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): A^op --> 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 {u_i: U_i --> X}, such that each u_i is an equalizer of a pair of maps (r_i, s_i): X --> V_i with finitely copresentable V_i. Let V be the products of V_i. Then V is also finitely copresentable by (4.1.2.c). Let r: X --> V be the map induced by the maps {r_i}, and let s: X --> V be the map induced by the maps {s_i}. 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 {V_i} of regular subobjects is contained in V, then a finite intersection of objects in {V_i} 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 {w_i: W_i --> X} of strong subobjects is contained in v. Let {w_j: W_j --> X} be the collection of finite intersections of w_i. Then w is the limit of the cofiltered systems {w_j}. Since rv = sv, we have rw = sw. Since C is finitely copresentable, there is an object w_j: W_j --> X such that rw_j = sw_j. Since v is the equalizer of (r, s), w_j is contained in v. Since w_j is a finite intersection of objects in {W_i}, 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 {V_i} of regular subobjects of X whose intersection is contained in V. Since the intersection of {V_i  U} is in V and V is finitely cogenerated in U, we can find a finite set {V_1s  U}  whose intersection is in V. Since U is finitely cogenerated and the intersection of {V_i} is in U, we can find a finite set {V_2s} whose intersection is in U. Then the intersection of the finite set {V_1s, V_2s} is contained in the intersection of {V_1s  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.