# Questions tagged [adjoint-functors]

The adjoint-functors tag has no usage guidance.

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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 ...

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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 ...

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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 ...

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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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### 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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### 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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156 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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123 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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### 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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### 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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### 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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155 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Four questions about C12 generators and adjunction

Consider the abelian cyclic group of twelve elements, C12, with generators
$$ C12^* = (e^1, e^5, e^7, e^{11}).$$
The cayley diagram generated by $e^1$ is a circle of twelve directed links for the ...

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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 ...

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### “2-Sheafification” with Values in non $Cat$ categories?

Let $X$ be a 2-site and consider the category of 2-presheaves over $X$, which will be denoted as $Pshv(X;Cat)$. These are $Cat$-valued 2-functors, where $Cat$ is the 2-category of categories. There is ...

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### Does the inclusion of presheaves into families of sets have a left adjoint?

Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this ...

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### Are there non-trivial infinite chains of adjoint functors?

There are self-adjoint functors $A \dashv A$. There are also functors $A$ that are both left- and right-adjoint to another functor $B$. $$A \dashv B \dashv A$$
There are also finite cyclic chains of ...

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### Uniqueness of $\infty$-adjoints

Adjoints in a 2-category are essentially unique, in the following strong sense. If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ ...

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### What do you get when you apply a universal cocone to a colimit functor

Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \...

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### Question about Enriched Categories and Functors

How would one describe the process of enriching a category C over some monoidal category D? Is there some functor between them that adds structure to the hom-sets?

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### Left adjoint of pullback

In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:
Indeed, the UMP of pullbacks essentially states that composition along
any function α is left adjoint to pullback ...

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### Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

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### Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.
Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...

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### Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...

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### On the coherence theorem for bicategories

The coherence theorem for bicategories, as usually stated, reads
Any bicategory $B$ is biequivalent to a (strict) 2-category.
It is possible to give an explicit construction of the strictification ...

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### Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...

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### Map of adjunctions

The following question must have been asked dozens of times, but I do not recall any non-trivial results.
Consider an adjoint square where the arrows indicate directions of $F, G, H, K$.
$\require{...

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### Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...

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### Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...

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### Cofree Lie Coalgebra

I have problems finding anything about the cofree Lie coalgebra functor
$\mathcal{L}ie^c$ out there.
Basically all I found was that it appears in Harrison cohomology and that,
given a $\mathbb{Z}$-...