category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the inserter category is a variation of the

comma category In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objec ...

where the two functors are required to have the same domain category.

Definition

If ''C'' and ''D'' are two categories and ''F'' and ''G'' are two functors from ''C'' to ''D'', the inserter category Ins(''F'', ''G'') is the category whose objects are pairs (''X'', ''f'') where ''X'' is an object of ''C'' and ''f'' is a morphism in ''D'' from ''F''(''X'') to ''G''(''X'') and whose morphisms from (''X'', ''f'') to (''Y'', ''g'') are morphisms ''h'' in ''C'' from ''X'' to ''Y'' such that

G(h) \circ f = g \circ F(h)

Properties

If ''C'' and ''D'' are locally presentable, ''F'' and ''G'' are functors from ''C'' to ''D'', and either ''F'' is cocontinuous or ''G'' is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

; then the inserter category Ins(''F'', ''G'') is also locally presentable.

References

Category theory {{cattheory-stub