nLab
power

Idea

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

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

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

$$V(v_1,V(v_2,v_3))\simeq V(v_2,V(v_1,v_3)).$$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 $C(-,-):C\times C\to V$. One says that $C$ is powered over $V$ if it is in addition equipped also with a mixed operation $⋔:V\times C\to C$ such that $⋔(v,c)$ 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

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

Definition

Definition

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

$$C(c_1,⋔(v,c_2))\cong V(v,C(c_1,c_2))$$C(c_1, \pitchfork(v,c_2)) \cong V(v, C(c_1,c_2))

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 $⋔(v_1,v_2):=[v_1,v_2]$.

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

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

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

  The defining natural isomorphism

  $${\mathrm{Hom}}_C(c_1,⋔(S,c_2))\simeq {\mathrm{Hom}}_{\mathrm{Set}}(S,{\mathrm{Hom}}_C(c_1,c_2))$$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).

Related concepts

• copower, (∞,1)-copower

• power

References

Section 3.7 of

• Max Kelly, Basic concepts of enriched category theory (tac ,pdf)