# Tagged Questions

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### How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
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### Is there an explicit construction of a free coalgebra?

I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the ...
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### Proving the Special Adjoint Functor Theorem from the general Adjoint Functor Theorem

Often, when dealing with adjoint functor theorems, people go about proving each one separately, from first principles if you will (this is the course taken in MacLane). However, the names suggest ...
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My first question here. Accordingly to M. Barr "Coequalizers and free triples" by a free triple (or free monad) generated by an endofunctor $R: X\rightarrow{X}$ we mean a triple $T=(T,\eta,\nu)$ and ...
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### Kan extensions and the yoneda embedding.

[Edit: For a category $C$ let $C^\wedge$ denote the category of presheaves on $C$.] Let $f:C\to D$ be a functor. Precomposition with $f^{op}$ induces a functor $f^\wedge:D^\wedge \to C^\wedge$. This ...
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### Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint?

(Alternate title: Find the Adjoint: Hopf Algebra edition) I was chatting with Jonah about his question Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$. It's very closely related ...
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### Does every cocontinuous functor between categories of presheaves on small categories have a right adjoint?

Let C, E be small categories, let Ĉ = SetCop, and let F:Ĉ → Ê be cocontinuous. I think F will always have a right adjoint when C, E are small, but not necessarily if they'...
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### “adjoint” =?= “inverse of composite endofunctor is uniform bi-composition”

Understanding adjoints has always been (and continues to be) a bit of a struggle for me. Today I stumbled upon a property of adjoint functors which seemed extremely intuitive to me. I was wondering ...
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Was wondering if someone could help. I am just about getting to grips with cateogry theory and adjoints. I have a query though, reading through some material. Does the forgetful functor from the ...
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Let $F_0 : C \to D$ be a functor. If it exists, let $G_0 : D \to C$ be its left adjoint. If it exists, let $F_1 : C \to D$ be its left adjoint. And so forth. In situations where the infinite ...
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### 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 ...
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### 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 ...
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### 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 structure....
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### 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 ...
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### 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 ...
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### 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} \...
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### 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, ...
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(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 ...
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### 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 (...
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### 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 ...
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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 ...