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38
votes
24answers
7k views

The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come. ...
16
votes
3answers
1k views

Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...
12
votes
0answers
692 views

Frobenius upper shriek/flat of a dualizing complex

Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite. ...
11
votes
1answer
313 views

What would be an infinity-groupoid analogue of the duality between sets and complete atomic boolean algebras?

Consider the object classifier of the $\infty$-topos of $\infty$-groupoids. For the role it plays in homotopy type theory as the type of types, let’s denote it as $Type = \coprod_{[F]} B Aut(F)$, the ...
11
votes
1answer
361 views

Which categories are the categories of models of a Lawvere theory?

Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
11
votes
1answer
194 views

Looking for concrete description of a category derived from abelian groups

The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}_{fp}$. This is not so special, since the ...
10
votes
1answer
882 views

Questions about spectra of rings of continuous functions

I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
9
votes
2answers
348 views

Is there any relationship between the topologies of the clique complex and the independence complex?

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial ...
8
votes
4answers
489 views

Duality between K-theory and K-homology in the non-spin^c case.

I posted this question on Math.SE (http://math.stackexchange.com/questions/409444/), but got no answer. So I repost it here. Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times ...
8
votes
2answers
1k views

Characterizing the Dual of $W_0^{s,p}$

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and ...
8
votes
0answers
260 views

Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples? Definition. Let $\mathcal{C}$ be a ...
7
votes
5answers
458 views

Dualizable classifying spaces

If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable. Are ...
7
votes
1answer
157 views

Dual of Banach-valued $L^p$ [duplicate]

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...
7
votes
1answer
104 views

Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714). Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
6
votes
3answers
468 views

Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...
6
votes
0answers
112 views

Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
5
votes
3answers
630 views

opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$. Is ${op}$ the instance in Cat of a more ...
5
votes
3answers
291 views

Why are possibility and necessity dual?

Hello, Recently, I'm studying modal logic for my master's thesis, and my research background is category theory. So, I naturally have a question that why it is said that necessity (box) and ...
5
votes
4answers
662 views

Existence of dominating measure for weak*-compact set of measures

I have posted the following question also here a longer time ago, but due to no answers I thought it might fit better to MO. Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a ...
5
votes
2answers
386 views

A name for a weak topology

Let $V$ be a real vector space and let $V'$ be the algebraic dual of $V$, i.e. the space of all the linear functionals $V\to\mathbb{R}$. Then there exists the weakest topology $\tau$ which makes all ...
5
votes
1answer
135 views

Dual cell structures on manifolds

Suppose that $M$ is a compact manifold without boundary (smooth if you like), and suppose further that $M$ is equipped with a regular CW-complex structure. Denote the face poset of this CW-complex by ...
5
votes
1answer
732 views

connections between Grothendieck's and Serre's duality

Hi, I would like to show that if $f: X \rightarrow Y=Spec \, \mathbb{C}$, where $X$ is a nonsingular complex projective variety, is the projection to a point, then the complex $f^! \mathcal{O}_Y$, ...
5
votes
1answer
388 views

Base change of trace for Gorenstein or Cohen-Macaulay morphisms

This is basically a question of functoriality for base change of CM morphisms. EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. ...
5
votes
0answers
433 views

Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s?

Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so ...
4
votes
2answers
934 views

Examples for “nice” Boolean algebras that are not complete or not atomic

A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms). Boolean Algebras that are complete as ...
4
votes
2answers
457 views

Tensor and Hom objects for finite flat group schemes

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to ...
4
votes
1answer
219 views

Equivalent Norms for the Dual of Sobolev / Bessel Spaces

Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb ...
4
votes
1answer
119 views

Self-dual surfaces in $\mathbb P^3$ with isolated singularities

I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties: the smooth quadric; Kummer surfaces; The surface with the equation ...
4
votes
1answer
181 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
4
votes
1answer
500 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
4
votes
2answers
437 views

Continuous Transportation Problem

Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the ...
4
votes
0answers
34 views

Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space. A poset $(P,\leq)$ is called ...
4
votes
0answers
190 views

Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
4
votes
0answers
149 views

Duality between K-theory and K-homology in the non-compact, spin$^c$ case

Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong ...
4
votes
0answers
264 views

Dual of a weighted projective space

I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...
4
votes
0answers
165 views

Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem $$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$ where the functions $f_1$ and $f_2$ are concave but ...
3
votes
2answers
506 views

Alexander duality theorem for CW-complexes and stable homotopy theory

In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$ for finite CW-complexes with a "nice ...
3
votes
2answers
561 views

Does equality of Hodge star and symplectic star imply Kähler structure?

Question The question asked is: On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does ...
3
votes
1answer
269 views

Electromagnetic duality symmetry

This question arose while reading http://prd.aps.org/abstract/PRD/v13/i6/p1592_1 (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well ...
3
votes
1answer
235 views

Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ? In ...
3
votes
0answers
119 views

Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?

Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in ...
3
votes
0answers
536 views

Infinite Linear Programming

I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
2
votes
1answer
116 views

Simultaneously extending the functionals of a subspace of a Banach space to the whole space

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. If $\varphi\in Y^*$, then Hahn-Banach allows us to extend $\varphi$ to a $\tilde\varphi\in X^*$, such that $\|\tilde\varphi\|=\|\varphi\|$. ...
2
votes
2answers
232 views

Why is the norm map dual to restriction under Tate local duality?

Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are ...
2
votes
1answer
400 views

A category being self-dual vs. it being a groupoid

What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference? To go from a self-duality $F$ on a ...
2
votes
1answer
120 views

Predual of a subspace

Let $E$ be a Banach space, let $d\ge 1$ be an integer. Let $\mathcal G$ be a weakly closed subspace of $(E^*)^d$ with finite codimension. I would like know if the space $\mathcal G$ is a dual space ...
2
votes
1answer
192 views

What is a de Vries algebra?

I've come across a set of slides by Guram Bezhanishvili where he claims the category of compact hausdorff spaces is related by duality to de Vries algebras. What are they?
2
votes
2answers
1k views

Dual Norm For Sum of 2-Norms

What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$ $\|\mathbf{x}\| = ...
2
votes
1answer
259 views

What is the dual of a pre-injective map?

In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...
2
votes
2answers
526 views

How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional Lp-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $L^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $L^p$ norm, so the unit sphere in $V_s$ ...