ~ ξ, ζ exist in L G , since the bottomdiagrams are cartesian in L G and the outer diagrams commute. Theorem 3.8.
The category
M od ( L G ) of L G -models is equivalent tothe category G rpd of groupoids.Proof. Following the proof of theorem 3.5, we have that for any model M there exists a graph G ∈ G r such that M ◦ J = G r ( i , G ).We refer to the proof of theorem 3.5 for what concerns those diagramsalready appearing there. The first and second diagrams in definition 3.7 yield the followingdiagram in S et G M ( ι ) / / t op BBBBBBBB G s op (cid:15) (cid:15) G M ( ι ) / / s op BBBBBBBB G t op (cid:15) (cid:15) G , G which say that the “inverse map” M ( ι ) assigns to any arrow f ∈ G anarrow f − := M ( ι )( f ) ∈ G such that s ( f − ) = t ( f ) and t ( f − ) = s ( f ).Applying M to the the commutative diagram defining ξ we obtainanother commutative diagram G M ( ι ) id ' ' M ( ξ )=( M ( ι ) ,id ) % % G × G G s ′ op (cid:15) (cid:15) t ′ op / / G s op (cid:15) (cid:15) G t op / / G M ( ξ ) is ( M ( m ) , id ), since the bottom diagram is cartesian in Set andthe outer diagram commutes (by the second axiom). By analogousconsiderations, we have that M ( ζ ) = ( id, M ( ι )).Therefore the third diagram yields the commutative diagram G M ( ι ) ,id ) / / t op (cid:15) (cid:15) G × G G M ( m ) (cid:15) (cid:15) G id,M ( ι )) o o s op (cid:15) (cid:15) G M ( e ) / / G G M ( e ) o o which says that f ◦ f − = id t ( f ) and f − ◦ f = id s ( f ) .These, together with what proved in theorem 3.5, say that ( G, M ( m ) , M ( e ) , M ( ι ))is a groupoid.For the converse, as in the proof of theorem 3.5, given a groupoid G ,using the inclusion G rpd ⊂ C at to apply the forgetful functor U to G ,define the functor M : L G → Set by the following: M ( G ) = Gr ( G, U ( G )) for G ∈ ob ( L G ) = ob ( Gr opf ), M ( α ) = Gr ( α, U ( G )) for morphisms α in Gr f , M ( m ) : U ( G ) × U ( G ) U ( G ) → U ( G ) , ( f, g ) g ◦ f , M ( e ) : U ( G ) → U ( G ) , a id a . M ( ι ) : U ( G ) → U ( G ) , f f − . ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 11
Then all diagrams commute. Finally, one can check that two con-structions are mutually inverse. (cid:3) Finitely presentable categories and models
We want now to prove that finitely presentable objects are justfinitely presentable models for a Lawvere G r -theory.In this section, L will denote a Lawvere G r -theory where Gr is con-sidered as a category, i.e., a Set -category. Recall that an object C ina category C is finitely presentable if the representable functor C ( C, − ) : C → S et preserves filtered colimits. Definition 4.1.
A model M ∈ M od ( L ) is finitely presentable whenthere exist G and H in G r f such that M is the coequalizer F ( H ) α / / β / / F ( G ) q / / M We call this a finite presentation of M . Proposition 4.2.
M od ( L ) is a reflective subcategory of [ L , S et ] .Proof. See [LR11]. (cid:3)
This implies in particular that
M od ( L ) is complete and cocomplete. Lemma 4.3. L ( G, − ) = F ( iG ) for G ∈ G r f .Proof. Our statement says that for a model
MM od ( L )( L ( G, − ) , M ) = G r ( iG, U ( M ))but this follows from proposition 4.1 of [NP09]. (cid:3) Proposition 4.4.
Free models on finite graphs form a dense family ofgenerators of
M od ( L ) .Proof. By proposition 4.2
M od ( L ) is a reflective subcategory of [ L , S et ];in [ L , S et ] every model M is the colimit of representable functors L ( J G, − ) for G finite; these, on the other hand, are in M od ( L ) as,by lemma 4.3, L ( G, − ) = F ( iG ) for G ∈ G r f ; so the colimit M existsin M od ( L ). (cid:3) Proposition 4.5. If M is a finitely presentable model, then it admitsa presentation (a coequalizer as in definition 4.1) such that the q , asgraph morphism, admits a section s , that is, q ◦ s = id M in G rF ( H ) α / / β / / F ( G ) q / / M s m m Proof.
Let M be a finitely presentable model and take a presentationof it F ( H ′ ) α ′ / / β ′ / / F ( G ′ ) q ′ / / M. Consider the following adjunctions of α ′ , β ′ H ′ α ′′ / / β ′′ / / U F ( G ′ ) . Let R be the smallest equivalence relation containing < α ′ ( v ) , β ′ ( v ) > ,for v ∈ | H ′ | , and, since | U F ( G ′ ) | = | G ′ | , let r : G ′ → G ′ /R be thequotient morphism. Applying F we get a morphism F ( r ) : F ( G ′ ) → F ( G ′ /R ). Note that F ( r ) is an epimorphism, because r is and F isleft-adjoint to U . We can now define a morphism ¯ q : F ( G ′ /R ) → M :it acts on an equivalence class of F ( G ′ /R as q acts on a representa-tive, and this is well-defined because of how R is defined; it acts onmorphisms precisely as q does, as R is an equivalence relation just onobjects. F ( G ′ /R ) ¯ q (cid:15) (cid:15) p / / NF ( H ′ ) α ′ / / β ′ / / F ( G ′ ) F ( r ) rrrrrrrrrr q ′ / / M t : : uuuuuuuuuuu We have that ¯ q ◦ F ( r ) ◦ α = q ◦ α = q ◦ β = ¯ q ◦ F ( r ) ◦ β and we wantto show that F ( H ) F ( r ) ◦ α / / F ( r ) ◦ β / / F ( G/R ) ¯ q / / M s n n is a coequalizer. It remains to prove the universal property. So let( N, p ) such that p ◦ F ( r ) ◦ α = p ◦ F ( r ) ◦ β . By universality we havea unique morphism t : M → N such that p ◦ F ( r ) = t ◦ q ′ . Since q ′ = ¯ q ◦ F ( r ) we have that p ◦ F ( r ) = t ◦ ¯ q ◦ F ( r ), and, since F ( r )is an epimorphism, we get that p = t ◦ ¯ q . Observe now that, since F ( G/R ) and M are graphs with same vertexes, there exists a section s : M → F ( G/R ) to ¯ q . Note finally that H ′ is finite by assumption and F ( G/R ) is finite since G is and R just identifies some vertexes. (cid:3) Proposition 4.6.
The finitely presentable models form a dense familyof generators in mod ( L ) , stable under finite colimits, and every modelis a filtered colimit of finitely presentable ones. ATEGORIES AS MODELS ON A SUITABLE ALGEBRAIC THEORY 13
Proof.
The proof with parallel that proposition 3.8.12 in [Bo94]. Let F be the full subcategory of finitely presentable models. For a model M consider the overcategory F /M and the forgetful functor φ : F /M → M od ( L ). Following [Bo94] and using proposition 4.5, we have colimit φ =( M, s ( F,f ) ), where s ( F,f ) = f : φ (( F, f )) = F → M .That the colimit above is cofiltered, that is, that F/M is cofiltered,follows from the fact that F is stable in M od ( L ) under finite colimits.Let us prove this. Following [Bo94], we soon have that F is stableunder finite coproducts. It is stable also under coequalizers. The proofis again similar to that in [Bo94], however we need to apply proposition4.4. Suppose P and Q are finitely presentable, let u, v : P → Q be twomorphism, and let ( R, r ) be the coequalizer: we want to prove that R is also finitely presentable. Since P and Q are finitely presentable wecan consider the diagram F ( H ) b (cid:15) (cid:15) a (cid:15) (cid:15) F ( K ) d (cid:15) (cid:15) c (cid:15) (cid:15) F ( G ) x / / y / / p (cid:15) (cid:15) (cid:15) (cid:15) F ( J ) q (cid:15) (cid:15) (cid:15) (cid:15) P u / / v / / Q r / / / / s J J R the existence of the lifts x and y of respectively u and v is a consequenceof proposition 4.4, since we can choose a presentation of Q admitting asection s : Q → F ( J ) of q . The proof follows now as in citeB, showingthat R admits indeed a presentation F ( G ∐ K ) x ∐ c / / y ∐ d / / F ( J ) r ◦ q / / R (cid:3) Lemma 4.7.
Free models on finite graphs are finitely presentable mod-els.Proof.
Let F ( G ) be a free model with G finite and consider a cofilteredcolimit X = colim X i , then by adjointness M od ( L )( F ( G ) , colim X i ) = G r ( G, U (colim X i )since U , being finitary monadic (see proposition 3.3) preserves filteredcolimits, we have G r ( G, U (colim X i ) = colim G r ( G, U ( X i )) finally, since G is finitely presentablecolim G r ( G, U ( X i )) = colim M od ( L )( F ( G ) , X i )thus free finitely presentable models are finitely presentable objects. (cid:3) Before enouncing the main result, the following one is expected, hav-ing started our construction with finitely presentable categories:
Proposition 4.8.
M od ( L ) is locally finitely presentable.Proof. M od ( L ) is cocomplete by proposition 4.2. Free generators arefinitely presentable by lemma 4.7 and by proposition 4.5 form a dense,thus strong, family of generators. (cid:3) We conclude with the main result:
Theorem 4.9.
Finitely presentable models correspond to finitely pre-sentable categories.Proof.
Let M a finitely presentable model and take a presentation F ( H ) / / / / F ( G ) / / M since F ( H ) and F ( G ) are finite presentable objects, and since these arestable under finite colimits, it follows that M is a finitely presentableobject.For the converse, suppose that for M ∈ M od ( L ) we have an isomor-phism M od ( L )( M, colim X i ) ∼ = colim M od ( L )( M, X i )for any filtered colimit X = colim X i . By proposition 4.6, M is afiltered colimit of finitely presentable ones: ( M, s ( F,f ) ) = colim φ ( F, f );so, substituting, we obtain
M od ( L )( M, M ) ∼ = colim M od ( L )( M, φ ( F, f ))Let f : M → F be the morphism corresponding to the identity on M : together with s ( F,f ) expresses M as a retract of P and so M asa coequalizer of ( id F , f ◦ s ( F,f ) ) : F → F . By proposition 4.6, M isfinitely presentable. (cid:3) References [Bo94] F. Borceux,
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