nLab
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 magmoid should be the oidification of a magma.

Definition

With a set of morphisms

Let QQ be a quiver with a collection of objects Ob(Q)Ob(Q), a set of morphisms Mor(Q)Mor(Q), a function s:Mor(Q)Ob(Q)s: Mor(Q) \to Ob(Q) called the source or domain and a function t:Mor(Q)Ob(Q)t: Mor(Q) \to Ob(Q) called the target or codomain. A magmoid is a quiver QQ with a partial binary operation

()():Mor(Q)×Mor(Q)Mor(Q)+1\left(-\right)\circ\left(-\right): Mor(Q) \times Mor(Q) \to Mor(Q) + 1

with coproduct injections inj:Mor(Q)Mor(Q)+1inj:Mor(Q) \to Mor(Q) + 1 and pt:1Mor(Q)+1pt: 1 \to Mor(Q) + 1, such that for f,gMor(Q)f, g \in Mor(Q), there is a function hh in the inverse image of inj(h)=fginj(h) = f \circ g if s(f)=t(g)s(f) = t(g), and fg=pt()f \circ g = pt(\bullet) otherwise.

With a family of sets of morphisms

Let QQ be a quiver with a collection of objects Ob(Q)Ob(Q) and a Set-valued functor Mor:Ob(Q)×Ob(Q)SetMor: Ob(Q) \times Ob(Q) \to Set for all objects a,bOb(Q)a, b \in Ob(Q). A magmoid is a quiver QQ with a binary operation

()():Mor(b,c)×Mor(a,b)Mor(a,c)\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)

for all a,b,cOb(Q)a,b,c \in Ob(Q).

Weak and strict magmoids

A weak magmoid is a magmoid QQ whose collection of objects Ob(Q)Ob(Q) form a groupoid, while a strict magmoid is a magmoid QQ whose collection of objects Ob(Q)Ob(Q) form a set.

Enriched magmoids

Let VV be a monoidal category (or a monoidal (infinity,1)-category) and let QQ be a VV-enriched quiver, with a collection of objects Ob(Q)Ob(Q) and a VV-valued functor Mor:Ob(Q)×Ob(Q)VMor: Ob(Q) \times Ob(Q) \to V for all objects a,bOb(Q)a, b \in Ob(Q). A VV-enriched magmoid is a VV-enriched quiver QQ with a binary operation

()():Mor(b,c)×Mor(a,b)Mor(a,c)\left(-\right)\circ\left(-\right): Mor(b,c) \times Mor(a,b) \to Mor(a,c)

for all a,b,cOb(Q)a,b,c \in Ob(Q).

Examples

A transitive relation is a magmoid enriched on truth values, or a magmoid MM where thete is at most one morphism from every object aa to object bb in MM

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 07:43:31. See the history of this page for a list of all contributions to it.