nLab linear magmoid

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.

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

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

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

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