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For $\mathsf{V}$ a closed monoidal category, it is canonically powered (or cotensored) and copowered (or tensored) over itself with respect to the internal hom and tensor product.

Likewise, any (co)complete category is canonically (co)powered over $\mathsf{Sets}$ in the obvious way.

Are there any other good examples of $\mathsf{V}$-enriched categories which are powered and copowered over some closed monoidal category $\mathsf{V}$?

For example, which abelian categories are (co)powered over abelian groups? Is the abelian category of sheaves of abelian groups (co)powered, for instance?

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    $\begingroup$ Yes, lots. When studying any enriched category, one of the first things you should do is think about what the tensors / powers look like. In fact, $V$-categories with $V$-cotensors can alternatively be defined as categories equipped with an action of $V$ (where $V$ is considered as a monoid in $\mathsf{Cat}$); the action is by cotensoring. So I guess the question is: which enriching categories are you interested in? $\endgroup$ – Tim Campion Apr 12 '17 at 22:23
  • $\begingroup$ I'm just starting to learn some enriched category theory, so I don't have a particular enriching category in mind yet. I suppose taking $\mathsf{V}$ to be some abelian monoidal category such as modules over a commutative ring or chain complexes of modules would be most useful for me. I was just hoping to see some examples of (co)powers that come up in practice, since the only two examples listed on the nLab page are the ones mentioned in my post. $\endgroup$ – ಠ_ಠ Apr 12 '17 at 22:33
  • $\begingroup$ Try simplicial sets or chain complexes $\endgroup$ – Tim Campion Apr 12 '17 at 23:57
  • $\begingroup$ Is there a canonical (co)tensor when we work with categories enriched over simplicial sets or chain complexes? $\endgroup$ – ಠ_ಠ Apr 13 '17 at 0:02
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    $\begingroup$ Abelian sheaves are tensored and cotensored as $Ab$-categories. One way of seeing this is by noting abelian presheaves are tensored and cotensored, and using the fact that reflective full subcategories (in the enriched sense) have the same types of enriched limits and colimits as exist in the ambient enriched category. This is covered in Kelly's book (Basic Concepts of Enriched Category Theory). $\endgroup$ – Todd Trimble Apr 15 '17 at 11:23
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To answer the question raised in the most recent edit: abelian sheaves are tensored and powered over abelian groups.

First of all, abelian presheaves have tensors and powers that are computed pointwise. By "abelian presheaves" I mean the $\textbf{Ab}$-category of functors $F: C \to \textbf{Ab}$ where $C$ is a small (ordinary) category, or somewhat more generally the $\textbf{Ab}$-category of $\textbf{Ab}$-functors $F: C \to \textbf{Ab}$ where $C$ is small and $\textbf{Ab}$-enriched (we can always regard a small $\textbf{Set}$-category $C$ as a small $\textbf{Ab}$-category, by defining $C(a, b) = F\hom(a, b)$, the free abelian group generated by the set $\hom(a, b)$). If $A$ is an abelian group, then the power $F^A$ is given by the "pointwise" formula $F^A(c) = F(c)^A$, i.e., $F^A(c) = \textbf{Ab}(A, F(c))$. This is a straightforward calculation having to do with commutation of limits. We have for each $G: C \to \textbf{Ab}$ an isomorphism $\textbf{Ab}^C(G, F^A) \cong \textbf{Ab}(A, \textbf{Ab}^C(G, F))$ since

$$\begin{array}{rcl} \textbf{Ab}^C(G, F^A) & \cong & \int_c \textbf{Ab}(Gc, F^A(c)) \\ & \cong & \int_c \textbf{Ab}(Gc, \textbf{Ab}(A, Fc)) \\ & \cong & \int_c \textbf{Ab}(Gc \otimes A, Fc) \\ & \cong & \int_c \textbf{Ab}(A, \textbf{Ab}(Gc, Fc)) \\ & \cong & \textbf{Ab}(A, \int_c \textbf{Ab}(Gc, Fc) \\ & \cong & \textbf{Ab}(A, \textbf{Ab}^C(G, F)). \end{array}$$

A very similar sort of calculation shows that $\textbf{Ab}$-tensors of abelian sheaves are also computed in pointwise (or objectwise) fashion.

As for sheaves: if $F$ is an abelian sheaf (for whatever topology), then for any abelian group $A$ the presheaf $F^A$ is a sheaf. Abstractly, if $F$ is an algebra of the (enriched) sheafification monad $\sigma$ on $\textbf{Ab}^C$, then so is $F^A$. For we have a universal element $\mathbb{Z} \to \textbf{Ab}(A, \textbf{Ab}^C(F^A, F))$ and a map

$$\textbf{Ab}^C(F^A, F) \to \textbf{Ab}^C(\sigma(F^A), \sigma(F)).$$

Putting these together we get $\mathbb{Z} \to \textbf{Ab}(A, \textbf{Ab}^C(\sigma(F^A), \sigma(F)) \cong \textbf{Ab}^C(\sigma(F^A), \sigma(F)^A)$, i.e., a map $\sigma(F^A) \to \sigma(F)^A$. Together with the algebra structure $\alpha: \sigma(F) \to F$, we produce an algebra structure on $F^A$ by an evident composite

$$\sigma(F^A) \to \sigma(F)^A \stackrel{\alpha^A}{\to} F^A$$

where $\alpha^A$ for any $\alpha: G \to F$ is defined by exploiting maps

$$\mathbb{Z} \stackrel{\text{canon}}{\to} \textbf{Ab}(A, \textbf{Ab}^C(G^A, G)) \stackrel{\textbf{Ab}(1, \textbf{Ab}^C(1, \alpha))}{\to} \textbf{Ab}(A, \textbf{Ab}^C(G^A, F)) \cong \textbf{Ab}^C(G^A, F^A)$$

(cf. similar constructions mentioned at Morphisms of cotensors).


Nothing in this answer so far has much to do with the details of abelian groups or sheafification or anything like that; it's really just pure enriched category theory. A more general result is that if $\mathcal{V}$ is a suitable base of enrichment (complete, cocomplete, symmetric monoidal closed) and if $C$ is a small $\mathcal{V}$-category, then $\mathcal{V}^C$ has $\mathcal{V}$-powers and $\mathcal{V}$-tensors (in fact is $\mathcal{V}$- complete and cocomplete), and also if $\mathcal{C}$ is $\mathcal{V}$-complete and $\sigma$ is a $\mathcal{V}$-monad on $\mathcal{C}$, then the $\mathcal{V}$-category of algebras $\mathcal{C}^\sigma$ inherits $\mathcal{V}$-weighted limits from $\mathcal{C}$. Just as you would expect by analogy from ordinary category theory. Putting these results together covers a huge range of examples that arise in the wild.


As for tensoring on abelian sheaves: this is a special case of (weighted) colimits on sheaves, where the recipe is well-known: sheafify the (pointwise) colimit taken in presheaves. Thus the formula for a sheaf $F$ should be given by $(A \cdot F)(c) = a(A \otimes iF(c))$, where $i$ is the full inclusion of sheaves into presheaves and $a$ is its left adjoint, reflecting presheaves back into sheaves. This construction applies more generally to idempotent monads $\sigma = ai$ where $i$ is a full inclusion and $a \dashv i$; however, it doesn't work for general (enriched) monads $\sigma$ (and in fact the construction of colimits in categories of algebras is a pretty big topic in its own right). But anyway, if $G$ is a sheaf, then we have

$$\text{Sh}(A \cdot F, G) \cong \text{Sh}(a(A \otimes iF), G) \cong \textbf{Ab}^C(A \otimes iF, iG) \cong \textbf{Ab}(A, \textbf{Ab}^C(iF, iG)) \cong \textbf{Ab}(A, \text{Sh}(F, G))$$

where the last isomorphism invokes (enriched) full faithfulness.


Although much of the theory of weighted limits and colimits in enriched category theory is a smooth generalization of the theory of ordinary (conical) limits and colimits from ordinary category theory, it must be said that powers and tensors add an important enriched ingredient, in the sense that

  • Categories that are complete and cocomplete in the ordinary sense need not admit (enriched) powers and tensors, but

  • If a category is complete in the ordinary sense and admits enriched powers, then it is complete in the enriched sense of admitting all weighted limits. Similarly if a category is cocomplete in the ordinary sense and admits tensors, then it is cocomplete in the enriched sense of admitting all weighted colimits.

For an example of the first point: consider the inclusion of monoids $M$ into categories, taking $M$ to the usual one-object category $BM$ whose morphisms are elements of $M$. Monoids are thus $\textbf{Cat}$-enriched by the formula $\textbf{Cat}(BM, BN)$. Certainly $\textbf{Mon}$ admits all ordinary limits (they are computed as in $\textbf{Set}$). However, $\textbf{Mon}$ does not admit $\textbf{Cat}$-powers. For instance if $\mathbf{2}$ is the arrow category, then the objects of $(BM)^\mathbf{2}$ correspond to elements $m \in M$, hence have more than one object (and in fact need not be even equivalent as categories to monoids, when one considers that morphisms $a: m \to n$ in $(BM)^\mathbf{2}$ are elements $a \in M$ such that $am = na$). This is enough to show $\textbf{Mon}$ doesn't have copowers, because the full inclusion $B: \textbf{Mon} \to \textbf{Cat}$ reflects any weighted limit in $\textbf{Cat}$.

Essentially all of this material can be found in Kelly's book, Basic Concepts of Enriched Category Theory.

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Given a group $G$ the set of functions $Set(X,G)$ admits a pointwise group structure -- in this way, the category of groups is cotensored as a $Set$-category. Likewise any of the standard algebraic categories.

Similarly when you encounter a functor category $[C,D]$ equipped with some ``pointwise" structure inherited from $D$ you are in the presence of a 2-category (or $Cat$-enriched category) admitting cotensors.

For example, the 2-category of (symmetric) monoidal categories / categories with limits/ categories with colimits (and of any any suitable flavour of morphism between them) admit cotensors -- although they often don't admit all weighted limits.

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  • $\begingroup$ Thanks for your reply! So I assume for the 2-category of (co)complete categories the (co) powering is given by taking (co)limits? That's a nice example to keep in mind. $\endgroup$ – ಠ_ಠ Apr 14 '17 at 0:33
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And now for something entirely different :D

The category $\mathbf{Set}^{\mathbb C^\circ}$ of presheaves of sets on a small category $\mathbb C$ is monoidal closed wrt cartesian products (i. e. cartesian closed) - in fact it is a topos.

Now it happens that the category $\mathbf{Set}^{\mathbb C}$ of presheaves of sets on the opposite category $\mathbb C^\circ$ is enriched, powered and copowered over $\mathbf{Set}^{\mathbb C^\circ}$.

Strangely enough, easiest to see all three is to start from poweredness. For $P\in\mathbf{Set}^{\mathbb C^\circ}$ and $Q\in\mathbf{Set}^{\mathbb C}$, the power $Q^P\in\mathbf{Set}^{\mathbb C}$ is given by $$ Q^P(c)=Q(c)^{P(c)}. $$ Given that, the enrichment $\mathrm{HOM}$ of $\mathbf{Set}^{\mathbb C}$ over $\mathbf{Set}^{\mathbb C^\circ}$ becomes uniquely determined given that we must have $$ \hom_{\mathbf{Set}^{\mathbb C}}(Q',Q^P)\approx\hom_{\mathbf{Set}^{\mathbb C^\circ}}(P,\mathrm{HOM}(Q',Q)): $$ taking here $P=\hom_{\mathbb C}(-,c)$ gives $$ \mathrm{HOM}(Q',Q)(c)=\hom_{\mathbf{Set}^{\mathbb C}}\left(Q',Q(-)^{\hom_{\mathbb C}(-,c)}\right)=\varprojlim_x Q(x)^{Q'(x)\times\hom_{\mathbb C}(x,c)}. $$ Now the $P$-fold copowers $P\cdot Q'$ can be easily found from $$\hom_{\mathbf{Set}^{\mathbb C}}(P\cdot Q',Q)\approx\hom_{\mathbf{Set}^{\mathbb C^\circ}}(P,\mathrm{HOM}(Q',Q))\approx\hom_{\mathbf{Set}^{\mathbb C}}(Q',Q^P).$$

All this has been observed by Lawvere, drawn from the analogy with measure theory, distributions, etc. - he calls the above $\mathrm{HOM}$ the Radon-Nikodym derivative.

There is a similar enrichment of left $B$-modules over right $B$-modules, for any bialgebra $B$. It is closely related to a bunch of constructions like pseudolinear algebra by Bakalov-d'Andrea-Kac, chiral enrichments of D-modules by Beilinson-Drinfeld, Borcherds' categories holding his quantum vertex algebras, Soibelman's meromorphic tensor categories, ...

Here again the powers are the easiest ones to see: for a right module $P$ and a left module $Q$, the power $Q^P$ is the space of all linear maps from $P$ to $Q$, with the left module structure given using the comultiplication of $B$.

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The category of sheaves of abelian groups on a space $X$ is powered and copowered over abelian groups as follows:

let $t: X \to \bullet$ denote the terminal map to the 1-point space. Then the direct image is the global sections functor $t_* = \Gamma(X, -)$ and the inverse image $t^{-1}$ constructs the constant sheaf with given stalk. Let $A_X := t^{-1} A$ denote the constant sheaf with stalk an abelian group $A$.

Then by the tensor-Hom and image adjunctions, for sheaves of abelian groups $\mathscr{F}$, $\mathscr{G}$ we have $$\mathrm{Hom}(\mathscr{F}, \mathscr{Hom}(A_X, \mathscr{G})) \cong \mathrm{Hom}(A_X \otimes\mathscr{F}, \mathscr{G}) \cong \mathrm{Hom}(A_X, \mathscr{Hom}(\mathscr{F}, \mathscr{G})) \cong \mathrm{Hom}(A, \mathrm{Hom}(\mathscr{F}, \mathscr{G})).$$

So sheaves of abelian groups are copowered over abelian groups by tensoring with the corresponding constant sheaf of abelian groups. They are powered by Hom-ing out of the constant sheaf.

The same argument applies for presheaves of abelian groups, mutatis mutandis.

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  • $\begingroup$ I was in the midst of writing up an answer, as requested, when I saw this. $\endgroup$ – Todd Trimble Apr 15 '17 at 16:20
  • $\begingroup$ Sorry! I appreciate that you took the time to type up such a detailed and insightful answer, and I upvoted your work. $\endgroup$ – ಠ_ಠ Apr 15 '17 at 21:25
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    $\begingroup$ Sorry in return, for sounding crabby! I'm glad you worked out this answer on your own. $\endgroup$ – Todd Trimble Apr 15 '17 at 22:11

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