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(When) does a morphism of monad induce adjoint functors between categories of algebras?

For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor ...
sysyphusV's user avatar
3 votes
2 answers
257 views

Directed colimit of fully faithful functors

Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
MikeTrooper's user avatar
8 votes
0 answers
172 views

When is the Eilenberg-Moore category of a relative monad between two topoi a topos?

In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint. Now how does this ...
Ilk's user avatar
  • 749
4 votes
1 answer
168 views

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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5 votes
0 answers
183 views

When are topoi of coalgebras atomic?

A geometric morphism is atomic if its inverse image is logical. Now consider a Grothendieck topos $\varepsilon$ and its terminal geometric morphism $\Gamma : \varepsilon \rightarrow Set$, topos is ...
Ilk's user avatar
  • 749
2 votes
0 answers
78 views

What can be said about the free-forgetful adjunction of monad algebras with respect to topoi?

For a monad T on a topos E, if T has a right adjoint, then the Eilenberg-Moore category of algebras of T is equivalent to the co-Eilenberg-Moore category of co-algebras for the right adjoint comonad ...
Ilk's user avatar
  • 749
3 votes
0 answers
50 views

A new(?) kind of 2-adjunction for relating cartesian closed functors using dinatural hexagons

$\newcommand{\A}{\operatorname{A}} \newcommand{\B}{\operatorname{B}} \newcommand{\Cat}{\mathcal{Cat}} \newcommand{\Cart}{\mathcal{Cart}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\operatorname{F}} \...
Johan Thiborg-Ericson's user avatar
1 vote
1 answer
94 views

When would a left admissible triangulated subcategory be admissible

I'm walking through the proof of [1, Thm 16 at pp. 515] and am stuck at the first sentence after equation (12), where the author states that the decomposition (12) is semiorthogonal when $a\geq 0$. ...
Noto_Ootori's user avatar
6 votes
0 answers
160 views

Drinfeld center of non-rigid closed monoidal categories

Context. The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\...
Max Demirdilek's user avatar
2 votes
0 answers
120 views

Universal property of Isbell duality

Let's take $\mathrm{C}$ be a category, let's have an adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves(C)} \leftrightarrows \mathrm{Presheaves(C)}$. One such adjunction is ...
Ilk's user avatar
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7 votes
1 answer
149 views

The change-of-monoid adjunction between categories of modules induced by a morphism of monoids

Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...
Angelos's user avatar
  • 73
3 votes
1 answer
130 views

Reference request: the free adjunction being free as an $(\infty, 2)$-category?

This question is a particular case of Tim Campion's question. Let $\mathrm{Adj}$ be the strict $2$-category corepresenting adjunctions, i.e., the free strict $2$-category generated by two objects $x, ...
nrkm's user avatar
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2 votes
0 answers
196 views

Proving that the functor induced by some inclusion functor has a left adjoint

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}\subseteq \operatorname{Proj}(\mathcal{A})$ be a full additive subcategory of $\mathcal{A}$. We define the full subcategory $\mathcal{B(A)}$ of ...
Juan C. Cala's user avatar
8 votes
1 answer
279 views

Reflective functors?

Let $C_0\subseteq C$ and $D_0\subseteq D$ be reflective subcategories with reflection functors $r_A$ and $r_B$. For any functor $F:C\to D$, we may consider the natural transformation $r_BF\eta_A:r_BF\...
John Pardon's user avatar
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8 votes
0 answers
175 views

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
  • 405
3 votes
0 answers
128 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
3 votes
0 answers
74 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
  • 193
10 votes
1 answer
306 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
9 votes
0 answers
190 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
  • 9,047
5 votes
1 answer
224 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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10 votes
0 answers
503 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) ...
Ilk's user avatar
  • 749
1 vote
1 answer
328 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
  • 227
3 votes
2 answers
944 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
95 views

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
2 votes
1 answer
105 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
  • 8,885
4 votes
0 answers
129 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
11 votes
0 answers
360 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
  • 8,885
2 votes
1 answer
224 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
  • 8,727
1 vote
0 answers
210 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
  • 8,727
0 votes
1 answer
148 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
  • 8,727
0 votes
0 answers
205 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
  • 173
5 votes
0 answers
93 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
135 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
  • 9,047
4 votes
1 answer
745 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
569 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
9 votes
1 answer
326 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
  • 8,885
2 votes
1 answer
218 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
  • 9,047
1 vote
0 answers
125 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
71 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
  • 9,047
4 votes
0 answers
319 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
  • 9,047
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 ...
user avatar
1 vote
1 answer
327 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
6 votes
1 answer
318 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
  • 8,885
3 votes
1 answer
206 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
  • 8,885
5 votes
1 answer
131 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
  • 8,885
9 votes
0 answers
608 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
219 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
  • 1,195
4 votes
1 answer
1k 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
  • 1,144
6 votes
0 answers
246 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
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