Few questions on Kan Extensions

Question

Consider a small category $\mathcal{A}$ and its free completion with finite limits $\hat{\mathcal{A}}$.

• Is the inclusion $\mathcal{A} \to \hat{\mathcal{A}}$ fully faithful?
• Does the inclusion preserve finite limits?
• What about colimits?

Now let's say I have a functor $\mathcal{A} \to$ Set which is faithful.

• Can I extend it to a functor $\hat{\mathcal{A}} \to$ Set?

• What limits is this extension preserving?

• Is the extension faithful?
• Do you have a reference for this kind of questions?

Answer 1

$\require{AMScd}$

Consider a small category $\mathcal{A}$ and its free completion with finite limits $\hat{\mathcal{A}}$.

• Is the inclusion $\mathcal{A} \to \hat{\mathcal{A}}$ fully faithful?

Yes; you have this diagram $$ \begin{CD} {\cal A} @>j>> \hat{\cal A}\\ @| @VVyV\\ {\cal A} @>>y> PSh(\cal A) \end{CD} $$ where $y$'s are Yoneda embeddings; these two are fully faithful, hence $j$ must be fully faithful (let's wave my hands and tell you that this follows from the cancellation property of the right class of the (bo,ff)-factorization system on $\bf Cat$).

• Does the inclusion preserve finite limits?
• What about colimits?

I bet you want to take a look at this nLab page; not an answer, but it might produce interesting refinements of your question.

Now let's say I have a functor $\mathcal{A} \to$ Set which is faithful.

• Can I extend it to a functor $\hat{\mathcal{A}} \to$ Set?

There is a formal sleight of hand that proves that the functor ${\bf CtsCat} \to {\bf Cat}$ that includes small-complete categories in categories has a left biadjoint. This left biadjoint has a name, it's the good old (dualization of the contravariant) yoneda embedding $y : {\cal A} \to [{\cal A}, {\bf Set}]^\text{op}$.

This means (since $\bf Set$ is complete) that any functor $F : {\cal A}\to \bf Set$ can be extended to a continuous functor $[{\cal A}, {\bf Set}]^\text{op} \to \bf Set$; I'd think about this being an extension, maybe even $Ran_yF$? Does this functor have a left adjoint (I'd bet $X\mapsto(A\mapsto {\bf Set}(X,FA))$; it's an easy coend juggling and dualization: these derivations are a bit blind, but it should go as follows: $$ \begin{align*} {\bf Set}(X, \text{Ran}_yF(P)) &\cong {\bf Set}\left(X, \int_A FA^{\hom(P, yA)}\right)\\ & \cong\int_A {\bf Set}(X, FA^{PA})\\ &\cong {\bf Set}(PA, {\bf Set}(X,FA))\\ &\cong [{\cal A},{\bf Set}](P, [X,F]) \end{align*} $$

• Is the extension faithful?

If all I just said is true, a right adjoint is faithful precisely if the component of the counit is an objectwise epimorphism.

• Do you have a reference for this kind of questions?

The things I said come from the $n$Lab, from the introduction to this paper and from a few dirty computations.