**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** or **magmal category** is a category $C$ with a functor

$\otimes: C \times C \to C$

from the product category of $C$ with itself.

If there is a specified object $I$, and natural isomorphisms $\lambda_A:\colon A\otimes I\stackrel{\simeq}{\to} A$ and $\rho_A:\colon I\otimes A\stackrel{\simeq}{\to} A$ satisfying $\rho_I = \lambda_I\colon I\otimes I\stackrel{\simeq}{\to} I$, then it is a **magmoidal category with unit**.

Note that this coherence condition appeared in Mac Lane‘s original definition of a monoidal category, but was proven to follow from the other coherence conditions for a monoidal category by Max Kelly. (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 $A\otimes (I\otimes B)$ and $(A\otimes I)\otimes B$.)

For example, monoidal categories such as braided monoidal categories, symmetric monoidal categories, cartesian monoidal categories, cocartesian monoidal categories, and rig categories are all magmoidal categories.

Every magma is a discrete magmoidal category.

An Ackermann groupoid is a particular sort of poset that is magmoidal (where ‘groupoid’ is here a synonym for ‘magma’).

- David Roberts,
*Substructural fixed-point theorems and the diagonal argument: theme and variations*, Compositionality**5**issue 8 (2023) doi:10.32408/compositionality-5-8,

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

The terminology **magmal category** first appears in:

- Ross Street,
*Wood fusion for magmal comonads*, arXiv:2311.07088 (2023).

Last revised on February 4, 2024 at 12:30:25. See the history of this page for a list of all contributions to it.