Questions tagged [adjoint-functors]

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3 votes
2 answers
240 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 ...
8 votes
0 answers
149 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 ...
4 votes
1 answer
148 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 ...
5 votes
0 answers
164 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 ...
7 votes
3 answers
383 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. ...
2 votes
0 answers
73 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 ...
5 votes
1 answer
308 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 $...
9 votes
1 answer
316 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 $\...
5 votes
1 answer
129 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 ...
1 vote
1 answer
88 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 $a\geq 0$. ...
6 votes
0 answers
150 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-\...
3 votes
0 answers
47 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}} \...
2 votes
0 answers
114 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 ...
10 votes
0 answers
494 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) ...
7 votes
1 answer
142 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 ...
3 votes
1 answer
117 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, ...
2 votes
0 answers
185 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 ...
11 votes
0 answers
334 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 ...
8 votes
1 answer
276 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\...
75 votes
8 answers
17k views

What is an intuitive view of adjoints? (version 1: category theory)

In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is. I know the definition (several ...
8 votes
0 answers
165 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 ...
10 votes
1 answer
300 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 ...
3 votes
0 answers
127 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 ...
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 ...
9 votes
0 answers
188 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 ...
2 votes
2 answers
936 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 $\mathbf{Set}$ ...
4 votes
1 answer
2k views

How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
5 votes
1 answer
222 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 ...
5 votes
1 answer
498 views

Do generalizations of adjoint functors, such as adjunctions in 2-categories, and multivariable adjunctions, have formulations in terms of something like universal morphisms?

I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about how to go from the ...
1 vote
1 answer
287 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 ...
1 vote
1 answer
532 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\...
3 votes
0 answers
94 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, ...
2 votes
1 answer
100 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 ...
4 votes
0 answers
124 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 \...
2 votes
1 answer
221 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 ...
1 vote
0 answers
202 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 ...
0 votes
1 answer
147 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 ...
0 votes
0 answers
189 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 $...
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 ...
4 votes
1 answer
742 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 ...
2 votes
0 answers
134 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 ...
5 votes
1 answer
222 views

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

$\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\tensor}{\otimes}$ $\newcommand{\colim}{\rm colim}$ $\...
2 votes
1 answer
217 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 ...
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 ...
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 ...
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 ...
4 votes
0 answers
295 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'\...
7 votes
3 answers
443 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 ...
1 vote
1 answer
316 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 ...
3 votes
1 answer
204 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 ...