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

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