nLab
loopoid

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 loopoid should be the oidification of a loop.

Definition

A loopoid 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, where every span

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

in QQ has 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 where every cospan

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

in QQ has 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 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 18:55:53. See the history of this page for a list of all contributions to it.