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9
votes
5answers
1k views

Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

I am looking for a counter-example of two functors F : C -> D and G : D->C such that 1) F is left adjoint to G 2) F is right adjoint to G 3) F is not an equivalence (ie F is not a quasi-inverse of ...
2
votes
3answers
514 views

How to construct pair of adjoint functors from category A to category A_D(category of diagrams)

I wonder whether following statements holds If A is an abelian category(or quasi abelian category) having enough projectives, then category of pointed diagram(which means diagram has final object,or ...
19
votes
2answers
955 views

When and why do universal objects have extra properties?

I'm interested in situations where universal objects come with more structure than their definitions suggest. A classic case of this is where the free abelian group on one element has a ring ...
28
votes
5answers
3k views

What is an intuitive view of adjoints? (version 1: category theory)

In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is. I know the definition (several ...
11
votes
3answers
530 views

Can different modules have the same symmetric algebra? (answered: no)

Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$. I'm wondering if anything is lost in just ...
3
votes
7answers
563 views

What is the “right” definition of the free abelian group on a set?

One way to define the free abelian group on a set $S$ is as $F(S) := \mbox{Hom}_{\text{Set}}(S, \mathbb{Z})$ with the group operation coming by composition with the map $\mathbb{Z} \times \mathbb{Z} ...
55
votes
9answers
6k 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
995 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
2k 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
445 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. ...