nLab magmoidal category

Idea

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.

Definition

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 $(A\otimes I)\otimes B$.)

Examples

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’).

Related concepts

• magma

• magmoidal groupoid

• categorification

• magmoid

References

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

  (arXiv:2110.00239).

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).