Questions tagged [adjoint-functors]
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196
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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 ...
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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 ...
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3
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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 ...
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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 ...
- 61.5k
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Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...
- 556
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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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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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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 ...
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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 ...
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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 ...
- 8,707
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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 ...
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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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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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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Left and right Kan extensions
Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between small categories. We define the functor
\begin{align*}
f:\hat{\mathcal{D}}&\longrightarrow\hat{\mathcal{C}} \\
G&\longmapsto G\circ F^{\...
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Adjoints for radical and socle functors
Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
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Surmounting set-theoretical difficulties in algebraic geometry
The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
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Reference request for (co-)free constructions
Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces ...
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Do adjoints of enriched functors preserve the enriched structure?
Is there is any reason in general for adjoints of enriched functors to preserve the enriched structure of categories?
The specific example I'm thinking of is the following:
Fix a commutative ring $R$...
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Existence of free functor to Banach spaces
Is there a "non-trivial" characterization of the concrete categories admitting and adjoint pair of functors $F \dashv G$ were $G$ is defined on the category sBan of separable Banach spaces and bounded ...
- 4,969
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CG spaces from the perspective of sheaves over compact Hausdorff spaces
A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....
user30211
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Does every functor between Grothendieck categories have adjoints?
Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits.
Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories.
Does $F$ have a right adjoint? ...
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Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie
I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
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When does the forgetful functor from modules to vector spaces have a right adjoint?
Given any algebra $R,$ when does the forgetful functor
$R\text{-}Mod \rightarrow Vec$
have a right adjoint?
Does this imply any finiteness conditions on R?
Is there a book/paper discussing this?
I'...
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Adjoints to forcing
Forcing over a partial order $P$ can be viewed in a category theoretic sense as constructing the presheaf topos ${\bf Set}^{P^{op}}$ over the partial order (viewed as a category) then passing through ...
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Semantics-structure adjunction
In the discussion on the nLab article for monadic adjunctions, John Baez suggests and Mike Shulman confirms that the relationship between adjunctions and monads itself constitutes an adjunction called ...
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Upgrade adjunction to equivalence
I'm studying category theory by myself and I just came across this sentence from Wikipedia:
An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and ...
user131962
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For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?
Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.
For which ...
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How are the unit/counit of a Hopf algebra and of an categorical adjunction related?
For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have ...
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Commutativity between functors on sheaves of abelian groups
I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
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Adjunctions between Groupoids and Hilbert spaces
I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
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Question on Eilenberg-Watts theorem
I'm not sure if this is a research level question, but:
Let $F:Rep_A \to Rep_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has ...
- 1,641
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Explicit expression of the unstraightening functor
Hard as I tried, I couldn't find a proof of Remark 2.2.2.11 in Higher Topos Theory, or prove it myself. It seems to need an explicit formulation for the unstraightening functor, so my question is: is ...
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Adjoints of scalar extension and scalar coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts):
$h^*$: Scalar extension by ...
- 6,670
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"Equivalence" is to "group" as "adjoint" is to ....?
The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces ...
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