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 colimits? (i.e., ordinary $Set$-based colimits with an enhanced $V$-universal property, although the latter is automatic if powers also exist), 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
Max Kelly, section 3.7 of Basic concepts of enriched category theory (tac ,pdf)
Francis Borceux, Vol 2, Section 6.5 of Handbook of Categorical Algebra, Cambridge University Press (1994)
Last revised on January 29, 2019 at 16:38:45. See the history of this page for a list of all contributions to it.