# Questions tagged [adjoint-functors]

The adjoint-functors tag has no usage guidance.

184
questions

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### Symmetric monoidal structures on the functor taking presheaves

Let $\mathrm{Cat}_\infty$ be the $\infty$-category of small $\infty$-categories, $\mathrm{Pr}^\mathrm{L}$ be the $\infty$-category of presentable $\infty$-categories, and $\mathcal P(\mathcal C)$ be ...

3
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### Which spectra have a homotopy-universal connective quotient?

Prefatory remark: This is a repost of a previous question, to which Tyler Lawson supplied a lovely $\infty$-categorical answer. The example that motivated the question was specifically about the ...

2
votes

0
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### Cohesive structure of Cahiers and Dubuc topoi

The inclusion of commutative rings into supercommutative rings has two adjoints, one projecting out the even part and the other quotienting out the ideal generated by odd elements. After passing to ...

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1
answer

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### Which spectra have a universal connective quotient?

Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E_\infty$-group ...

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### Unit of the inverse/direct image sheaf adjunction in terms of étale spaces

$\def\sF{\mathcal{F}}
\def\sect{\operatorname{Sect}}$[I am reposting here this question from MSE, since thus far I received no answers there and maybe here I attract the attention of other people that ...

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### Adjunctions that are easier to prove in one direction

It is well known that there are (at least) $4$ equivalent characterizations of an adjunction:
An antiparallel pair of functors $F:\mathcal{C}\rightleftarrows\mathcal{D}:G$ together with a natural ...

5
votes

1
answer

199
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### Lift a monad along a generic right adjoint

$\require{AMScd}$We have a neat way to lift a monad along a monadic right adjoint, through a distributive law: in a setting like
$$
\begin{CD}
X @. X \\
@VUVV @VVUV\\
C @>>T> C
\end{CD}$$
if ...

8
votes

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### Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...

1
vote

1
answer

171
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### Sufficient condition for right exact functor to be a left adjoint

Disclaimer: I first tried to ask this question on stackexchange https://math.stackexchange.com/questions/4577320/sufficient-conditions-for-a-right-exact-functor-to-be-a-left-adjoint but I did not get ...

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votes

2
answers

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### Which functors preserve the number of connected components?

The categories $\mathbf{Top}$ of topological spaces, $\mathbf{sSet}$ of simplicial sets and $\mathbf{Cat}$ of small categories are all equipped with a functor $\pi_0$ into the category $\mathbf{Set}$ ...

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### When do geometric morphisms lead to periodic adjoints?

This may be a naïve question but I've been unable to locate a reference that addresses it. Any thoughts are appreciated!
Let $f:\mathcal{E}\to\mathcal{S}$ be a cohesive morphism of toposes. That is, ...

3
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1
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### If a monad in a 2-category admits a terminal resolution, does it admit an Eilenberg–Moore object?

Let $T = (t, \mu, \eta)$ be a monad on an object $A$ of a 2-category $\mathcal K$. In The formal theory of monads, Street proves (Theorem 3) that if $l \dashv r$ is the canonical adjunction associated ...

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### Adjoints on power sets and a connection to quantifiers as adjoints

While working through the Awodey book on Category Theory, we stumbled upon exercise 9.8.
The situation there is that you have $f : A \to B$ in Sets, and consider $\text{im}\, f : \mathcal P (A) \to \...

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### A right adjoint preserves Phi-colimits if and only if the left adjoint does what?

Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We ...

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vote

1
answer

196
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### How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?

Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...

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### Do we have a left adjoint of $i^*$ for a closed immersion $i$?

Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in ...

0
votes

1
answer

133
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### What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$?

Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega_X]=[0]$ where $\omega_X$ is the canonical ...

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### On the spectrum of a compact pertubation of a skew-adjoint operator

Let $A\colon \text{dom}(A)\subset H \to H$ ($H$ is a Hilbert space) be a skew-adjoint (i.e. $A^{*}=-A$), closed and densely defined operator. Then the essential spectrum is the set of spectral values $...

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### Left adjoints for functors out of a Deligne-Kelly tensor product

Let $k$ be a field, and let $\mathcal{C}$ be a locally finitely presentable $k$-linear categories. The Deligne-Kelly tensor product $\mathcal{C}\boxtimes\mathcal{C}$ is the 2-universal locally ...

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votes

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### EM functor from monads to adjunctions

What is the action on $1$-cells of the functor sending a monad to its EM adjunction? What about the Kleisli adjunction?
Let $A$ be the walking adjunction. Recall that an adjunction is the same thing ...

4
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1
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687
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### Adjunction between topological spaces and condensed sets

I am trying to prove that the functor
\begin{align*}
\mathrm{Top} &\longrightarrow \mathrm{Cond}(\mathrm{Set}) \\
X &\longmapsto \underline{X}
\end{align*}
admits a left adjoint and it is the ...

1
vote

1
answer

351
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### Formula for the left adjoint of the nerve functor?

I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim_{\Delta^n\...

8
votes

1
answer

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### Algebraically-free monadicity theorem

The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\...

2
votes

1
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189
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### Uniqueness of comparison functors

Given an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ with unit $\eta$ and counit $\epsilon$, we naturally have a monad $(G\circ F,\eta,G\epsilon_F)$ on $\mathcal{C}$ and a comparison ...

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### Reference for "adjunction up to twisting by autoequivalences"

Does anyone have any references on the following type of thing, which one might call "adjunction up to autoequivalences"?
We have functors $F \colon C \to D$ and $F' \colon D \to C$, but ...

3
votes

0
answers

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### Adjoints to the forgetful functor from the $2$-category of monads

For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.
There is an obvious forgetful ...

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### Morphisms of adjunctions

In Mac Lane a morphism of adjunctions $$(F\dashv G:\mathcal{C}\leftrightarrows\mathcal{D},\eta:1_\mathcal{C}\Rightarrow G\circ F,\epsilon:F\circ G\Rightarrow1_\mathcal{D})$$ $$\longrightarrow$$ $$(F'\...

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votes

3
answers

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### Is this space the Stone–Čech compactification?

Let $\mathbb{S}$ be the Sierpiński space, the two pointed space $\{ 0, 1 \}$ with open sets $\{0 \}$, $\emptyset$, $\{ 0, 1 \}$. We give $\{ 0, 1 \}$ a partial order where $0 < 1$.
Let $X$ be a ...

1
vote

1
answer

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### Adjoints of exact functors between semisimple abelian categories

Motivated by the answer to this question, I will ask the following question: Let $\mathcal{A}$ and $\mathcal{B}$ be small semisimple abelian categories. Let $U:\mathcal{A} \to \mathcal{B}$ be a ...

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1
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### Adjunctions with respect to profunctors

Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...

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1
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184
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### Characterisation of functors whose left adjoint is Kleisli

This question is inspired by Characterization of functors whose right adjoint is monadic?.
Let $F : \mathbf C \rightleftarrows \mathbf D : U$ be an adjunction, and suppose that we want to establish ...

3
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### Adjoining extensions in bicategories

Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...

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### Does Stokes theorem have anything to do with adjoint functors?

I notice some similarity between Stokes theorem in differential geometry and the definition of adjoint functors:
in both cases, there is a 2-placed function (the $\operatorname{hom}$ functor, or the ...

3
votes

0
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### Confused about the definition of the Kahn-Priddy map

The Kahn-Priddy map is defined in various papers as follows:
Define $i:\mathbb {RP}^{n-1}\rightarrow O(n)$ by taking $\ell\in \mathbb {RP}^{n-1}$ to the reflection across $\ell^\perp.$ Define $j:O(n)\...

4
votes

1
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### Proof of derived tensor-hom adjunction

This is a cross-post from math.stackexchange, since I didn't get any answers there.
As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-...

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2
answers

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### Minimal set of assumptions for set theory in order to do basic category theory

Consider a normal first course on category theory (say up to and including the statement and proof) of the adjoint functor theorem (AFT). What are the minimal assumptions for the definition of a set ...

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### Different levels of isomorphism/equivalence/adjunction between bicategories

What are all the different levels of 'isomorphism/equivalence/adjunction' we can have between bicategories? Do any of them 'collapse' to one-another?
When working with $1$-categories, we have four ...

6
votes

1
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363
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### $\infty$-natural transformations and adjunctions

I'm having troubles proving these two related statements, which are immediate for 1-categories and should of course be true for $\infty$-categories:
Given a natural transformation $\alpha: f \...

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### Lax algebras for pseudomonads and monads in Kleisli bicategories for the induced pseudocomonad

In Day–Street's Lax monoids, pseudo-operads, and convolution, they remark without proof:
There are general principles involved here. Suppose $(T, m, j)$ is a pseudomonad on any bicategory $\mathcal K$...

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### Is the restriction of an injective sheaf on a closed subscheme still injective?

Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$.
Question. Is $i^*\mathcal{I}$ still an ...

3
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1
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### Does the right adjoint of a comonad induce the following comodule map?

Let $\mathcal{C}$ be a category and $\mathcal{G}=(G,\delta, \epsilon)$ be a comonad on $\mathcal{C}$. Here $G: \mathcal{C}\to \mathcal{C}$ is a functor, $\delta: G\to G^2$ and $\epsilon: G\to id_{\...

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1
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### Reference request: Who first proved that right adjoints preserve limits?

One of the most famous and unifying theorems in category theory is that right adjoints preserve limits. I wonder: Who was the first one to prove this fact?
The notion of adjoint functors is, of course,...

3
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### All functors "are" left adjoints, and applications?

Throughout this thread, let us assume smallness.
All functors "are" left adjoints
Let $D \xrightarrow{F} C$ be any functor, which induces
$$ D \xrightarrow{F} \hat{C}$$
by compositing the ...

6
votes

2
answers

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### Yves Diers's thesis ("Catégories localisables")

I am looking for a copy of Yves Diers's 1977 thesis Catégories localisables, which is the original reference for "multi-" category theory, such as multi-adjoints, multi-colimits, and so on. ...

7
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### Prof and the completion of Cat under right adjoints

In Bénabou's Les distributeurs, in which the bicategory of profunctors is introduced, Bénabou remarks (page 17, quoted below) that $\mathbf{Prof}$ may be viewed as the construction of a bicategory ...

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### Characterization of functors whose right adjoint is monadic?

Let $F: \mathcal A^\to_\leftarrow \mathcal B: U$ be an adjunction, and suppose we want to know whether the comparision functor $\mathcal B \to Alg^{UF}$ is an equivalence, where $Alg^{UF}$ is the ...

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### Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...

0
votes

1
answer

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### Sober spaces vs. spatial frames-a big picture

For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...

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votes

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### Which direction of the adjoint functor theorem is most useful?

In the daily life of a working mathematician which direction of the adjoint functor theorem is more useful? Unpacking, does one find it more useful to:
a) prove that a functor admits an adjoint and ...

1
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0
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### $\Gamma: \mathcal C \to \text{Fun}(\mathcal Z, \mathcal C)$ has a left adjoint iff all $F \in \text{Fun}(\mathcal Z, \mathcal C)$ has a colimit

Let $\mathcal Z$ be a small category and $\mathcal C$ any category. We then consider the category $\text{Fun}(\mathcal Z, \mathcal C)$ the category of functors from $\mathcal Z$ to $\mathcal C$ and ...