nLab unital magmoid

Contents

Context

Category theory

Algebra

Categorification

Contents

Idea

Just as a groupoid is the oidification of a group and a ringoid is the oidification of a ring, a unital magmoid should be the oidification of a unital magma.

Definition

A unital magmoid QQ is a magmoid where every object aOb(Q)a \in Ob(Q) has an identity morphism id a:aaid_a: a \to a, such that for any morphism f:abf:a \to b, fid a=ff \circ id_a = f, and for any morphism g:cag:c \to a, id ag=gid_a \circ g = g.

A unital magmoid is invertible if for every pair of objects a,bOb(Q)a,b \in Ob(Q) and for every morphism f:abf:a \to b, there exists an inverse morphism g:bag:b \to a such that fg=id bf \circ g = id_b and gf=id ag \circ f = id_a.

Examples

See also

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 June 7, 2022 at 17:21:25. See the history of this page for a list of all contributions to it.