nLab linear magmoid

Contents

Context

Enriched category theory

Algebra

Categorification

Contents

Idea

A linear magmoid is a magmoids whose hom-sets are all vector spaces (or modules) and whose composition operation is bilinear. This concept is an oidification of the concept of nonassociative nonunital algebra.

Definitions

Fix a commutative ring KK. (Often we want KK to be a field, such as the field \mathbb{C} of complex numbers.)

A KK-linear magmoid is a magmoid enriched over KK\,Mod, the monoidal category of KK-modules with the usual tensor product. (Note that we usually speak of KK\,Vect instead of KModK\,Mod when KK is a field.)

Examples

algebraic structureoidification
truth valuepreorder
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on May 23, 2021 at 12:21:16. See the history of this page for a list of all contributions to it.