# nLab powered and copowered category

Contents

### Context

#### Enriched category theory

enriched category theory

# Contents

## Definition

Throughout, let $\mathcal{V}$ be a Bénabou cosmos for enriched category theory.

###### Definition

(tensoring and cotensoring)

Let $\mathcal{C}$ be a $\mathcal{V}$-enriched category.

1. A powering (or cotensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

1. $[-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}$
2. for each $v \in \mathcal{V}$ a natural isomorphism of the form

(1)$\mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2])$
2. A copowering (or tensoring) of $\mathcal{C}$ over $\mathcal{V}$ is

1. $(-)\otimes(-) \;\colon\; \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}$
2. for each $v \in \mathcal{V}$ a natural isomorphism of the form

(2)$\mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) )$

If $\mathcal{C}$ is equipped with a (co-)powering it is called (co-)powered over $\mathcal{V}$.

## Properties

###### Proposition

(tensoring left adjoint to cotensoring)

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ (Def. ), then for fixed $v \in \mathcal{V}$ the operations of tensoring with $v$ and of cotensoring with $\mathcal{V}$ form a pair of adjoint functors

$\mathcal{C} \underoverset {\underset{ [v,-] }{\longrightarrow}} {\overset{ v \otimes (-) }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}$
###### Proof

The hom-isomorphism characterizing the pair of adjoint functors is provided by the composition of the natural isomorphisms (1) and (2):

$\array{ \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2]) }$

It follows that:

###### Proposition

(in tensored and cotensored categories initial/terminal objects are enriched initial/terminal)

If $\mathcal{C}$ is both tensored and cotensored over $\mathcal{V}$ then

1. an initial object $\emptyset$ of the underlying category of $\mathcal{C}$ is also enriched initial, in that the hom-object out of it is the terminal object $\ast$ of $\mathcal{V}$

$\mathcal{C}(\emptyset, c) \;\simeq\; \ast$
2. a terminal object $\ast$ of the underlying category of $\mathcal{C}$ is also enriched terminal, in that the hom-object into it is the terminal object of $\mathcal{V}$:

$\mathcal{C}(c, \ast) \;\simeq\; \ast$
###### Proof

We discuss the first claim, the second is formally dual.

By prop. , tensoring is a left adjoint. Since left adjoints preserve colimits, and since an initial object is the colimit over the empty diagram, it follows that

$v \otimes \emptyset \;\simeq\; \emptyset$

for all $v \in \mathcal{V}$, in particular for $\emptyset \in \mathcal{V}$. Therefore the natural isomorphism (2) implies for all $v \in \mathcal{V}$ that

$\mathcal{C}(\emptyset, c) \;\simeq\; \mathcal{C}( \emptyset \otimes \emptyset, c ) \;\simeq\; \mathcal{V}( \emptyset, \mathcal{C}(\emptyset, c) ) \;\simeq\; \ast$

where in the last step we used that the internal hom $\mathcal{V}(-,-) = [-,-]$ in $\mathcal{V}$ sends colimits in its first argument to limits (this prop.) and used that a terminal object is the limit over the empty diagram.

## Examples

###### Example

The symmetric closed monoidal category $\mathcal{V}$ is tensored and cotensored over itself, with tensoring being its tensor product and powering being its internal hom.

###### Example

For $\mathcal{C}$ a small $\mathcal{V}$-enriched category, the $\mathcal{V}$-enriched category of enriched presheaves $[\mathcal{C}^{op}, \mathcal{V}]$ is tensored and cotensored (this Prop.)

Last revised on August 1, 2018 at 08:28:12. See the history of this page for a list of all contributions to it.