powering in nLab

> 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.

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#### Enriched category theory
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[[!include enriched category theory contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
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## Idea

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

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

for all [[objects]] $v_i \in V$ ([prop.](closed+monoidal+category#TensorHomIsoInternalizes)). If we regard $V$ as a $V$-[[enriched category]] we write $V(v_1,v_2) \mathrel{:=} [v_1,v_2]$ and this reads

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

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 comes with natural isomorphisms of the form

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

## Definition 

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###### Definition

Let $V$ be a [[closed monoidal category|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.


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###### 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]].

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## Properties

* Powers are a special sort of [[weighted limit]]: in particular, where the domain is the unit $V$-category. Conversely, all weighted limits can be constructed from powers together with [[conical limit]]s.  The dual colimit notion of a power is a [[copower]].


## Examples

### In 1-category theory

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

* Every [[locally small category]] $C$ ($V = (Set,\times)$ ) with all [[product]]s  is powered over [[Set]]: the powering operation

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

  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]]).

* In a [[2-category]] $\mathcal{K}$ (seen as a $\mathbf{Cat}$-enriched category), powers by the walking arrow $\downarrow$ are ways to internalize 'generalized arrows' of a given object $A:\mathcal{K}$.
Specifically, ${A^\downarrow} := {\downarrow \pitchfork A}$, called the **object of arrows** of $A$ is, when it exists, an object such that:
$$
\mathcal{K}(X, A^\downarrow) \simeq \mathbf{Cat}(\downarrow, \mathcal{K}(X,A)) = \mathcal{K}(X,A)^\downarrow.
$$
Thus [[generalized elements]] of $A^\downarrow$ correspond to 2-cells between generalized elements of $A$, explaining why $A^\downarrow$ can be considered a 'view from the inside' of the internal structure of $A$.


[[!include powering of ∞-toposes over ∞-groupoids -- section]]


## Related concepts

* [[tensored and cotensored category]]

* [[copower]], [[(∞,1)-copower]]

* [[pullback-power]]


## References

Textbook accounts:

* [[Max Kelly]], section 3.7 of: _Basic concepts of enriched category theory_ ([tac](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html) ,[pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

* {#Borceux94} [[Francis Borceux]], Vol 2, Section 6.5 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* [[Emily Riehl]], §3.7 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) &lbrack;[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)&rbrack;



[[!redirects powerings]]

[[!redirects power]]
[[!redirects powers]]

[[!redirects cotensor]]
[[!redirects cotensoring]]

[[!redirects cotensors]]
[[!redirects cotensored category]]