Let $C$ be a cocomplete category. Suppose that $X : A \to B$ is a functor, where $A$ is small. Then every functor $F : A \to C$ admits a left Kan extension $\mathrm{Lan}_X(F) : B \to C$, defined by mapping $b \in B$ to $\mathrm{colim}_{X(a) \to b} F(a)$. The action on morphisms is as follows: If $\iota_{\sigma} : F(a) \to \mathrm{Lan}_X(F)(b)$ denotes the colimit inclusion induced by $\sigma : X(a) \to b$, then for a morphism $f : b \to b'$ we have $\mathrm{Lan}_X(F)(f) \circ \iota_{\sigma} = \iota_{f \sigma}$. This is well-known and easy to check.

Now I would like to work with $k$-linear (i.e. $\mathsf{Mod(k)}$-enriched) categories instead. So assume that $A,B,C$ are $k$-linear categories and likewise $X,F$ are $k$-linear functors. Then it doesn't seem to be the case that $\mathrm{Lan}_X(F)$, as defined above, is a $k$-linear functor again, right? So how can we explicitly construct a $k$-linear functor $\mathrm{Lan}_X(F)$ which serves as a linear left Kan extension (which I would define by a natural isomorphism of $k$-modules $\hom(\mathrm{Lan}_X(F),T) \cong \hom(F,T \circ X)$, where $T : B \to C$ is a $k$-linear functor)?

I'm aware that Kelly treats enriched Kan extensions in his book on enriched categories. But it doesn't seem to help me. I would like to know a down-to-earth construction without delving into general enriched category theory. After all, linear categories should be a very basic example.

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    $\begingroup$ There is a standard weighted colimit formula that always works (as explained by Kelly), and in some cases the weighted colimit can be reduced to an ordinary colimit. $\endgroup$ – Zhen Lin Aug 25 '13 at 9:46
  • $\begingroup$ Yes I kind of know that (without understanding it). I would like to see a "formula" for $\mathrm{Lan}_X(F)$ with which I can work. $\endgroup$ – Martin Brandenburg Aug 25 '13 at 9:47
  • $\begingroup$ I could try to explain that, but are you looking for something else? $\endgroup$ – Zhen Lin Aug 25 '13 at 9:47
  • $\begingroup$ Okay the formula ncatlab.org/nlab/show/end#kan_extension_40 means that for $k$-linear categories we should define $\mathrm{Lan}_X(F) := \int^{a \in A} \hom(X(a),-) \otimes_k F(a)$, right? Does this coend have the usual description, or does the enrichment play a role here? $\endgroup$ – Martin Brandenburg Aug 25 '13 at 9:53
  • $\begingroup$ If $A$ is an ordinary category and the target category is cocomplete in the ordinary sense and also tensored and cotensored, then (IIRC) the coend can be computed in the ordinary way. If $A$ is also enriched then it has to be an enriched coend. $\endgroup$ – Zhen Lin Aug 25 '13 at 10:00

Let $\mathcal{V}$ be a Bénabou cosmos (complete and cocomplete symmetric monoidal closed category), let $\mathcal{A}$ be a small $\mathcal{V}$-category, let $\mathcal{B}$ be any $\mathcal{V}$-category, and let $\mathcal{C}$ be a $\mathcal{V}$-category that is cocomplete in the ordinary sense as well as tensored and cotensored. Let $X : \mathcal{A} \to \mathcal{B}$ and $F : \mathcal{A} \to \mathcal{C}$ be $\mathcal{V}$-functors. A pointwise left Kan extension (which Kelly refers to simply as ‘left Kan extension’) is a left Kan extension that is preserved by representable functors, in the sense that $$\mathcal{C} (\operatorname{Lan}_X F, C) \cong \operatorname{Ran}_X \mathcal{C} (F, C)$$ and the right hand side can be described by the following end in $\mathcal{V}$: $$\operatorname{Ran}_X \mathcal{C} (F, C) \cong \int_{A : \mathcal{A}} \mathcal{V} (\mathcal{B} (X A, -), \mathcal{C}(F A, C))$$ Note that the right hand side, evaluated at $B$ in $\mathcal{B}$, is the definition of the hom-object $$[\mathcal{A}, \mathcal{V}](\mathcal{B}(X, B), \mathcal{C}(F, C))$$ and thus we obtain the following formulae in $\mathcal{C}$: $$(\operatorname{Lan}_X F) B \cong \mathcal{B}(X, B) \star_{\mathcal{A}} F \cong \int^{A : \mathcal{A}} \mathcal{B}(X A, B) \otimes F A$$

Our problem is now to describe $\mathcal{V}$-enriched ends/coends. Let $H : \mathcal{A}^\mathrm{op} \otimes \mathcal{A} \to \mathcal{C}$ be a $\mathcal{V}$-functor. Then, for $A$ and $A'$ in $\mathcal{A}$, we obtain a pair of morphisms in $\mathcal{V}$, \begin{align*} \mathcal{A} (A', A) \otimes H (A, A') & \to H (A, A) \\ f \otimes z & \mapsto H (A, f) (z) \\ \mathcal{A} (A', A) \otimes H (A, A') & \to H (A', A') \\ f \otimes z & \mapsto H (f, A') (z) \end{align*} and the coend of $H$ is computed by the following coequaliser diagram: $$\coprod_{(A, A') \in \operatorname{ob} \mathcal{A}} \mathcal{A} (A', A) \otimes H (A, A') \rightrightarrows \coprod_{A \in \operatorname{ob} \mathcal{A}} H (A, A) \to \int^{A : \mathcal{A}} H (A, A)$$ Note that because $\mathcal{C}$ is cotensored, ordinary colimits and conical colimits coincide.

The question of whether enriched left Kan extensions can be computed using only (possibly iterated) colimits over ordinary diagrams essentially boils down to whether every $\mathcal{V}$-presheaf $\mathcal{A}^\mathrm{op} \to \mathcal{V}$ can be expressed as a (possibly iterated) colimit over an ordinary diagram of representable $\mathcal{V}$-presheaves. According to Kelly (§3.9), this is the case when $\mathcal{V}$ is $R\text{-}\mathbf{Mod}$, but I don't see why immediately.


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