The adjoint-functors tag has no wiki summary.

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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, ...

**27**

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**4**answers

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### 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 ...

**26**

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**5**answers

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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 ...

**22**

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**2**answers

740 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 ...

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**2**answers

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### 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.
...

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**2**answers

942 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 ...

**17**

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**4**answers

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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 ...

**17**

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**4**answers

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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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**2**answers

963 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 ...

**11**

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### 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 ...

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**3**answers

567 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 ...

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474 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 ...

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**3**answers

529 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 ...

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**3**answers

384 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?

**10**

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**1**answer

379 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 ...

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972 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 ...

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**4**answers

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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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**1**answer

290 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 ...

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**2**answers

729 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 ...

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285 views

### bar-cobar or cobar-bar

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

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**1**answer

193 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 ...

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279 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 $F\dashv G$ ...

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**1**answer

286 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 ...

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**1**answer

366 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 ...

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286 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 ...

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**4**answers

722 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 ...

**4**

votes

**2**answers

435 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 ...

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**1**answer

314 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 ...

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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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### 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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**1**answer

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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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**2**answers

295 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 ...

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**1**answer

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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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### “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 ...

**3**

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**1**answer

125 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 ...

**3**

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**1**answer

226 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 ...

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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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**1**answer

759 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 ...

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329 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?

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**1**answer

188 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 ...

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**1**answer

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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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**1**answer

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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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507 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 ...

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**1**answer

150 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 ...

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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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203 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 ...

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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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### 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 ...

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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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### 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 ...