Given a category and an object , the under category (also called coslice category) (alternative notations include and 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.
If is an initial object in , then is isomorphic to .
If has a terminal object , then the coslice is known as the category of pointed objects in . For instance:
the category of pointed sets, is the coslice of Set under the singleton set,
the category of pointed topological spaces is the coslice of Top under the point space,
the category of pointed simplicial sets is the coslice of sSet under the 0-simplex.
If is a monoidal category with tensor unit , then the coslice is also known as the category of pointed objects in a monoidal category. For instance:
the category of pointed abelian groups is the coslice of Ab under the additive group of integers,
the category of pointed modules is the coslice of Mod under the additive module of the ground ring
the category of bi-pointed sets is the coslice of , the category of pointed sets, under the boolean domain.
the category of pointed endofunctors is a coslice of and endo-functor category under the identity functor.
Generally, for any one may think of the -coslice category as the category of “-pointed objects”.
The category of commutative algebras over a field is the coslice under of the category CRing of commutative rings.
If is a category with all limits, then a limit in any of its under categories is computed as a limit in the underlying category .
In detail:
Let be any functor.
Then, the limit over in is the image under the evident projection of the limit over itself:
and is uniquely characterized by .
Over a morphism in the limiting cone over (which exists by assumption) looks like
By the universal property of the limit this has a unique lift to a cone in the under category over :
It therefore remains to show that this is indeed a limiting cone over . Again, this is immediate from the universal property of the limit in . For let be another cone over in , then is another cone over in and we get in a universal morphism
A glance at the diagram above shows that the composite constitutes a morphism of cones in into the limiting cone over . Hence it must equal our morphism , by the universal property of , and hence the above diagram does commute as indicated.
This shows that the morphism which was the unique one giving a cone morphism on does lift to a cone morphism in , which is then necessarily unique, too. This demonstrates the required universal property of and thus identifies it with .
One often says “ reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if is monadic (i.e., has a left adjoint such that the canonical comparison functor is an equivalence), then both reflects and preserves limits. In the present case, the projection is monadic, is essentially the category of algebras for the monad , at least if admits binary coproducts. (Added later: the proof is even simpler: if is the underlying functor for the category of algebras of an endofunctor on (as opposed to algebras of a monad), then reflects and preserves limits; then apply this to the endofunctor above.)
under category
Discussion in the generality of -categories:
Last revised on July 1, 2024 at 10:24:12. See the history of this page for a list of all contributions to it.