nLab absorption category

Contents

Context

Algebra

Category theory

Categorification

Contents

Idea

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

Definition

An absorption category or annihilation category CC is a category where for every two objects a,bOb(C)a, b \in Ob(C) there is a morphism 0 ab:ab0_{a\to b}: a \to b, such that for any objects a,b,c,dOb(C)a, b, c, d \in Ob(C), for any morphism f:bcf:b \to c, f0 ab=0 acf \circ 0_{a\to b} = 0_{a\to c}, and for any morphism g:dag:d \to a, 0 abg=0 db0_{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

algebraic structureoidification
truth valuepreorder
magmamagmoid
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 June 18, 2021 at 14:33:33. See the history of this page for a list of all contributions to it.