In a closed monoidal category the tensor product and internal hom are related by the defining natural isomorphism
The notion of copowering generalizes this to the situation where a category does not act on itself by tensors, but where another category acts on .
The dual notion is that of powering.
Let be a closed monoidal category serving as the cosmos for enrichment.
In a -enriched category , the copower of an object by an object is an object with a (-enriched-)natural isomorphism
where
denotes the -valued hom-object of ,
denotes correspondingly the internal hom of (i.e. the hom-object with respect to its canonical self-enrichment).
(terminology)
Copowers are frequently called tensors and a -category having all copowers is called tensored, while the word “copower” is reserved for the case . 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 be a Bénabou cosmos.
In regarded as a -enriched category over itself, the copower is just the given tensor product of .
If is -copowered and is -small, then the -enriched functor category is -copowered.
If is -copowered and is a -reflective full embedding, then is also copowered.
(copowering of small categories over set)
Every locally small category with all coproducts is canonically copowered over Set: the copowering functor
sends to the coproduct of -many copies of :
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 is the category of monoids (or that of groups). In this case, the copower of a monoid by a set is the free product of copies of , which can more concretely be described as a “one-sided version” of the tensor product of commutative monoids. Indeed, is the monoid consisting of
for each and each , where is the unit of . Here, for each and each , we write for the equivalence class of .
for each ;
which is independent of , as for all .
Explicitly, is isomorphic to the above monoid via the isomorphism sending to the element of given by the word , where is the element in the -th copy of in the expression .
The universal property of the copower states that a morphism of monoids from to a monoid is the same data as a “left-bilinear” map of sets , satisfying
for each .
The copower also endows with skew monoidal structures and , given by
for each , where and are the underlying sets of and . While monoids in with respect to the tensor product of commutative monoids are semirings, monoids in with respect to and 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.