Coslice (under) categories
Given a category and an object , the under category (also called coslice category) (also written and sometimes, confusingly, ) is the category whose
objects are morphisms in starting at ;
morphisms are commuting triangles
The under category is a kind of comma category; it is the strict pullback
in Cat, where
is the interval category ;
is the internal hom in Cat, which here is the arrow category ;
the functor is evaluation at the left end of the interval;
, the point, is the terminal category, the 0th oriental, the 0-globe;
the right vertical morphism maps the single object of the point to the object .
The left vertical morphism is the forgetful morphism which forgets the tip of the triangles mentioned above.
The dual notion is an over category.
, the category of pointed sets, is the undercategory , where is the singleton set.
The category of commutative algebras over a field is the category Ring of commutative rings under .
Revised on January 9, 2011 18:00:45
by Urs Schreiber