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53
votes
9answers
5k views

How do I check if a functor has a (left/right) adjoint?

Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, ...
14
votes
2answers
963 views

What's an example of an “adjunction up to adjunction”?

(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...
8
votes
4answers
1k views

When does the sheaf direct image functor f_* have a right adjoint?

Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring). Under what ...
4
votes
2answers
435 views

Is there a free digraph associated to a graph?

A little bit of background: A graph G is, of course, a set of vertices V(G) and a multiset of edges, which are unordered pairs of (not necessarily distinct) vertices. We say that two vertices v_1, v_2 ...
11
votes
6answers
1k views

Can adjoint linear transformations be naturally realized as adjoint functors?

Last week Yan Zhang asked me the following: is there a way to realize vector spaces as categories so that adjoint functors between pairs of vector spaces become adjoint linear operators in the usual ...
27
votes
4answers
2k views

Does a scheme have a “separification”?

Background: (1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful ...
19
votes
2answers
1k views

Is every functor a composition of adjoint functors?

My understanding of Ben's answer to this question is that even though associated graded is not an adjoint functor, it's not too bad because it is a composition of a right adjoint and a left adjoint. ...