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 valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
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
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

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