nLab alternative 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, an alternative magmoid should be the oidification of a alternative magma.

Definition

An alternative magmoid is a magmoid QQ such that for every two objects a,bOb(Q)a,b \in Ob(Q) and for every morphism f:abf:a \to b and endomorphisms g:aag:a \to a and h:bbh:b \to b,

f(gg)=(fg)g f \circ (g \circ g) = (f \circ g) \circ g \,

and

h(hf)=(hh)f h \circ (h \circ f) = (h \circ h) \circ f \,

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 11:05:39. See the history of this page for a list of all contributions to it.