The adjoint-functors tag has no usage guidance.

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### Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

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### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

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### About cartesian closure of lax.functors categories

Let $\mathscr{A}$ a category and $F, G, H: \mathscr{A}^{op}\to CAT$ lax.functors. I wish find a possible "natural correspondence" between categories: $[F\times G, H]_O \leftrightarrow [F, H^G]_O$ ...

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### Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...

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### 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, ...

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### 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 ...

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### 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 $F, G, H, K$.
...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 : ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### bar-cobar or cobar-bar

What is the standard or best reference for the adjointnes of bar and cobar constructions?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### [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 $F\dashv G$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 $I:C^{op}\to C$, ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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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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### 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 ...

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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 ...