# Questions tagged [adjoint-functors]

The adjoint-functors tag has no usage guidance.

123
questions

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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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61 views

### 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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103 views

### 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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231 views

### 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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386 views

### 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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125 views

### 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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263 views

### 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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1k views

### 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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68 views

### 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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69 views

### 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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181 views

### 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 ...

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487 views

### 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 $K$....

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164 views

### 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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545 views

### 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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298 views

### 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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248 views

### 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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383 views

### 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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**4**answers

749 views

### 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 ...

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118 views

### 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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139 views

### 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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179 views

### 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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245 views

### 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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180 views

### 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 ...

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271 views

### 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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429 views

### 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 ...

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384 views

### “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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144 views

### Can the cobordism hypothesis be formulated as a statement about adjoint functors?

I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors.
For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} ...

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315 views

### On the History of Double Adjoints

This is my first post here, as someone from Mathsstack suggested this might me a more suitable forum for this specific question.
I have been reading some texts by Joaquim Lambek on formal languages, ...

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180 views

### Left adjoint pseudofunctor commutes with pseudocolimits

I'm looking for a reference for this seemingly basic fact: assume I have a 2-functor $G : {\cal X}\to {\cal Y}$ and assume I can define a left 2-adjoint $F$ for it, which is nevertheless only a ...

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134 views

### Is “square” functor monomorphic on objects?

I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is monomorphic on objects, in other words, that if there exists an isomorphism $A\times ...

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227 views

### Adjoints for the functor ${\bf Top}\to {\bf Conv}$

Let $X$ be a set and let $\Phi(X)$ denote the collection of filters on $X$. For $x\in X$ we denote by $P_x$ the filter $P_x=\{A\subseteq X:x\in A\}$. A convergence space is a pair $(X,\to)$, where $X$ ...

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158 views

### Translating parabolic induction as $\Lambda G^F/U^F\otimes_{\Lambda L^F}-$ to $\hom_{\Lambda L^F}(\Lambda U^F/G^F,-)$?

Suppose $P=L\ltimes U$ is an $F$-stable parabolic subgroup of a finite group of Lie type $G$, with $F$-stable Levi complement $L$. Here $F$ is a Frobenius endomorphism, and $G^F$ is the subgroup of ...

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### Right adjoint completions

Forgive me if this question is not well thought out. I don't know how else to ask it.
The nlab page on completion gives some examples of completions which are left adjoints. These completions are "...

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87 views

### About small $\omega$-orthogonality classes and Gabriel-Ulmer duality

I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\...

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159 views

### About small-orthogonality classes of a locally presentable category

Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
categories. $\mathcal{A}$ reflective and closed under filtered
colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
$...

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350 views

### Categories which are both monadic and comonadic over another category

I heard a professor say that $\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same.
What does it mean, intuitively, that a category is ...

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927 views

### If a right adjoint to the product functor exists, must it be the diagonal?

Let $C$ be a category with binary products. The product functor $\times : C^2 \to C$ is right adjoint to the diagonal $\Delta: C \to C^2$. If $C$ has biproducts, then $\times$ is also left adjoint to $...

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102 views

### Fundamental monoid pertaining to adjunctions

Marco Grandis has been working to collect and formalize the ideas of directed homotopy theory (his main work on the subject has been listed in the references at the nLab page on the subject: directed ...

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230 views

### link between completion of the universal enveloping algebra and an endomorphism of functor

My question could be resume in the following way :
Let $\mathfrak{t} \to \mathrm{End}(V)$ a representation of an abelian Lie algebra into an infinite dimensional vector space.
What can we say ...

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162 views

### Free functors having a left adjoint

If $(T,\eta,\mu)$ is a monad over $C$ and $C^T$ is the category of $T$-algebras, is well known that the forgetful functor $U:C^T\to C$ has a left adjoint $F:C\to C^T$. Moreover, under certain ...

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### General description of transition arrows of covering morphisms in family fibrations

For sets and functions, I think the following data are equivalent:
A function $g:A\times B\to B$ such that $(\pi_1,g):A\times B\to A\times B$ is a bijection;
a function $A\to \mathrm{Aut}B$.
Proof. ...

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### Faithfulness of Right adjoint to Kan extension

Let $C$ be a category, $D$ be a Grothendieck topos, and suppose we have a fully faithful, left-exact functor $F:C\rightarrow D$. Let $Lan_{y}F:PShv(C)\rightarrow D$ be the Yoneda extension of $F$. ...

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### Is there a construction capturing indexed families of adjunctions?

I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which ...

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196 views

### Intrinsic notions of étale?

The usual notion of trivial covering morphism is in a sense intrinsic to the adjunction $\Pi_0\dashv H$ between connected components and discrete spaces: a continuous map $f$ is a trivial covering ...

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### Left adjoint to inclusion of Connected Groupoids into Groupoids

Let $Gpd$ denote the category of groupoids and functors. Let $Gpd_{con}$ denote the subcategory spanned by connected groupoids, i.e for every $x,y\in Ob(Gpd_{con})$, there is at least one morphism $x\...

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784 views

### Why quasi-inverse functors are adjoint pairs?

Let $C$,$D$ be two category, and $F:C \rightarrow D$, $G:D \rightarrow C$ are two functors such that $FG \simeq Id_{D}$ and $GF \simeq Id_{C}$, Show that $(F,G)$ is an adjoint pair.
To show this, we ...

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195 views

### Free commutative monoid monad

Has the monad induced by the free commutative monoid functor already been studied anywhere? Does it have any particular properties (other than not being cartesian)?
I would prefer a reference on ...

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### Left adjoint for categories of commutative monoids?

The $n$Lab writes (prop. 2.2 in https://ncatlab.org/nlab/show/category+of+monoids) :
Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the ...

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### Finite well-completeness and the small object argument?

I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.
First of all, some ...