# Can universal constructions be described as fixed points of functors on Cat?

Suppose $Cat$ is the category of (small) categories and functors. My question is the following:

Can we characterize different categorical notions which are given by universal properties (i.e. "$\mathcal{C}$ has a terminal object", "$\mathcal{C}$ has binary products", "$\mathcal{C}$ is a Cartesian closed category", etc... by results such as: $$\mathcal{C} \text{ has all binary products if and only if } \mathcal{F}(\mathcal{C}) \simeq \mathcal{C}$$ where $\mathcal{F} : Cat \to Cat$ is some (large) functor, thought of, in this case for instance, as a functor which takes $\mathcal{C}$ and "closes" it under binary products.

This seems like a natural generalization from e.x. the construction of algebraic closures in field theory to categories, but I have yet to see any results like this. Have people considered such characterizations of categorical notions in the literature? Is such a characterization possible always, never, or only in certain instances?

• Perhaps completion is what you are looking for? – asdq Feb 6 '18 at 21:02
• @asdq If $F : \mathcal{C} \to \mathcal{C}^*$ (e.x. the Yoneda embedding $y : \mathcal{C} \to Set^{\mathcal{C}^{op}}$) is a completion, is it the case that $(\mathcal{C}^*)^* \simeq \mathcal{C}^*$? If this is the case, then I imagine that completions can probably be formalized in this way, but I haven't seen this anywhere in the literature. – Nathan BeDell Feb 6 '18 at 21:24
• No, this is not the case, already for size reasons. – Kevin Carlson Feb 6 '18 at 23:09
• The particular example of binary products could probably be realized in a category of limit sketches, since the sketch structure remembers which products are intentional and should be preserved by functors. – Hurkyl Feb 7 '18 at 5:12

A common situation is that we have an endofunctor $F$ on a category $C$ together with a natural map $\eta: 1\to F$, which could be dualized. Then we say an object $x$ is $F$-complete if $\eta_x$ is an isomorphism. For a reasonable notion of completion, we should assume $Fx$ is always complete. A natural way to enforce this is to assume that $\eta$ is the unit of a monad structure on $F$ whose multiplication is invertible; a completion is a kind of free construction, so we should expect a monad to appear. These extra assumptions make $F$ the monad induced by the inclusion of some reflective subcategory.