# nLab power

Contents

This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Limits and colimits

limits and colimits

# Contents

## Idea

In a closed symmetric monoidal category $V$ the internal hom $[-,-] : V^{op} \times V \to V$ satisfies the natural isomorphism

$[v_1,[v_2,v_3]] \simeq [v_2,[v_1,v_3]]$

for all objects $v_i \in V$ (prop.). If we regard $V$ as a $V$-enriched category we write $V(v_1,v_2) := [v_1,v_2]$ and this reads

$V(v_1,V(v_2,v_3)) \simeq V(v_2,V(v_1,v_3)) \,.$

If we now pass more generally to any $V$-enriched category $C$ then we still have the enriched hom object functor $C(-,-) : C^{op} \times C \to V$. One says that $C$ is powered over $V$ if it is in addition equipped also with a mixed operation $\pitchfork : V^{op} \times C \to C$ such that $\pitchfork(v,c)$ behaves as if it were a hom of the object $v \in V$ into the object $c \in C$ in that it satisfies the natural isomorphism

$C(c_1,\pitchfork(v,c_2)) \simeq V(v,C(c_1,c_2)) \,.$

## Definition

###### Definition

Let $V$ be a closed monoidal category. In a $V$-enriched category $C$, the power of an object $y\in C$ by an object $v\in V$ is an object $\pitchfork(v,y) \in C$ with a natural isomorphism

$C(x, \pitchfork(v,y)) \cong V(v, C(x,y))$

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

We say that $C$ is powered or cotensored over $V$ if all such power objects exist.

###### Remark

Powers are frequently called cotensors and a $V$-category having all powers is called cotensored, while the word “power” 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

• Powers are a special sort of weighted limits. Conversely, all weighted limits can be constructed from powers together with conical limits. The dual colimit notion of a power is a copower.

## Examples

• $V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) := [v_1,v_2]$.

• Every locally small category $C$ ($V = (Set,\times)$ ) with all products is powered over Set: the powering operation

$\pitchfork(S,c) := \prod_{s\in S} c$

of an object $c$ by a set $S$ forms the $|S|$-fold cartesian product of $c$ with itself, where $|S|$ is the cardinality of $S$.

The defining natural isomorphism

$Hom_C(c_1,\pitchfork(S,c_2))\simeq Hom_{Set}(S,Hom_C(c_1,c_2))$

is effectively the definition of the product (see limit).