nLab magmoid

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

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

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

With a family of sets of morphisms

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

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

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

Weak and strict magmoids

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

Enriched magmoids

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

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

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

Examples

A transitive relation is a magmoid enriched in truth values, or a magmoid $M$ where there is at most one morphism from every object $a$ to another object $b$ in $M$.

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

Related concepts

• magmoidal category
• H-magmoid

 References

The terminology seems to have been introduced in:

• A. Arnold, M. Dauchet, Théorie des magmoïdes (I), RAIRO – Informatique Théorique 12 3 (1978) 235-257 [numdam:ITA_1978__12_3_235_0, eudml:92079]

• A. Arnold, M. Dauchet, Théorie des magmoïdes (II), RAIRO – Informatique Théorique 13 2 (1979) 135-154 [numdam:ITA_1979__13_2_135_0]

Further discussion:

• Miklos Bartha, p. 101 in: An algebraic model of synchronous systems, Information and Computation 97 1 (1992) 97-131

• Dan Jonsson, Def. 2.2 in: On Group-Like Magmoids [arXiv:1902.06109]

• Fiona Torzewska, §2.1 of: Topological quantum field theories and homotopy cobordisms [arXiv:2208.14504]