The incarnation of internal homs/mapping spaces in the context of pointed objects.
Let be a closed monoidal category with finite limits.
For two pointed objects in , their pointed mapping space
(the “object of basepoint-preserving maps”), is the pullback
where the morphism is induced from the point , and the morphism is the adjunct to .
Regard as a pointed object with basepoint induced by the map whose adjunct is .
Let be a closed monoidal category with finite limits and with finite colimits.
For every pointed object the operation of forming the pointed mapping space out of , and the operation of forming the smash product with , form a pair of adjoint functors
This makes itself a closed monoidal category, which is symmetric if is. The tensor unit is for the unit for the monoidal structure on .
(Elmendorf-Mandell 07, lemma 4.20)
The case when is cartesian, or at least semicartesian, is most common in the literature.
If is monoidal but not closed, the same definition of the smash product makes monoidal as long as the tensor product of preserves finite colimits in each variable separately.
If not, the smash product can fail to be associative. For instance, the smash product on the ordinary category Top (without any niceness conditions imposed) is not associative.
Last revised on January 28, 2020 at 15:58:47. See the history of this page for a list of all contributions to it.