nLab quasigroupoid

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 quasigroupoid should be the oidification of a quasigroup.

Definition

A quasigroupoid is a magmoid QQ such that for every span

s f g x y \array{ && s \\ & {}^{f}\swarrow && \searrow^{g} \\ x &&&& y }

in QQ, there exists morphisms i:xyi:x\to y and j:yxj:y \to x such that if=gi \circ f = g and jg=fj \circ g = f, and for every cospan

a b f g c \array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }

in QQ, there exists morphisms d:abd:a\to b and e:bae:b \to a such that gd=fg \circ d = f and fe=gf \circ e = g.

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 19:20:33. See the history of this page for a list of all contributions to it.