nLab alternative magmoid

Idea

Just as a groupoid is the oidification of a group and a ringoid is the oidification of a ring, an alternative magmoid should be the oidification of a alternative magma.

Definition

An alternative magmoid is a magmoid $Q$ such that for every two objects $a,b \in Ob(Q)$ and for every morphism $f:a \to b$ and endomorphisms $g:a \to a$ and $h:b \to b$,

$$f \circ (g \circ g) = (f \circ g) \circ g \,$$

and

$$h \circ (h \circ f) = (h \circ h) \circ f \,$$

Examples

• Every category is an alternative magmoid.

• A one-object alternative magmoid is a alternative magma.

• A Mod-enriched alternative magmoid is called a alternative linear magmoid.

Related concepts

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