Questions tagged [adjoint-functors]
The adjoint-functors tag has no usage guidance.
197
questions
8
votes
1
answer
277
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\...
- 18.3k
8
votes
0
answers
155
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 ...
- 699
8
votes
0
answers
172
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 ...
- 405
7
votes
1
answer
1k
views
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,...
- 79
7
votes
3
answers
447
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 ...
- 8,755
7
votes
2
answers
984
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 ...
- 9,009
7
votes
2
answers
175
views
(When) does a morphism of monad induce adjoint functors between categories of algebras?
For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor ...
- 73
7
votes
3
answers
384
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. ...
- 8,755
7
votes
2
answers
973
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 $K$....
user30211
7
votes
1
answer
145
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 ...
- 73
7
votes
1
answer
629
views
[Reference Request] The Definition of Adjoint Functors between dg-categories
Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...
- 8,707
7
votes
0
answers
615
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{...
- 10.7k
7
votes
0
answers
390
views
bar-cobar or cobar-bar
What is the standard or best reference for the adjointnes of bar and cobar constructions?
- 3,850
7
votes
0
answers
629
views
Constructing pointwise Kan extensions as adjoints to some functor
Background
I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because I'...
- 695
7
votes
0
answers
1k
views
Proving the Special Adjoint Functor Theorem from the general Adjoint Functor Theorem
Often, when dealing with adjoint functor theorems, people go about proving each one separately, from first principles if you will (this is the course taken in MacLane). However, the names suggest ...
- 2,289
6
votes
2
answers
480
views
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
$$ \cdots \...
- 5,038
6
votes
2
answers
452
views
Adjoints for radical and socle functors
Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
- 343
6
votes
1
answer
921
views
Adjunctions: Algebras of the induced monad VS. Coalgebras of the induced comonad.
Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent?
The example i have ...
- 3,161
6
votes
2
answers
704
views
Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
- 2,421
6
votes
1
answer
490
views
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 ...
- 203
6
votes
1
answer
414
views
$\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 \...
- 1,071
6
votes
4
answers
2k
views
Kan extensions and the yoneda embedding.
[Edit: For a category $C$ let $C^\wedge$ denote the category of presheaves on $C$.]
Let $f:C\to D$ be a functor. Precomposition with $f^{op}$ induces a functor
$f^\wedge:D^\wedge \to C^\wedge$. This ...
- 3,161
6
votes
1
answer
270
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? ...
- 131
6
votes
1
answer
2k
views
Adjunctions between derived functors
Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...
- 456
6
votes
1
answer
456
views
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}$-...
- 2,014
6
votes
0
answers
153
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-\...
- 766
6
votes
0
answers
246
views
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 ...
- 9,009
5
votes
1
answer
391
views
Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 46.8k
5
votes
1
answer
327
views
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$ ...
- 65.1k
5
votes
1
answer
223
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}$$
if ...
- 13.2k
5
votes
1
answer
130
views
Adjoining extensions in bicategories
Given a bicategory $\mathcal K$, is there a universal construction of a bicategory $\mathcal K'$ and faithful locally fully faithful pseudofunctor $\mathcal K \hookrightarrow \mathcal K'$ such that ...
- 8,755
5
votes
1
answer
222
views
Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...
- 556
5
votes
1
answer
667
views
"adjoint" =?= "inverse of composite endofunctor is uniform bi-composition"
Understanding adjoints has always been (and continues to be) a bit of a struggle for me.
Today I stumbled upon a property of adjoint functors which seemed extremely intuitive to me. I was wondering ...
- 3,247
5
votes
1
answer
107
views
About small $\omega$-orthogonality classes and Gabriel-Ulmer duality
I am reading the paper http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\...
- 3,759
5
votes
1
answer
150
views
Finite well-completeness and the small object argument?
I'm reading a few papers on reflective factorization systems and I've just noticed they're all mentioning a procedure which seems very similar to the small object argument.
First of all, some ...
- 10.3k
5
votes
1
answer
312
views
Adjunctions with respect to profunctors
Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $...
- 8,755
5
votes
1
answer
176
views
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 ...
- 10.3k
5
votes
1
answer
510
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 ...
- 695
5
votes
0
answers
170
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 ...
- 699
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 ...
5
votes
0
answers
117
views
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 ...
- 2,014
5
votes
0
answers
157
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 ...
- 10.3k
5
votes
0
answers
211
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 ...
- 171
5
votes
0
answers
281
views
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{...
- 4,116
4
votes
1
answer
1k
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 ...
- 59
4
votes
2
answers
509
views
Why is $Lex(\mathcal{A},\mathcal{Ab})$ abelian? Does $Lex(\mathcal{A},\mathcal{Ab})\rightarrow Func(\mathcal{A},\mathcal{Ab})$ admit a left-adjoint?
What is the best way to show, that $Lex(\mathcal{A},\mathcal{Ab})$ is abelian, where $\mathcal{A}$ is an abelian category and $\mathcal{Ab}$ is the category of abelian groups from scratch?
There is ...
- 41
4
votes
1
answer
515
views
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 ...
4
votes
1
answer
355
views
A Question on Auto-Adjoint functors
Let $F: \mathscr{C}\to \mathscr{C}^{op}$, with an adjoint $G$, and $\eta: 1_\mathscr{C} \Rightarrow G\circ F $ and $\varepsilon: F\circ G\Rightarrow 1_{\mathscr{C}^{op}}$ with components (in $\...
- 4,125
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. ...
- 415
4
votes
1
answer
253
views
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 ...
- 915