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 in truth values, or a magmoid MM where there is at most one morphism from every object aa to another object bb in MM.

algebraic structureoidification
truth valuepreorder
magmamagmoid
pointed magma with an endofunctionsetoid/Bishop set
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 January 20, 2023 at 04:00:55. See the history of this page for a list of all contributions to it.