The adjoint-functors tag has no wiki summary.
2
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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
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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
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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
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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
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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
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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
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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
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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
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1answer
542 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
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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
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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
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2answers
702 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. ...
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4answers
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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 ...
4
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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
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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
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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 ...
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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
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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
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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
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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 ...
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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 ...
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5answers
931 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
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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 ...
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5answers
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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 ...
11
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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
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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
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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, ...
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2answers
922 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 ...
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4answers
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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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2answers
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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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6answers
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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 ...
