4.1. Locally Finitely Copresentable Categories Recall that a set of objects in a category A is called a  cogenerating set provided that for a pair of maps (r1, r2): X --> T such that r1 r2 there exists an object R  and a map g: T --> R such that gr1 gr2. If is a small set we say that is a small cogenerating set. If consists of a single object T then T is called a cogenerator for A. A cogenerating set is called strong provided that for each object X and each proper quotient of X there exists a map X --> R with R  which does not factorize through that quotient.  Suppose A is complete. Then a small set of objects is a cogenerating set iff every object is a subobject of a product of objects in ; is a strong cogenerating set  iff every object is a strong subobject of a product of objects in . We say a set of objects is a regular cogenerating set if every object is a regular subobject of a product of objects in .  An object X of a category A is finitely copresentable provided that the functor hom(~, X): Aop --> Set preserves directed colimits (or filtered colimits).  Definition 4.1.1. A category A is called locally finitely copresentable if it is cocomplete and has a small set of finitely copresentable objects such that every object is an inverse limit of objects in the set.  The dual notion is a locally finitely presentable category.  We summarize the properties of a locally finitely copresentable category A which will be needed below (see [Gabriel and Ulmer 1971], [Johnstone 1982], [Adamek and Rosicky 1994], or [Borceux 1994, Vol II]):  Remark 4.1.2. (a) A is complete and cocomplete.  (b) Each object X has a small set of subobjects. Since the category is complete, this implies that any intersection of subobjects exist. Consequently, any intersection of strong subobjects exist, i.e. A is perfect.  (c) The full subcategory Fin(A) of finitely copresentable objects is closed under finite limits and is essentially small (i.e. with a small skeleton).  (d) Any map has an epi-strong-mono factorization (see (1.2.2.c)).  (e) A complete category is locally finitely copresentable iff every object is an inverse limit of finitely copresentable objects and there exists, up to isomorphism, only a small set of finitely copresentable object (i.e. the full subcategory Fin(A) of finitely copresentable objects is essentially small).  (f) A complete category is locally finitely copresentable iff it has a small strong cogenerating set formed by finitely copresentable objects.  (e) If C is an essentially small category with finite limits we denote by Cart(C. Set) the category of finite-limit-preserving functors from C to Set. Then the opposite Cart(C, Set)op of Cart(C, Set) is locally finitely copresentable and C is equivalent to the subcategory of finitely copresentable objects of Cart(C, Set)op. Any locally finitely copresentable category A is equivalent to Cart(Fin(A). Set)op.  Let A be a locally finitely copresentable category. A regular subobject (or regular mono) of an object is called finitely cogenerated if it is the equalizer of a pair of maps with a finitely copresentable object as codomain.  Proposition 4.1.3. (a) Any regular subobject is an intersection of finitely cogenerated regular subobjects.  (b) Any pullback of a finitely cogenerated regular subobject is finitely cogenerated.  (c) Any finite intersection of finitely cogenerated regular subobjects is finitely cogenerated.  Proof. (a) The collection of finitely copresentable objects is a set of cogenerators for A. Thus the assertion follows from (1.7.6).  (b) follows from the definition.  (c) Suppose u is a finite intersection of finitely cogenerated regular subobjects {ui: Ui --> X}, such that each ui is an equalizer of a pair of maps (ri, si): X --> Vi with finitely copresentable Vi. Let V be the products of Vi. Then V is also finitely copresentable by (4.1.2.c). Let r: X --> V be the map induced by the maps {ri}, and let s: X --> V be the map induced by the maps {si}. Then u is the equalizer of (r, s): X --> V with a finitely copresentable codomain. Thus u is finitely cogenerated.  Proposition 4.1.4. The following conditions are equivalent for a regular subobject V of an object X:  (a) V is finitely cogenerated.  (b) If the intersection of a collection {Vi} of regular subobjects is contained in V, then a finite intersection of objects in {Vi} is contained in V.  Proof. First assume v: V --> X is the equalizer of a pair of maps (r, s): X --> C with a finitely copresentable codomain C. Suppose the intersection w: W --> X of a collection {wi: Wi --> X} of strong subobjects is contained in v. Let {wj: Wj --> X} be the collection of finite intersections of wi. Then w is the limit of the cofiltered systems {wj}. Since rv = sv, we have rw = sw. Since C is finitely copresentable, there is an object wj: Wj --> X such that rwj = swj. Since v is the equalizer of (r, s), wj is contained in v. Since wj is a finite intersection of objects in {Wi}, this prove that (a) implies (b).  Conversely, assume v: V --> X satisfies the condition (b). Since v is an intersection of finitely cogenerated regular subobjects by (4.1.3.a), it is a finite intersection of such subobjects. Thus v is finitely cogenerated by (4.1.3.c).  Proposition 4.1.5. Any composite of finitely cogenerated monos is finitely cogenerated.  Proof. Suppose U is a finitely cogenerated regular subobject of an object X and V a finitely cogenerated regular subobject of U. Consider a collection {Vi} of regular subobjects of X whose intersection is contained in V. Since the intersection of {Vi U} is in V and V is finitely cogenerated in U, we can find a finite set {V1s U}  whose intersection is in V. Since U is finitely cogenerated and the intersection of {Vi} is in U, we can find a finite set {V2s} whose intersection is in U. Then the intersection of the finite set {V1s, V2s} is contained in the intersection of {V1s U}, thus is in V. This shows that V is finitely cogenerated in X by (4.1.4).    Recall that a non-empty partially ordered set (i.e. a poset) is called directed  if each pair of elements has an upper bound. An element s of a poset (S, ) is called finite (or compact) provided that for each directed set T S with s  T there exists t T such that s t. An algebraic lattice is a poset (S, ) which is cocomplete and every element is a directed join of finite elements.  Proposition 4.1.6. The dual of the lattice of regular subobjects of an object is an algebraic lattice.  Proof. It follows from (4.1.4) that a finitely cogenerated subobject is a compact element in the dual of the lattice of regular subobjects of an object. The assertion then follows from (4.1.3.a).   Suppose is a  small strong cogenerating set formed by finitely copresentable objects. A regular subobject (or regular mono) of an object is called -principal if it is the equalizer of a pair of maps with a finitely copresentable object in as codomain. The proof of the following proposition is similar to that of (4.1.3):  Proposition 4.1.7. (a) Each -principal regular subobject is finitely cogenerated.  (b) Any regular subobject is an intersection of -principal regular subobjects.  (c) Any pullback of a -principal regular subobject is -principal.     [Next Section][Content][References][Notations][Home] 