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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 ...
W. Rether's user avatar
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3 votes
0 answers
107 views

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 ...
Theo Johnson-Freyd's user avatar
2 votes
0 answers
58 views

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 ...
NDewolf's user avatar
  • 183
9 votes
1 answer
260 views

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 ...
Theo Johnson-Freyd's user avatar
0 votes
0 answers
45 views

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 ...
Elías Guisado Villalgordo's user avatar
9 votes
0 answers
180 views

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 ...
Alec Rhea's user avatar
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5 votes
1 answer
199 views

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 ...
fosco's user avatar
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8 votes
0 answers
247 views

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) ...
Timo's user avatar
  • 399
1 vote
1 answer
171 views

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 ...
Adelhart's user avatar
  • 185
2 votes
2 answers
829 views

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 $\mathbf{Set}$ ...
Samuel Adrian Antz's user avatar
3 votes
0 answers
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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, ...
Andrew Dudzik's user avatar
3 votes
1 answer
89 views

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 ...
varkor's user avatar
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4 votes
0 answers
100 views

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 \...
JoJoModding's user avatar
9 votes
0 answers
135 views

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 ...
varkor's user avatar
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1 vote
1 answer
196 views

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 ...
Zhaoting Wei's user avatar
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1 vote
0 answers
158 views

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 ...
Zhaoting Wei's user avatar
  • 8,417
0 votes
1 answer
133 views

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 ...
Zhaoting Wei's user avatar
  • 8,417
0 votes
0 answers
118 views

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 $...
user99432's user avatar
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5 votes
0 answers
63 views

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 ...
Thibault Décoppet's user avatar
2 votes
0 answers
112 views

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 ...
Alec Rhea's user avatar
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4 votes
1 answer
687 views

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 ...
Luiz Felipe Garcia's user avatar
1 vote
1 answer
351 views

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\...
Samuel Adrian Antz's user avatar
8 votes
1 answer
224 views

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 $\...
varkor's user avatar
  • 6,532
2 votes
1 answer
189 views

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 ...
Alec Rhea's user avatar
  • 8,294
1 vote
0 answers
121 views

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 ...
Jo Mo's user avatar
  • 338
3 votes
0 answers
64 views

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 ...
Alec Rhea's user avatar
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3 votes
0 answers
199 views

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'\...
Alec Rhea's user avatar
  • 8,294
8 votes
3 answers
1k views

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 ...
Cayley-Hamilton's user avatar
1 vote
1 answer
233 views

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 ...
Tim Montegue's user avatar
4 votes
1 answer
216 views

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 $...
varkor's user avatar
  • 6,532
2 votes
1 answer
184 views

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 ...
varkor's user avatar
  • 6,532
3 votes
0 answers
74 views

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 ...
varkor's user avatar
  • 6,532
9 votes
0 answers
527 views

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 ...
Benaya's user avatar
  • 91
3 votes
0 answers
191 views

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)\...
Nikhil Sahoo's user avatar
  • 1,107
4 votes
1 answer
803 views

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 $(S,V)$-...
rollover's user avatar
  • 203
15 votes
2 answers
1k views

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 ...
Jake Wetlock's user avatar
6 votes
0 answers
227 views

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 ...
Alec Rhea's user avatar
  • 8,294
6 votes
1 answer
363 views

$\infty$-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 \...
Giulio Lo Monaco's user avatar
3 votes
0 answers
43 views

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 $\mathcal K$...
varkor's user avatar
  • 6,532
4 votes
0 answers
202 views

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 ...
Stabilo's user avatar
  • 1,345
3 votes
1 answer
81 views

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_{\...
Zhaoting Wei's user avatar
  • 8,417
7 votes
1 answer
1k views

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,...
user907616's user avatar
3 votes
0 answers
257 views

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 ...
Student's user avatar
  • 4,560
6 votes
2 answers
302 views

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. ...
varkor's user avatar
  • 6,532
7 votes
3 answers
407 views

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 ...
varkor's user avatar
  • 6,532
10 votes
1 answer
413 views

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 ...
Tim Campion's user avatar
  • 55.4k
5 votes
1 answer
207 views

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

$\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\tensor}{\otimes}$ $\newcommand{\colim}{\rm colim}$ $\...
Bastiaan Cnossen's user avatar
0 votes
1 answer
149 views

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 ...
truebaran's user avatar
  • 8,736
6 votes
1 answer
436 views

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 ...
Dick Johnson's user avatar
1 vote
0 answers
86 views

$\Gamma: \mathcal C \to \text{Fun}(\mathcal Z, \mathcal C)$ 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 ...
Falcon's user avatar
  • 280