In a closed monoidal category $C$ the tensor product $a \otimes b$ and internal hom $[b,c]$ are related by the defining natural isomorphism
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.
Let $V$ be a closed 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
where $C(-,-)$ is the $V$-valued hom-functor of $C$ and $V(-,-)$ is the internal hom of $V$.
Copowers are frequently called tensors and a $V$-category having all copowers is called tensored, while the word “copower” is reserved for the case $V=Set$. 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.
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 (i.e., ordinary $Set$-based colimits), assuming these exist. The dual limit notion of a copower is a power.
Every locally small category $C$ with all coproducts is canonically copowered over Set: the copowering functor
sends $(S,b)$ to $|S|$-many copies of $b \in C$:
The defining natural isomorphism in this situation is then just the fact that the hom functor sends colimits in its first argument to limits:
The next two classes of examples are covered in Kelly’s book (see references).
If $B$ is $V$-copowered and $A$ is $V$-small, then the $V$-functor category $B^A$ is $V$-copowered.
If $C$ is $V$-copowered and $B \to C$ is a $V$-reflective full embedding, then $B$ is also copowered.
copower, (∞,1)-copower
Section 3.7 of
Section 6.5 of