Enriched category theory

Limits and colimits



In a closed monoidal category CC the tensor product aba \otimes b and internal hom [b,c][b,c] are related by the defining natural isomorphism

C(ab,c)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 CC does not act on itself by tensors, but where another category VV acts on CC.

The dual notion is that of powering.



Let VV be a closed monoidal category. In a VV-enriched category CC, the copower of an object xCx\in C by an object kVk\in V is an object kxCk\odot x \in C with a natural isomorphism

C(kx,y)V(k,C(x,y)) C(k\odot x, y) \cong V(k, C(x,y))

where C(,)C(-,-) is the VV-valued hom-functor of CC and V(,)V(-,-) is the internal hom of VV.


Copowers are frequently called tensors and a VV-category having all copowers is called tensored, while the word “copower” is reserved for the case V=SetV=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 VV-category VV, the copower is just the tensor product of VV.

  • Copowers are a special sort of weighted colimit. Conversely, all weighted colimits can be constructed from copowers together with conical colimits? (i.e., ordinary SetSet-based colimits with an enhanced VV-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 CC with all coproducts is canonically copowered over Set: the copowering functor

    :Set×CC \otimes : Set \times C \to C

    sends (S,b)(S,b) to |S||S|-many copies of bCb \in C:

    Sb:= sSb. 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( sSb,c) sSC(b,c)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)) \,.

The next two classes of examples are covered in Kelly’s book (see references).

  • If BB is VV-copowered and AA is VV-small, then the VV-functor category B AB^A is VV-copowered.

  • If CC is VV-copowered and BCB \to C is a VV-reflective full embedding, then BB is also copowered.


Last revised on January 29, 2019 at 16:38:45. See the history of this page for a list of all contributions to it.