The adjoint-functors tag has no wiki summary.
13
votes
7answers
765 views
Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?
In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...
14
votes
1answer
433 views
Lurie's approach to the bar-cobar adjunction
I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...
3
votes
0answers
164 views
Map of adjunctions
The following question must have been asked dozens of times, but I do not recall any non-trivial results.
Consider an adjoint square where the arrows indicate directions of $F, G, H, K$.
...
3
votes
1answer
172 views
Functoriality of the adjoint functor construction?
Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
6
votes
2answers
195 views
Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
3
votes
0answers
69 views
Cofree Lie Coalgebra
I have problems finding anything about the cofree Lie coalgebra functor
$\mathcal{L}ie^c$ out there.
Basically all I found was that it appears in Harrison cohomology and that,
given a ...
3
votes
1answer
183 views
When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the ...
1
vote
1answer
115 views
Are the pullback functors of adjoint functors also adjoint?
Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by pre-composition, are these two also adjoint ...
1
vote
1answer
284 views
Category which has no non-trivial adjoint functors
Does there exist a category C which such that there is no functor $F:C \rightarrow D$ with $D\not\cong C$ which has a left (or right) adjoint?
7
votes
1answer
236 views
The category of categories and adjunctions
What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
3
votes
1answer
138 views
How to construct a free 2-group on a groupoid?
Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor ...
-1
votes
2answers
181 views
Basic category theory: Universality of adjunction unit is justified by Yoneda Proposition in Mac Lane's text [closed]
Background
I am reviewing some category theory, which I did not learn too well the first time around. One text I am using is Mac Lane's. Near the beginning of the chapter on adjunctions (pg 80),
he ...
2
votes
1answer
101 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
299 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
212 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
173 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
440 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
238 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
413 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
156 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 ...
3
votes
2answers
319 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 ...
7
votes
0answers
303 views
bar-cobar or cobar-bar
What is the standard or best reference for the adjointnes of bar and cobar constructions?
2
votes
1answer
160 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
323 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
170 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
197 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
306 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
407 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
1answer
310 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
239 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 ...
10
votes
1answer
404 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 ...
24
votes
2answers
824 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 ...
2
votes
1answer
667 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
1answer
499 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 ...
1
vote
1answer
538 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
332 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
350 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
485 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
541 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
786 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
681 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. ...
26
votes
5answers
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
743 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
606 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 ...
4
votes
4answers
781 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
429 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 ...
3
votes
1answer
443 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
872 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
581 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 ...
