In
category theory, a branch of
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 ob ...
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
.
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 g ...
; then the inserter category Ins(''F'', ''G'') is also locally presentable.
References
Category theory
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