Not to be confused with an M-category.
Let be a category and let be a class of monomorphisms in . (Often, will be the right class in an orthogonal factorization system.) We say that is -complete if it admits all (even large) intersections of -subobjects. This means that it admits all (even large) wide pullbacks of families of -morphisms, and such pullbacks are again in . (If is the right class of an OFS, then any intersection of -morphisms which exists is automatically in .)
If is the class of all monomorphisms, we may say mono-complete for -complete.
If is -well-powered, then no large limits are required in the definition of -completeness. Therefore, if is well-powered and complete, it is -complete whenever is the right class in an OFS. Dually, if is well-copowered and cocomplete, it is -cocomplete whenever is the left class in an OFS.
For similar reasons, the category FinSet is mono-complete and epi-cocomplete—although it is not complete or cocomplete, it is finitely complete and cocomplete, and its subobject lattices and quotient lattices are likewise essentially finite?.
If is a topological concrete category over a category which is mono-complete or epi-complete, then is also mono-complete or epi-complete. For the faithful forgetful functor preserves and reflects monos and epis, and so the initial -structure on an intersection of underlying monos in gives an intersection in and the final -structure on a cointersection? of underlying epis in gives a cointersection in .
Let be a class of maps in a category , and assume that
Then there is an orthogonal factorization system , with .
Given , let be the intersection of all -morphisms through which factors. Then by the universal property of this intersection, we have for some ; thus it suffices to show . Suppose given a commutative square
with . By pulling back to (since pullbacks of -morphisms exist), we may assume that and is the identity. But now the composite is an -morphism through which factors, so by definition, factors through it. Thus is an isomorphism and so the lifting problem can be solved.
In fact, it is easy to see that the same proof constructs a factorization structure for sinks.
Note that if is already part of a prefactorization system, then any composite, pullback, or intersection of -morphisms which exists is automatically also in , since .
Let be a prefactorization system on a category , and assume that
Then is an orthogonal factorization system.
The following is a slight generalization of Theorem 3.3 of CHK. There it is stated only for the case strong monomorphisms, in which case a finitely complete and -complete category is called finitely well-complete.
Let be an adjunction, and assume that is finitely complete and -complete for some OFS , where consists of monomorphisms and contains the split monics. Define to be the class of maps inverted by , and ; then is an OFS on .
First of all, since belongs to a prefactorization system, it is closed under composites, pullbacks, and any intersections which exist. Therefore, if we define , then satisfies the hypotheses of Theorem 1, and so we have an OFS .
Moreover, it is useful to notice that : this is an easy consequence of the fact that if , then , since for each , so that is an isomorphism.
Now suppose given ; we want to construct an -factorization. Let be the pullback of along the unit . The naturality square for at shows that factors through , say .
Since is evidently in , so is ; thus it suffices to find an -factorization of .
Let be the -factorization of . Since , it suffices to show that . Note also that since is a first factor of the unit , by passing to adjuncts we find that is split monic: in the former diagram we have , so that the adjunct , hence also is a split monic. But is then also split monic, hence belongs to and thus also to (since it obviously belong to ). Therefore, since , the naturality square for at contains a lift: there is an such that in the diagram
and . Passing to adjuncts again, we find that is also split epic, since we can consider the diagram
and the commutativity
Hence is an isomorphism; thus as desired.
This is useful in the construction of reflective factorization systems.