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2
votes
1answer
80 views

Simple technical adjunction question

Suppose $F,G$ are adjoint, and $\epsilon:F\circ G\rightarrow Id$ is the counit. Is it always true that$$ Id_{FG}\epsilon=\epsilon Id_{FG} $$ as maps from $FGFG$ to $FG$? It's true if you precompose ...
5
votes
1answer
276 views

Not quite adjoint functors

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
2
votes
1answer
174 views

A Criterion for a morphism to be a counit of an Adjunction

Suppose we have two functors $F:C\leftrightarrow D:G$ and a morphism $\varepsilon:FG\rightarrow\operatorname{Id}_D$. I am looking for a way to check whether $\varepsilon$ is the counit of an ...
-1
votes
1answer
153 views

irreducible Classical Lie algebras [closed]

which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D_4$ and $E_6$ are irreducible?
1
vote
3answers
378 views

Another adjoint pair: Definable sets and set-builder formulas

I see adjointness between the two concepts of "being a definable set" and "being a set-builder formula": A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : ...
3
votes
1answer
221 views

Categories where morphisms are pairs of adjoint functors

Is there a particular name for those categories (of categories) where morphisms "come as adjoint pairs", as in the case of toposes and model categories? Is there any attempt to study them as general ...
10
votes
3answers
349 views

Is there a monad on Set whose algebras are Tychonoff spaces?

Compact Hausdorff spaces are algebras of the ultrafilter monad on Set. Is the category of Tychonoff spaces also monadic over Set?
2
votes
1answer
142 views

admissible subcategories over non algebraically closed fields

Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that ...
4
votes
2answers
284 views

Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?

What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch? There is ...
6
votes
0answers
270 views

bar-cobar or cobar-bar

What is the standard or best reference for the adjointnes of bar and cobar constructions?
2
votes
1answer
143 views

Is there an analog of adjoint functor theorem for adjunctions of two variables?

Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection $$ \mathscr ...
5
votes
0answers
264 views

Constructing pointwise Kan extensions as adjoints to some functor

Background I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because ...
3
votes
0answers
123 views

Do generalizations of adjoint functors, such as adjunctions in 2-categories, and multivariable adjunctions, have formulations in terms of something like universal morphisms?

I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about how to go from the ...
1
vote
1answer
191 views

Can we characterize endofunctors which admit a monad structure?

In answering this MO question, the issue was raised of characterizing when a given endofunctor $R:C\to C$ has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, i.e. which admit a ...
7
votes
1answer
280 views

Is every functor inducing a homotopy equivalence a composition of adjoint functors?

It was asked here whether every functor is a composition of adjoint functors. The answer is no, because all adjoint functors induce homotopy equivalences on the nerve, and we can construct functors ...
5
votes
1answer
343 views

Adjoint Functors as Initial Objects of Some Category

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
6
votes
0answers
236 views

[Reference Request] The Definition of Adjoint Functors between dg-categories

Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair? In my mind $F\dashv G$ ...
3
votes
1answer
224 views

A Question on Auto-Adjoint functors

Let $F: \mathscr{C}\to \mathscr{C}^{op}$, with an adjoint $G$, and $\eta: 1_\mathscr{C} \Rightarrow G\circ F $ and $\varepsilon: F\circ G\Rightarrow 1_{\mathscr{C}^{op}}$ with components (in ...
9
votes
1answer
358 views

Fullness of pullback functor in algebraic geometry

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can ...
22
votes
2answers
710 views

Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...
1
vote
1answer
541 views

Adjunctions between derived functors

Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...
1
vote
2answers
488 views

Relating eigenvectors of two self-adjoints operators

Suppose I have a self-adjoint operator $\mathbf{L}$ which I seperate in two parts which are themselves self-adjoint. I write this in terms of their eigenvalues/eigenvectors: $\mathbf{v} \Lambda ...
0
votes
1answer
442 views

a “self-dual” adjunction

Is there a name for $(U,\eta)$ such that $(\eta, \eta^{op}):U^{op}\dashv U$ (is an adjunction). To clarify — $C:category$, $(I,I^{op})$ is the contravariant isomorphism with $I:C^{op}\to C$, ...
4
votes
1answer
294 views

On the barycentric subdivision of a poset

Hi everybody, I'm interested in the barycentric subdivision of a poset $P$, defined as the face poset of the corresponding order complex $\mathcal F(\Delta(P))$ (or alternatively, the poset of chains ...
2
votes
2answers
321 views

Reference request: 2-Monads and 2-Adjunctions

Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T-,-)\cong \mathcal C(-,\mathrm{For}-).$$ Is the same true for 2-monads on a 2-category?
11
votes
3answers
470 views

Is every saft category cocomplete?

Here is a word that I think should be adopted by the category theorists. (If there is another synonymous word already in existence, please let me know.) Definition: A category $C$ is saft if every ...
3
votes
1answer
476 views

Adjunctions: Algebras of the induced monad VS. Coalgebras of the induced comonad.

Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent? The example i have ...
7
votes
2answers
701 views

An elementary question about adjunctions between presheaf categories preserving pullbacks.

A functor $C \to D$ between categories induces a morphism of presheaf categories $Pre(D) \to Pre(C)$. This functor has a left adjoint given by left Kan extension and I am interested in knowing when ...
1
vote
1answer
629 views

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. ...
17
votes
4answers
2k views

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 ...
4
votes
0answers
662 views

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 ...
2
votes
2answers
560 views

Free monad or monad defined from an adjunction.

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 ...
2
votes
0answers
201 views

Are the only universal (co-)universal conditions (co-)limits?

Reading this title, you may have thought there was a typo, but there isn't (well I don't think there is at least!). This question arises from a definition I formulated recently, and would like to ...
4
votes
4answers
683 views

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 ...
17
votes
4answers
1k views

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 ...
1
vote
2answers
394 views

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 ...
2
votes
1answer
421 views

“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 ...
2
votes
1answer
740 views

Left adjoint Grp->Set??

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 ...
11
votes
3answers
554 views

Iterated adjoint functors

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 ...
8
votes
5answers
930 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
498 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
929 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 ...
25
votes
5answers
2k 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
528 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
555 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} ...
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
921 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
418 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 ...