Questions tagged [adjoint-functors]

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

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When representables are adjoints

Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits. In general representables preserve limits, but the hypothesis ...
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What is the right adjoint to the functor $\sf{Psh}\to\sf{Set}$ which evaluates the presheaf on the whole space?

$\newcommand{\O}{\mathcal{O}}\newcommand{\T}{\mathcal{T}}\newcommand{\op}{^{\sf{op}}}\newcommand{\set}{\sf{Set}}\newcommand{\ps}{\sf{Psh}_{\T}}$Let $\T$ be a topological space and $\O(\T)$ the poset ...
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Why is a closed monidal category enriched over itself?

Let $M$ be a left-closed monidal category (Assume more such as symmetry if needed). Let $i$ be the unity object of $M$. Let $\alpha,\lambda,\rho$ be the associator, left unitor, right unitor of $M$, ...
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2 votes
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2-categorical universal property of the classifying category of a type theory

For example let us say we are in the setting of cartesian closed categories and the simply typed $\lambda$-calculus. Let $\mathtt{strCCCat}$ denote the $2$-category of strict cartesian closed ...
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Is my proof that an equivalence of categories induced and adjunction correct?

$\newcommand{\A}{\mathscr{A}}\newcommand{\B}{\mathscr{B}}$This is a quick question - my proof is very short, but I’m suspicious of it. Suppose the categories $\A,\B$ are equivalent via the functors $...
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How do I show that, if $fp=fq$ then $p=q$, in the context of $f$ being the counit of an adjunction and $p,q$ natural transformations?

$\newcommand{\A}{\mathscr{A}}\newcommand{\B}{\mathscr{B}}\newcommand{\I}{\mathscr{I}}\require{AMScd}$The titular statement is probably false, but it was the only way I could see to fit my question ...
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Show that $(L, \Delta)$ is an adjunction.

Consider a category $\mathcal{C}$ with binary coproducts, and the following functors: The coproduct functor $L: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ defined by $L\left(C, C^{\prime}...
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The difference between totally (large) cocontinuous functors and small cocontinuous functors

$\newcommand{\cat}{\mathbf}\newcommand{\op}{\mathrm{op}}\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\cSet}{\cat{Set}}$A category $\cat C$ is total if the Yoneda embedding $\cat C→[\cat C^{\op},\...
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Terminology: "One-sided adjoint equivalence"

1. Context One page 31 of Weakly distributive categories Cockett and Seely define tensor inverses in monoidal categories. Let me unpack their definition in different language: Let $(C, \otimes, I, a, ...
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Does $\mathrm{Fun}$ preserve adjunctions?

Let $u: A \to B$ be a left adjoint functor with right adjoint $v: B \to A$ and let $C$ be a further category. Is it true that $\mathrm{Fun}(C, u)$ is left adjoint to $\mathrm{Fun}(C, v)$? Similar ...
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Examples of a category without duals?

This may be a silly question, but are there any examples of categories that don't have duals (ie, neither for objects nor arrows)? I'm currently under the impression that it's impossible to not have ...
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Diagonal functor is the right-adjoint of the coproduct functor

I am starting to study category theory and I have to prove that the diagonal functor is the right-adjoint of the coproduct functor. I would like to write this using the Hom-set definition adjunction. ...
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$(X^Y)^Z\cong X^{(Y+Z)}$ or $(X^Y)^Z\cong X^{(Y\times Z)}$?

$\DeclareMathOperator\Hom{Hom}$I have the following exercise in my class of Category Theory: Prove that $\text{Hom}(Z,\Hom(Y,X))\cong \Hom(Y*Z, X)$ but I am not sure what $*$ is. I think that $*$ ...
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Left adjoint to forgetful functor from topological rings to topological abelian groups

Does the forgetful functor from the category of topological rings to the category of topological abelian groups, $U: \mathbf{TopRing} \to \mathbf{TopAb}$, have a left adjoint and if so, what is it? A ...
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Sheafification of presheaves with values in an abelian category

Let $\mathcal{A}$ be an abelian category, and let $X$ be a topological space. Can we always define a sheafification for any $\mathcal{A}$-valued presheaf over $X$? More precisely: let $\mathsf{PSh}_\...
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When does a representable functor have a right adjoint?

In Wikipedia I saw the result that, when category $\mathcal{C}$ has all small copowers, a functor $\mathcal{C}\overset{K}{\rightarrow}\text{Set}$ has a left adjoint if and only if it is representable. ...
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Does dimensional reduction of TQFTs have an adjoint?

If $Cob_n$ is $n$-dimensional cobordisms, and $Z \colon Cob_n \to \mathcal{V}$ is an $n$-dimensional TQFT (i.e. a symmetric monoidal functor - in particular, $\mathcal{V}$ is symmetric), then one can ...
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Is there a Hom-isomorphism-like representation of adjoint functors that does not duplicate the data of the functor maps?

I'm toying with adjunctions in a theorem prover, and searching for representations that are convenient for my specific task. One thing I dislike about the Hom-isomorphism representation is that a ...
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Is there a connection between free–forgetful adjunctions and tensor-hom adjunctions?

In the Wiki article on adjunction https://en.wikipedia.org/wiki/Adjoint_functors, there is a motivation section that talks about how adjunctions can be viewed as "Solutions to optimization ...
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CWM, Characterization of adjoints via limits

On page 234 of Categories for the working mathematician, Mr. Saunders Mac Lane says followings: Theorem 2. A functor $G:A\rightarrow X$ has a left adjoint if and only if both (1) $G$ preserves all ...
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From monads to comonads by the calculus of mates

If a comonad $D$ is left adjoint to an endofunctor $T$, then $T$ can be made into a monad: its unit and multiplication are given respectively by the mates of the counit and comultiplication of $D$. ...
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Adjunctions b/w constant diagram functor and limit/colimit functors for fixed index category

Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for ...
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1 answer
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Relation between right adjoints and counits

Assume that $F: \mathcal{C} \rightarrow \mathcal{D}$ is left adjoint to $G: \mathcal{D} \rightarrow \mathcal{C}$ with counit $\varepsilon: F G \Rightarrow$ id $_{\mathcal{D}}$ and unit $\eta$ : id $_{\...
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Group objects in category of pointed topological spaces?

A pointed space is a (non-empty) topological space with a choice of one of its points. Together with pointed spaces and continuous maps preserving base points is called the category of pointed spaces ...
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When is pulling back along an algebra morphism right adjoint to "scalar extension"?

Let $\mathfrak{M}$ be an arbitrary monoidal category, and let $A, B$ be algebras therein, together with an algebra morphism $f \colon A \to B$. The algebra morphism always induces a pullback functor, ...
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If monads come from adjunctions what do graded monads come from?

Basically can you generalize adjunction to "graded adjunction?" I have a hunch you would end up with something similar to some linear logic stuff. My thoughts are you can use monads for ...
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2 votes
1 answer
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When does transporting a monad along an adjunction "preserve" its category of modules?

If I have an adjunction $F \dashv G$ where $F \colon C \to D$, and $(N, m, u)$ is a monad on $D$, then I can define a monad $\widetilde{N}$ on $C$ via \begin{align} (GNF, \widetilde{m} = G m_F \...
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Right adjoints to forgetful functors from TopRing and TopGrp to Top

I believe the forgetful functor $F: \mathbf{TopGrp} \to \mathbf{Top}$ has a left adjoint. Does it also have a right adjoint? Does the forgetful functor $G: \mathbf{TopRing} \to \mathbf{Top}$ have a ...
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  • 139
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2 answers
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Mutual Left/Right Adjoints and the Triangle Identities

This has to do with definition 4.3.1 and exercise 4.3.i in Rhiel's "Categories Theory in Context". I am trying to determine the triangle identities for a pair of mutually left (or right) ...
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1 vote
1 answer
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$F$ left adjoint to $G \iff F,G$ define a functor from $\textbf{Arr}(\textbf{X}\times\textbf{A}) \to 2\times 1$ square CDs in $\textbf{Set}$?

Let $\textbf{A, X}$ be categories and $F:\textbf{X} \to \textbf{A}$ and $G: \textbf{A} \to \textbf{X}$. Then there is a map that takes an object in $\text{Arr}(\textbf{X}\times\textbf{A})$ (the arrow ...
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1 vote
2 answers
105 views

How do you derive the adjoint's "naturality" condition as seen in MacLane & Moerdijk's book "Sheaves in Geometry and Logic"?

It's tag (7) as pictured below. I also included the definition of adjoint that they use. I know that by definition of adjunction (using the natural homset isomorphism), we have two naturality ...
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4 votes
1 answer
166 views

Constructing counit in adjoint functor theorem for total categories

The theorem I am referring to is, Let $C,$ $D$ be locally small categories. Assume $C$ is a total category (i.e. the Yoneda functor $Y : C \to \operatorname{PreSh}(C)$ has a left adjoint $Y^L$). Let $...
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Localization of cocomplete categories and right orthogonality: does the equivalence always hold?

In Handbook of Categorical Algebra, Volume 1: Basic category theory, Borceux proves the following (around theorem 5.4.7 page 198). Definition. An object $x$ of a category $\newcommand{\cC}{\mathsf{C}}\...
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In what sense is the uniqueness of left adjoint isomorphism 'canonical'

In my category theory course, Peter Johnstone has written that for any two left adjoints $F$, $F'$ "there is a canonical natural isomorphism $F \to F'$" Explicitly, this isomorphism is that ...
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2 votes
1 answer
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Adjunction in Abelian Categories

Let $C,D$ be arbitrary categories and $F:C\to D, G:D\to C$ functors. We say $F,G$ are adjoint if for every $X\in C, Y \in D$ there is an isomorphism of Hom-Sets between $Hom_D(FX,Y)\cong Hom_C(X,GY)$,...
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1 vote
1 answer
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An adjunction between two preorders

Let $\mathcal{C}$ and $\mathcal{D}$ be two preorders, $L: \mathcal{C} \to \mathcal{D}$, $R: \mathcal{D} \to \mathcal{C}$ a pair of functors such that for all objects $c \in \mathcal{C}$ and $d \in \...
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0 votes
1 answer
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Free group construction via General Adjoint Functor Theorem in Riehl

In Emily Riehl, "Category theory in Context", Section 4.6: "Existence of adjoint functors", the free group on a set $S$ is constructed by specializing the proof of the general ...
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6 votes
1 answer
165 views

Exercise 1.5.xi from Emily Riehl's "Category Theory in Context" on properties of some functors

I've been going through Emily Riehl's textbook on categories, and struggle with the exercise 1.5.xi. Consider the functors $Ab \to Grp$ (inclusion), $Ring \to Ab$ (forgetting the multiplication), $(-)...
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Tensor product of adjoint maps

Let $V$ be a symmetric monoidal category and $D$ a monoidal category. Assume that $D$ is $V$-enriched with an action $V*D\to D$ satisfying the usual axioms and hom-objects $\underline{D}(X,Y)\in V$ ...
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1 vote
1 answer
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Category theory in Context, Proposition 4.4.5: Idemotence when Promoting equivalence to adjunction

Begin with an equivalence $F: C \to D$, $G: D \to C$ along with natural isomorphisms_ to the identities, so $\eta: 1_C \simeq GF$ and $\epsilon: FG \simeq 1_D$. The claim is that we can replace $\...
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1 vote
1 answer
59 views

Category Theory in Context, Proposition 4.4.5: Promoting equivalence to adjunction

Begin with an equivalence $F: C \to D$, $G: D \to C$ along with natural isomorphisms_ to the identities, so $\eta: 1_C \simeq GF$ and $\epsilon: FG \simeq 1_D$. The claim is that we can replace $\...
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1 vote
1 answer
65 views

Adjoint Functors: Unit is Right Invertible?

$\DeclareMathOperator{\Hom}{Hom}$ Suppose there is an adjunction $F: C \to D$, $G: D \to C$, $F \dashv G$. Let $\eta: 1_C \implies GF$ be the unit of the adjunction. Suppose there is a natural ...
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0 answers
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Adjoint for real functions

We want to find sufficient and necessary conditions for $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ to be adjoints with the usual order in $\mathbb{R}$, i.e $$f(x)\leq ...
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7 votes
1 answer
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Does the inclusion of schemes affine over $S$ to $S$-schemes have a left adjoint?

Let $S$ be a scheme. Consider the category $\mathrm{Aff}_{/S}$ of schemes affine over $S$, by which I mean $S$-schemes $X$ such that the structure morphism $X \to S$ is affine. For simplicity, I'll ...
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1 vote
1 answer
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Adjoints for the restriction of category-valued representations of groups

Setup. Let $G$ be a group and let $\mathscr{A}$ be a category. We denote the category of functors from $G$ to $\mathscr{A}$ by $[G, \mathscr{A}]$ and think of these functors as $\mathscr{A}$-valued ...
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2 votes
2 answers
138 views

How to *find* an adjoint functor?

Let $U\colon C\to D$ be a functor. How do people go about finding a left adjoint of $U$? Are there special techniques for that? I only know necessary conditions for the mere existence: right adjoints ...
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Adjoint functors between Set and Rel

Are there any particularly well known adjoint functors between the category Set and the category Rel, of relations ? If so what are they ?
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How is the pair of adjoint functors defined in this ncat article?

I'm trying to read the article about spaces and quantities on ncat. I find it very interesting, but I'm afraid I can't follow how technically they are defining the functors – either because it's ...
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1 vote
0 answers
58 views

Can we control the "degree" of accessibility of right adjoints between presentable $\infty$-Categories? (HTT 5.4.7.7)

Suppose $g: D \to C$ is a right adjoint between presentable $\infty$-categories. Then by the adjoint functor theorem, $g$ is accessible, i.e. there is a regular cardinal $\kappa$ such that both $C, D$ ...
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3 votes
1 answer
61 views

Clarifying the concept of the unit of an adjunction

$\newcommand{\Hom}{\operatorname{Hom}}\newcommand{\iso}{\texttt{iso}}\newcommand{\id}{\operatorname{id}}$ I'm reading Emily Riehl's "Category Theory in Context", Section 4.2, "The unit ...
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