Recall that a set of objects in a category A is called a cogenerating set provided that for a pair of maps (r_{1}, r_{2}): X > T such that r_{1} r_{2} there exists an object R and a map g: T > R such that gr_{1} gr_{2}. 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.
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.
Proof. (a) The collection of finitely copresentable objects is
a set of cogenerators for A. Thus the assertion follows from (1.7.6).
Proposition 4.1.4. The following conditions
are equivalent for a regular subobject V of an object X:
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).
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 nonempty 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.
