nLab
copower

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

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

$$C(k\odot x,y)\cong V(k,C(x,y))$$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 $V=\mathrm{Set}$. 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 colimit?s. 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

  $$\otimes :\mathrm{Set}\times C\to C$$\otimes : Set \times C \to C

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

  $$S\otimes b:=\coprod _{s\in S}b.$$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:

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

Related concepts

• copower, (∞,1)-copower

• power

• bitensoring

References

Section 3.7 of

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