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Questions tagged [adjoint-functors]

For questions about adjoint functors from category theory. Use in conjunction with the tag (category-theory).

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Categories internal to the category of convex sets

Let $\text{Conv}$ be the category of convex sets, as described by Proposition 1 of Jacobs. I want to understand the nature of categories internal to Conv. Any concrete description would be useful, but ...
Richard Southwell's user avatar
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Right adjoint of the forgetful functor from involution algebras

This question comes from Harold Simmons Introduction to Category Theory. The author defines an involution algebra to be a set together with an involution. We will write the involution $a^\circ$, so $\...
stillconfused's user avatar
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How much does "Set" determine "Cat"? (To what extent are "reflective supercategories" unique?)

Definition: Given a category $C$, call a category $A$ (if it exists) ("$A$" for "ambient") a reflective supercategory of $C$ if (1) there is a functor $\iota: C \to A$ that is ...
hasManyStupidQuestions's user avatar
2 votes
1 answer
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What is wrong with the following proof of the Adjoint Functor Theorem?

I'm reading about (versions of) the adjoint functor theorem and in particular looking at total categories. As I understand it, there is a theorem which says if $\mathcal{C}, \mathcal{D}$ are locally ...
ham_ham01's user avatar
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Show that if $G$ is the right adjoint of restriction-of-scalars along any ring homomorphisms $A\to B$, then $G$ is always a monomorphism

Let $A$ and $B$ be rings. Show that the right adjoint of restriction-of-scalars functor along any ring homomorphism $f: A \to B$ preserves injectives and show that the unit of this adjunction is ...
Squirrel-Power's user avatar
3 votes
1 answer
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Why does this specific adjoint chain come up for different categories?

Consider the three categories $\mathbf{Top}$ (of topological spaces with continuous maps), $\mathbf{Graph}$ (of simple graphs with maps that preserve or contract edges) and $\mathbf{Pros}$ (of pre-...
Tim Seifert's user avatar
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Left adjoint of an additive functor between triangulated categories that commute with shift

Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, ...
Snake Eyes's user avatar
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Are right-adjoints of a forgetful functor reflectors?

From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them ...
chickenNinja123's user avatar
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Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
Squirrel-Power's user avatar
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Let $M$ be an $A,B$-bimodule, $C$ ring. The functor $\text{Hom}_A(-,M)$, from the opp category of $A,C$-bimodules to $C,B$-bimodules, has left adjoint

Let $M$ be an $A,B$-bimodule and $C$ a ring. Show that the functor $\text{Hom}_A(-,M)$, from the opposite category of $A,C$-bimodules to $C,B$-bimodules, is a right adjoint. [Hint: its left adjoint ...
Squirrel-Power's user avatar
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Waldhausen's $S.$ construction as an adjoint

I am studying K-theory with Weibel's K-book and have just read the definition of the $S.$ construction for Waldhausen categories. I also recently watched a series of talks by Thomas Nikolaus which ...
DevVorb's user avatar
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Let $R$ be a commutative ring and $A,B,C$ be $R$-modules, $A$ finitely presented, $C$ flat. Then Hom$(A,B\otimes C)\cong\text{Hom}(A,B)\otimes C$

Let $R$ be a commutative ring with $A$, $B$, and $C$ all $R$--modules. Suppose that $A$ is finitely presented and $C$ is flat (that is, the functor ${-} \otimes C$ preserves short exact sequences). ...
Squirrel-Power's user avatar
7 votes
2 answers
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What's the point of naturality in the definition of an adjunction?

Two functors $F: 𝓒→𝓓$ and $G: 𝓓→𝓒$ are adjoint iff there is a family of isomorphisms \begin{align} η_{X, A}: 𝓓(FX, A)\tilde{\to} 𝓒(X, GA) \end{align} which is natural in $X$ and $A$ (i.e., $�...
Lukas Juhrich's user avatar
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Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
strat's user avatar
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On the right adjoint of the derived pushforward of a proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(...
strat's user avatar
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If $A$ and $B$ are dualizable objects in a monoidal category, is the unit of the one duality the inverse of the counit of the other duality?

I'm currently trying to wrap my head around dualizable objects in monoidal categories and I was wondering whether the following claim holds: Let $A$ and $B$ be dualizable objects in a monoidal ...
user11718766's user avatar
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3 answers
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Intuition behind 'adjointness' of adjoint functor for a lay person

For quite some time I, as a beginner in Category Theory, have been pondering about the intuition behind defining adjoint functors. I've gone through the other answers addressing this, but they seemed ...
Avi123's user avatar
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Why does duality of objects $A$, $A^\ast$ in a symmetric monoidal category imply an adjunction $(-) \otimes A \dashv (-) \otimes A^\ast$?

Let $\mathcal{C}$ be a symmetric monoidal category and let $A$ and $A^*$ be dual in the sense of Definition 2.1 in nLab. Dold & Puppe (1984) show (Thm 1.3) that the map $$ \text{Hom}(X, Y \otimes ...
user11718766's user avatar
1 vote
1 answer
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Concrete functors with left adjoint

Let $R$ be a commutative ring. The category $R$-Mod of modules over $R$ has a natural faithful functor into Set sending a $R$-module to its underlying set and a $R$-module morphism to its underlying ...
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Why is $\mathcal{C}$ equivalent to $\mathcal{C}^{\text{op}}$ when $\mathcal{C}$ is a compact category?

I came across the statement that a compact closed category $\mathcal{C}$ is equivalent to its dual category $\mathcal{C}^{\text{op}}$ (see e.g. this StackExchange post). This fact seems to be regarded ...
user11718766's user avatar
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1 answer
100 views

Forgetful functor $Z(C)\rightarrow C$ has Left Adjoint

The monoidal center $Z(C)$ of a monoidal category $C$ comes with a forgetful functor $F:Z(C)\rightarrow C$ defined $Z(X,\phi)=X.$ Does $F$ always admit a left adjoint? This is known (Section 3.2.) if $...
Peter's user avatar
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Left adjoint of an embedding

Let R be be an embedding functor from C to D, that is R is injective-on-objects and faithful, and let L be its left adjoint, is there some property of L, and a good name for it, which R being an ...
Ilk's user avatar
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1 answer
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Understanding the Difference Between Adjoint and Inverse Functors

I found myself a bit puzzled over the distinction between adjoint functors and inverse functors. They both seem to involve relationships between functors and categories, but I'm not clear on how they ...
whymgang's user avatar
3 votes
1 answer
103 views

Adjunction between convex spaces and Lawvere metric spaces

In Section 1.21 Meng describes a functor $\varphi$ from the category of convex spaces, $\text{Conv},$ to the category of Lawvere metric spaces, cost-Cat. Here cost-Cat can also be thought of as the ...
Richard Southwell's user avatar
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1 answer
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Existence of periodic adjunctions

Given any positive integer $n$, do there exist functors (endofunctors if $n$ is odd) $F_1, F_2, ..., F_n$ for which $F_1 \dashv F_2 \dashv ... \dashv F_n \dashv F_1$, but none of the $F_i$s (with ...
Geoffrey Trang's user avatar
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1 answer
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Category Theory: Natural Isomorphism of Bifunctors vs Argumentwise Natural Isomorphism

I'm inexperienced with category theory, and studying adjoint functors using the hom-functor definition. This has led me to a very basic question about bifunctors. Given bifunctors $F,G: \mathcal{C×D} \...
C V Astley's user avatar
1 vote
1 answer
81 views

Prove that $F$ is left adjoint to $G$ iff $G$ is right adjoint to $F$.

I started the definition of adjoint functor using universal morphisms. A functor $F:\mathcal C\to \mathcal D$ is a left adjoint functor if given any object $Y\in D$, there is an object $G(Y)$ in $\...
William's user avatar
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2 votes
2 answers
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Constructing a left adjoint to the restriction functor $U:\mathsf{C}^{\mathsf{J}}\to\mathsf{C}^{\operatorname{ob}\mathsf{J}}$

$\def\Lan{\operatorname{Lan}} \def\J{\mathsf{J}} \def\C{\mathsf{C}} \def\ob{\operatorname{ob}}$I am working through Exercise 5.5.v of Riehl's Category Theory in Context. The other question previously ...
Elías Guisado Villalgordo's user avatar
3 votes
1 answer
64 views

Left adjoint to forgetful functor from groups to groupoids, generalizing injective inclusions to free product of groups

Is there a left adjoint $F$ to the "forgetful" inclusion functor $U$ from the category of groups (interpreted as groupoids with one object $*$) to the category of groupoids? If so, then ...
I Eat Groups's user avatar
1 vote
0 answers
87 views

To prove two functors are left and right adjoint

In chapter 3 of the book the book Representations of $SL_2(\mathbb{F}_q)$ by Cédric Bonnafé: Harish-Chandra Induction, for two finite groups $\Gamma$, and $\Gamma'$, and $M$ a $(K[\Gamma],K[\Gamma'])$...
Damn it My Foot's user avatar
2 votes
2 answers
84 views

Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.

I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words): Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
user11718766's user avatar
2 votes
0 answers
80 views

Left and right adjoints of functor category inclusion

Question. Let $i:\mathcal{C}\hookrightarrow\mathcal{D}$ be a full subcategory. Assume we are given a cocomplete category $\mathcal{A}$. Show that the induced pre-composition functor $i^*:\mathrm{Fun}(\...
Robert's user avatar
  • 530
3 votes
2 answers
73 views

Examples of adjoint 2-functors

I hope this is not too vague, but I'm learning about 2-dimensional categories and I would like more examples of adjoint 2-functors to study. Could somebody tell me some interesting 2-adjunctions ? ...
Richard Southwell's user avatar
0 votes
1 answer
35 views

Two-variable adjunction

In Riehl's Category Theory in Context, the following result has proof left as an exercise: Proposition 4.3.6. Suppose that $F: \mathsf{A} \times \mathsf{B} \rightarrow \mathsf{C}$ is a bifunctor so ...
Elías Guisado Villalgordo's user avatar
0 votes
1 answer
58 views

Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits

I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
love and light's user avatar
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0 answers
40 views

Left and right exactness of adjoint functors

I am trying to find a proof of the following statement: Let $A$ and $B$ be abelian categories, and let $(F, G)$ be an adjoint pair of additive functors $F : A → B$ and $G: B → A$. Show that $F$ is ...
mNugget's user avatar
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4 votes
2 answers
114 views

Proving naturality in both arguments

I am having a hard time showing that two functors are adjoint. Consider as an example \begin{align} t&: \mathbf{Ab} \longrightarrow \mathbf{T},\\ i&: \mathbf{T} \longrightarrow \mathbf{AB}. \...
mNugget's user avatar
  • 493
3 votes
0 answers
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Dual Concept of a Well-Powered Category

I was studying the SAFT theorem (Special Adjoint Functor Theorem), using Leinster's Basic Category Theory and I had the homework to dualize it. So far I have the following, Consider a category $\...
babu's user avatar
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1 vote
0 answers
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$F : C \to D$ is an equivalence of categories with quasi-inverse $G: D \to C$. Does the pair $(F,G)$ necessarily form an adjunction?

Suppose $F : C \to D$ is an equivalence of categories with quasi-inverse $G: D \to C$. Does the pair $(F,G)$ necessarily form an adjunction? How would I prove this? I think I can use the unit-counit ...
some_math_guy's user avatar
1 vote
1 answer
54 views

Proving the existance and uniqueness of the unit of an adjunction

Let $L: C \to D$ be left adjoint to the functor $R: D \to C.$ For an object $X$ of $C$, we have an isomorphism $Hom_D(L(X),L(X)) \xrightarrow{\sim} Hom_C(X,(R\circ L)X)$. Show that there exists a ...
some_math_guy's user avatar
1 vote
1 answer
86 views

On the adjointness of the inclusion and the zero-homology functor

I am trying to get a better understanding of homology from a Category Theory perspective. Apparently, the following is a well-known fact: Fact. Let $R$ be a ring. Then, the inclusion functor $i: _{R}...
PDEsperate's user avatar
0 votes
2 answers
87 views

'Uniqueness' of adjoint functors

Let $F:\mathcal{A}\to\mathcal{B}$ be a left-adjoint functor. What 'choices' need to be made in order to construct a right-adjoint $G:\mathcal{B}\to\mathcal{A}$ for $F$ and a natural isomorphism $$\...
user829347's user avatar
  • 3,422
3 votes
1 answer
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Condition for a symmetric monoidal category to be closed

Let $\mathcal{C}$ be a symmetric monoidal category i.e. a category with a binary functor $\otimes$, a distinguished object $I$, and distinguished isomorphisms $A\otimes B\to B\otimes A$, $$(A\otimes B)...
user829347's user avatar
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3 votes
1 answer
63 views

Is $p^{-1}\mathcal F = p^!\mathcal F$ if $p: Y \to X$ is a covering map?

Suppose $p: X \to Y$ is an unramified (possibly infinite) covering map of complex manifolds. The functor $p^!: D^b(Y) \to D^b(X)$ is supposed to be the right-adjoint of $R p_!:D^b(X) \to D^b(Y)$, ...
red_trumpet's user avatar
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1 vote
1 answer
67 views

Proving bijectivity of the isomorphism when finding the right adjoint of functor F: Haus $\rightarrow$ to Top

Right adjoint of functor F: Haus $\rightarrow$ to Top Let $X$ be a topological space. We introduce a relation $\sim$ on X by declaring $x \sim y$ if for every continuous map $f : X \rightarrow H$ with ...
darkside's user avatar
  • 637
2 votes
1 answer
121 views

inclusion functor Haus $\rightarrow$ Top does not have a right adjoint

Let $X$ be a topological space. We introduce a relation $\sim$ on X by declaring $x \sim y$ if for every continuous map $f : X \rightarrow H$ with $H$ a Hausdorff space we have $f (x) = f (y)$. I have ...
darkside's user avatar
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1 vote
0 answers
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Categorical notion of finiteness

TLWR : I'd like to define either $\operatorname{Hom}_F : \text{Set} \times \text{Set}_* \to \text{Set}_*$ (set of functions with finite support) or simply $\mathcal{P}_F : \text{Set} \to \text{Set}$ (...
B. Pillet's user avatar
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0 answers
24 views

Is the James reduced the product the right adjoint to an appropriate forgetful functor

While reading Hatcher's algebraic topology book, the construction $J(X)$ is refered to as the "freest" H-space structure on a topological space $X$. I wanted to solidify this intuiton by ...
DevVorb's user avatar
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1 vote
0 answers
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Is there a concept of a "free Hilbert space on a topological space"?

We know that if you have a map $f: A \to H $ from a set $A$ to the unit sphere of a Hilbert space $H$, you can get a bounded linear map $\tilde{f}: L^2(A) \to H$. This construction can be view as ...
Chen's user avatar
  • 69
4 votes
1 answer
308 views

Left adjoint to the forgetful functor from Hilbert spaces to topological spaces

I'm trying to generalize a method which can solve a problem involving discrete group to continuous group. And the method involve construct a "free vector space" from discrete space. So I'm ...
Chen's user avatar
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