Under what conditions, one of completeness and cocompleteness can imply the other in a category?

Question

When I tried to prove the completeness and cocompleteness of the category of small categories $\mathbf{Cat}$, I thought that proving either one of them could imply the other by taking the dual arguments. But this is not true. Then I have been trying to think about if one of completeness and cocompleteness in $\mathbf{Cat}$ can imply the other.

My question is: In $\mathbf{Cat}$, does one of completeness and cocompleteness imply the other? Under what conditions for a category, one of completeness and cocompleteness can imply the other?

Answer 1

Suppose $C$ is a (locally small) complete category. Then (under enough axiom of choice) it is cocomplete if and only if for all small categories $I$, $\Delta : C\to C^I$ has a left adjoint, that is, if and only if $\Delta$ is a right adjoint.

But the adjoint functor theorem tells us that by completeness, it suffices that $\Delta$ preserve limits and satisfy the solution set condition. Preservation of limits under $\Delta$ is immediate, so the only thing to check is the solution set condition.

So a locally small complete category is cocomplete if and only if for any small category $I$, $\Delta : C\to C^I$ satisfies the solution set condition. You can even restrict to $I$ being either a discrete category, or the category on two parallel arrows. Then you can try to give more concrete interpretations to what the solution set condition means in these cases.

Of course, dualizing all this tells us when a cocomplete category is complete.

A well-known example of all this (where the proof is much simpler though) is that a small complete category is automatically cocomplete (and conversely). In fact, a small complete category is automatically a preorder, in which case cocompleteness follows easily from completeness from order-theoretic considerations.

Answer 2

One specific situation in which completeness and cocompleteness come together, which applies to $\mathbf{Cat}$, is for accessible categories. For simplicity, a finitely accessible category $\mathcal{A}$ is one with directed colimits (colimits of diagrams indexed by posets in which every finite set of elements admits an upper bound) exist, and in which every object may be written as a directed colimit of a small set $G$ of finitely presentable objects. Here a finitely presentable object $a$ is one such that the functor $\mathcal{A}(a,-)$ preserves directed colimits.

The lovely thing about accessible categories is that functors between them which preserve directed colimits automatically satisfy the solution set condition, intuitively because everything is determined by objects from the small set $G$. Thus the diagonal functors $\mathcal A\to \mathcal A^J$ have left adjoints whenever they have right adjoints, and vice versa.

Now, one can prove that $\mathbf{Cat}$ is finitely accessible relatively easily. A directed colimit of categories has object and morphism sets given by the directed colimit of the underlying sets. Now finite presentability in the categorical sense corresponds in $\mathbf{Cat}$ to a category with finitely many objects, in which every arrow is written as a composite of finitely many generating arrows, and in which only finitely many relations between composites of the generating arrows suffice to specify the category. Every category is certainly a directed colimit, even a union, of finitely presentable categories, in essentially the same way as a group or ring is the union of its finitely presentable subobjects.

Thus $\mathbf{Cat}$ is finitely accessible, and so as soon as you prove it's complete you must know it's cocomplete as well. This is in fact roughly the way I would prove cocompleteness of $\mathbf{Cat}$, since it's a bit difficult to construct coequalizers explicitly.