Questions tagged [adjoint-functors]
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167
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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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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 ...
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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 ...
2
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1
answer
166
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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 $\...
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1
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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 ...
- 338
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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 $$(F\dashv G:\mathcal{C}\leftrightarrows\mathcal{D},\eta:1_\mathcal{C}\Rightarrow G\circ F,\epsilon:F\circ G\Rightarrow1_\mathcal{D})$$ $$\longrightarrow$$ $$(F'\...
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Is this space the Stone–Čech compactification?
Let $\mathbb{S}$ be the Sierpiński space, the two pointed space $\{ 0, 1 \}$ with open sets $\{0 \}$, $\emptyset$, $\{ 0, 1 \}$. We give $\{ 0, 1 \}$ a partial order where $0 < 1$.
Let $X$ be a ...
- 4,283
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1
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Adjoints of exact functors between semisimple abelian categories
Motivated by the answer to this question, I will ask the following question: Let $\mathcal{A}$ and $\mathcal{B}$ be small semisimple abelian categories. Let $U:\mathcal{A} \to \mathcal{B}$ be a ...
- 395
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Adjunctions with respect to profunctors
Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...
- 5,035
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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 ...
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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 ...
- 5,035
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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 ...
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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)\...
- 945
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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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Proof that $e_{k,j}^{\tau}(q,p)=\overline{e_{j,k}^{\tau}(-q,-p)}$ in the spectral theory of the twisted Laplacian?
I leave some context for this question, this question is about the spectral analysis of the twisted Laplacian, I need some assistance in solving this problem.
A fundamental ingredient in the ...
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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 ...
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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$...
- 5,035
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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 ...
- 1,243
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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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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,...
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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 ...
- 4,005
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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. ...
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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 ...
- 5,035
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votes
1
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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 ...
- 49.2k
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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}$
$\...
- 519
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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 ...
- 8,660
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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 ...
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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 ...
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Can the category of S-local objects be reflective but not a localization by S?
This is cross-posted from MSE (and substantially re-written) after receiving no answers.
Suppose $\mathcal C$ is a category and $S \subseteq \operatorname{Mor}(\mathcal C)$ is some collection of ...
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votes
2
answers
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Monoidal categories whose tensor has a left adjoint
Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some ...
- 5,035
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votes
1
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Question about adjoint of forgetful functor from condensed abelian groups to condensed sets
There is a forgetful functor from condensed abelian groups to condensed sets. According to Scholze's notes, this has an adjoint $T \mapsto \mathbb{Z}[T]$ (which is the sheafification of the functor ...
5
votes
2
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A specific property of bi-adjunction
Let $$I: C \rightleftarrows D: F$$ be biadjoint [1] functors between categories $C, D$. That is, $I$ is the left and also the right adjoint of $F$ (thus vice versa). Put in notations, it's
$$ \cdots \...
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Does a natural transformation of functors induce a natural transformation between their right adjoints?
Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F$ and $G: \mathcal{C}\to \mathcal{D}$ be two functors. Suppose $F$ and $G$ have right adjoints $F^{\wedge}$ and $G^{\wedge}: \mathcal{D}\to ...
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Left/right adjoints to core/Cartesian inclusions
Let $p:E\to B$ be a fibration and let $Cart(E)$ denote the subcategory of $E$ with all objects but only Cartesian arrows. Since all isomorphisms in $E$ are Cartesian, we naturally have inclusion ...
- 6,241
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votes
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When is a finitary functor induced by Ind (co)continuous
Let $\mathbf C$ and $\mathbf D$ be small categories. $\mathrm{Ind}(\mathbf C)$ is an accessible category (by definition), and is locally finitely presentable (i.e. cocomplete, or equivalently complete)...
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votes
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Is the group Hopf algebra left and right adjoint?
Suppose that $G$ is a group and $k$ is a field. Then it is well known that the group ring (group algebra) functor $k[\bullet]$ is left adjoint to the group of units functor, the latter of which ...
- 1,914
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How to understand adjoint functors?
I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.
I have a good ...
- 683
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Has the covariant Hom-functor of the category of additive categories a left adjoint?
Let $\mathsf{Add}$ denote the (strict) 2-category of small additive categories and additive functors. Because categories of additive functors are itself additive, we have for each additive category $\...
- 111
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Admissibility of intersection of subcategories
Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\...
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Adjunctions capturing duality between ideals and saturated monoids in a commutative ring?
Let $R$ be a commutative ring. A saturated monoid in $R$ is a multiplicative submonoid $S\subset R$ which is closed under divisors, i.e $xy\in S\implies x\in S$. This is the converse of the analogous ...
- 9,655
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Oplax monoidal functors of $\infty$-categories
In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
- 8,035
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votes
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111
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Does the functor sending a DGA to its zeroth component admit a right adjoint?
Let $A$ be a ring and write $\underline{A}^\bullet$ for the associated trivial DGA. We have a functor
$$\mathrm{ev}_0\colon\mathbf{dgAlg}_{\underline{A}^\bullet}\longrightarrow\mathbf{Alg}_A$$
sending ...
- 6,346
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votes
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answers
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What is the left adjoint to base change of schemes?
Restriction of Scalars and Functoriality of Presheaves.
Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is associated to $\phi$ a natural functor from $\mathrm{Alg}_S$ to $\mathrm{...
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Notation for "the" left adjoint functor
As far as I know, there is no "official" notation for the left adjoint of a functor $F : \mathcal{C} \to \mathcal{D}$ if it exists. I have seen the notation $F^*$ sometimes, but this looks only nice ...
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Subfunctor of internal Hom
Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...
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