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 serving as the cosmos for enrichment.
In a $V$-enriched category $\mathbf{C}$, the copower of an object $c \in \mathbf{C}$ by an object $v \in V$ is an object $v \cdot c \in \mathbf{C}$ with a ($V$-enriched-)natural isomorphism
where
$\mathbf{C}(-,-)$ denotes the $V$-valued hom-object of $C$,
$\mathbf{V}(-,-)$ denotes correspondingly the internal hom of $V$ (i.e. the hom-object with respect to its canonical self-enrichment).
(terminology)
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.
Let $V$ be a Bénabou cosmos.
In $V$ regarded as a $V$-enriched category over itself, the copower is just the given tensor product of $V$.
If $B$ is $V$-copowered and $A$ is $V$-small, then the $V$-enriched 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.
(copowering of small categories over set)
Every locally small category $C$ with all coproducts is canonically copowered over Set: the copowering functor
sends $(S,b)$ to the coproduct of $|S|$-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:
(copowering of the category of monoids)
A particularly illuminating instance of Example occurs when $C$ is the category of monoids (or that of groups). In this case, the copower $X\cdot A$ of a monoid $A$ by a set $X$ is the free product of $X$ copies of $A$, which can more concretely be described as a “one-sided version” of the tensor product of commutative monoids. Indeed, $X\cdot A$ is the monoid consisting of
for each $x\in X$ and each $a,b\in A$, where $()$ is the unit of $Free(X\times A)$. Here, for each $x\in X$ and each $a\in A$, we write $x\cdot a$ for the equivalence class of $(x,a)$.
for each $x\cdot a,y\cdot b\in X\cdot A$;
which is independent of $x$, as $x\cdot 1_A=()=y\cdot 1_A$ for all $x,y\in A$.
Explicitly, $\coprod_{x\in X}A$ is isomorphic to the above monoid via the isomorphism sending $(x_{1}\cdot a_{1})\cdots(x_{n}\cdot a_{n})$ to the element of $\coprod_{x\in X}A$ given by the word $a^{(x_{1})}_{1}\cdots a^{(x_{n})}_{n}$, where $a^{(x_{i})}_{i}$ is the element $a_{i}$ in the $x_{i}$-th copy of $A$ in the expression $\coprod_{x\in X}A$.
The universal property of the copower $X\cdot A$ states that a morphism of monoids from $X\cdot A$ to a monoid $B$ is the same data as a “left-bilinear” map of sets $f\colon A\times X\to B$, satisfying
for each $x\in X$.
The copower $\cdot\colon Set\times Mon\to Mon$ also endows $Mon$ with skew monoidal structures $\triangleleft$ and $\triangleright$, given by
for each $A,B\in Obj(Mon)$, where $|A|$ and $|B|$ are the underlying sets of $A$ and $B$. While monoids in $CMon$ with respect to the tensor product of commutative monoids are semirings, monoids in $Mon$ with respect to $\triangleleft$ and $\triangleright$ recover left and right near-semirings.
Textbook accounts:
Max Kelly, Section 3.7 of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp. (ISBN:9780521287029);
republished as:
Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (tac:tr10, pdf)
Francis Borceux, Section 6.5 of: Handbook of Categorical Algebra Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)
Emily Riehl, §3.7 in: Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]
Last revised on October 31, 2023 at 16:09:07. See the history of this page for a list of all contributions to it.