Morita equivalence via Kan extension

Question

Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}_yf=F\colon \widehat{\mathcal C}\leftrightarrows \widehat{\mathcal A}$, so that $\cal A,C$ are Morita equivalent categories?

Answer 1

Expanding on Zhen's comment into an answer, the Cauchy completion of $A$ can be identified with the small-projective objects in $\widehat{A}$ (i.e. those where mapping out of them preserves all small colimits). Thus, an equivalence of presheaf categories must restrict to an equivalence of Cauchy completions. It follows that if $f:C\to \widehat{A}$ induces such an equivalence, it must land inside the small-projectives and be Cauchy-surjective onto them. In other words, it must be the case that an object of $\widehat{A}$ is small-projective iff it is a retract of an object in the image of $f$. Moreover, $f$ must be fully faithful, since $C$ embeds fully faithfully in $\widehat{C}$. Conversely, if $f$ satisfies these two conditions, then it induces an equivalence of Cauchy completions, hence of presheaf categories.

So the answer is, it is necessary and sufficient that

1. $f$ is fully faithful, and
2. an object of $\widehat{A}$ is small-projective iff it is a retract of an object in the image of $f$.

Answer 2

Adding my two cents to Mike's answer. Here is a way to show that $f$ establishes a Morita equivalence as described in the question iff it gives rise to an equivalence between the Cauchy completions. It depends on few facts some of which are not trivial.

Let $\mathbf{Prof}$ denote the bicategory whose objects are categories $A, B \cdots$ and whose morphisms are profunctors $F : A \nrightarrow B$.

A presheaf category $\hat{A}$ is $\mathbf{Prof}(A, I)$, where $I$ is the terminal category.

A functor $f : C \to \hat{A}$ can be regarded as a profunctor $F : A \nrightarrow C$, and $\mathrm{Lan}_yf$ is the functor $-\circ F : \mathbf{Prof}(C, I) \to \mathbf{Prof}(A, I)$.

Every cocontinuous functor between presheaf categories comes from a profunctor. It follows that $(-\circ F)$ is an equivalence of categories iff $F$ is an equivalence in $\mathbf{Prof}$.

In $\mathbf{Prof}$ there are equivalences $A \simeq \bar{A}$ between a category and its Cauchy completion. Composing $F$ from both sides with these equivalences we get a profunctor $\bar{F} : \bar{A} \nrightarrow \bar{C}$. Immediately, $F$ is an equivalence if and only if $\bar{F}$ is an equivalence.

Every morphism in $\mathbf{Prof}$ whose target is a Cauchy complete category and which has a right adjoint converges to a functor. Hence, if $\bar{F}$ is an equivalence, then it comes from a functor $\bar{A} \to \bar{C}$ which is equivalence in $\mathbf{Cat}$.