categorification
Just as a monoidal category is the categorification of a monoid and a rig category is the categorification of a rig, a magmoidal category should be the categorification of a magma.
A magmoidal category is a category with a functor
from the product category of with itself.
If there is a specified object , and natural isomorphisms and , then it is a magmoidal category with unit. Depending on the intended uses, it might be necessary to also impose the coherence condition that (which appeared in Mac Lane‘s original definition of a monoidal category, but proved redundant by Max Kelly). Note that the coherence condition usually placed on unitors in monoidal categories cannot even be stated in the context of a magmoidal category, since there is no associator to relate and .
For example, monoidal categories such as braided monoidal categories, symmetric monoidal categories, cartesian monoidal categories, cocartesian monoidal categories, and rig categories are all magma categories.
An Ackermann groupoid is a particular sort of poset that is magmoidal (where ‘groupoid’ is here a synonym for ‘magma’).
The concept seems to first be named in
Last revised on May 16, 2022 at 02:10:53. See the history of this page for a list of all contributions to it.