nLab absorption category

Idea

The concept of an absorption category should be the oidification of an absorption monoid.

Definition

An absorption category or annihilation category $C$ is a category where for every two objects $a, b \in Ob(C)$ there is a morphism $0_{a\to b}: a \to b$, such that for any objects $a, b, c, d \in Ob(C)$, for any morphism $f:b \to c$, $f \circ 0_{a\to b} = 0_{a\to c}$, and for any morphism $g:d \to a$, $0_{a\to b} \circ g = 0_{d\to b}$.

Such a structure is the same thing as a category enriched in the category of pointed sets, taking the monoidal product to be the smash product.

Examples

• Every ringoid and algebroid is an absorption category.

• An absorption monoid is an absorption category with only one object.

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