Given a small category , one can consider the category of presheaves valued in some category . Given some assumptions on , any functor of small categories induces two adjoint pairs
Here, is given by precomposition with , whereas and are the left and right (global) Kan extensions along .
In fact, more generally, any profunctor (i.e. or, after currying, ) gives rise to a single adjoint pair
( and are not standard notations) where is the Yoneda extension of .
A functor gives rise to two profunctors:
a companion profunctor , given by
After currying, this amounts to the functor .
a conjoint profunctor , given by
After currying, this amounts to the functor .
The former profunctor produces the adjoint pair , i.e. and . The latter profunctor produces the adjoint pair , i.e. and .
Let be a functor of small categories and some category. The restriction of scalars functor is given by the formula , i.e. mapping a presheaf to the composite
Suppose admits small colimits (resp. small limits). Then the functor admits a left adjoint (resp. right adjoint ).
Let be an adjoint pair and consider the induced functors and . One has
Note that all these claims are in fact equivalent.
If is fully faithful, then so are and .
A left adjoint functor is fully faithful precisely if is naturally isomorphic to the identity functor (by the unit). Dually, is fully faithful precisely if is naturally isomorphic to the identity (by the co-unit). Hence, it suffices to prove , which by uniqueness of the right adjoint immediately implies that and thus proves both claims.
Being a left adjoint, preserves colimits. Because every presheaf is a colimit of representable objects, it is sufficient to show that where is the Yoneda-embedding. We have
For functoriality of sheaves, see
The pseudofunctor (with as its action on morphisms) takes a category to its free cocompletion. As such, it has the structure of a weak 2-monad, and more specifically it is a prototypical example of a lax-idempotent 2-monad. It factors over its Eilenberg-Moore 2-category, the 2-category of (total? cocomplete?) categories, as a lax-idempotent 2-adjunction. The bind operation for is given by Yoneda extension and the Kleisli 2-category of is Prof.
Last revised on February 16, 2024 at 13:49:22. See the history of this page for a list of all contributions to it.