# nLab copower

Contents

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Limits and colimits

limits and colimits

# Contents

## Idea

In a closed monoidal category $C$ the tensor product $a \otimes b$ and internal hom $[b,c]$ are related by the defining natural isomorphism

$C(a \otimes b, c) \simeq C(a, [b,c]) \,.$

The notion of copowering generalizes this to the situation where a category $C$ does not act on itself by tensors, but where another category $V$ acts on $C$.

The dual notion is that of powering.

## Definition

###### Definition

Let $V$ be a closed monoidal category. In a $V$-enriched category $C$, the copower of an object $x\in C$ by an object $k\in V$ is an object $k\odot x \in C$ with a natural isomorphism

$C(k\odot x, y) \cong V(k, C(x,y))$

where $C(-,-)$ is the $V$-valued hom-functor of $C$ and $V(-,-)$ is the internal hom of $V$.

###### Remark

Copowers are frequently called tensors and a $V$-category having all copowers is called tensored, while the word “copower” is reserved for the case $V=Set$. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.

## Properties

• In the $V$-category $V$, the copower is just the tensor product of $V$.

• Copowers are a special sort of weighted colimit. Conversely, all weighted colimits can be constructed from copowers together with conical colimits? (i.e., ordinary $Set$-based colimits with an enhanced $V$-universal property, although the latter is automatic if powers also exist), assuming these exist. The dual limit notion of a copower is a power.

## Examples

• Every locally small category $C$ with all coproducts is canonically copowered over Set: the copowering functor

$\otimes : Set \times C \to C$

sends $(S,b)$ to $|S|$-many copies of $b \in C$:

$S \otimes b := \coprod_{s \in S} b \,.$

The defining natural isomorphism in this situation is then just the fact that the hom functor sends colimits in its first argument to limits:

$C(\coprod_{s \in S} b , c) \simeq \prod_{s \in S} C(b,c) \simeq Set(S, C(b,c)) \,.$

The next two classes of examples are covered in Kelly’s book (see references).

• If $B$ is $V$-copowered and $A$ is $V$-small, then the $V$-functor category $B^A$ is $V$-copowered.

• If $C$ is $V$-copowered and $B \to C$ is a $V$-reflective full embedding, then $B$ is also copowered.