nLab magmoidal category




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 CC with a functor

:C×CC\otimes: C \times C \to C

from the product category of CC with itself.

If there is a specified object II, and natural isomorphisms λ A::AIA\lambda_A:\colon A\otimes I\stackrel{\simeq}{\to} A and ρ A::IAA\rho_A:\colon I\otimes A\stackrel{\simeq}{\to} A, then it is a magmoidal category with unit. Depending on the intended uses, it might be necessary to also impose the coherence condition that ρ I=λ I:III\rho_I = \lambda_I\colon I\otimes I\stackrel{\simeq}{\to} I (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 A(IB)A\otimes (I\otimes B) and (AI)B(A\otimes I)\otimes B.


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

  • Alexei Davydov, Nuclei of categories with tensor products, Theory and Applications of Categories, Vol. 18, 2007, No. 16, pp 440-472. tac:18-16

Last revised on May 15, 2022 at 22:10:53. See the history of this page for a list of all contributions to it.