Questions tagged [duality]
Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.
312 questions
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Is the category of abstract $\sigma$-algebras the pro-category of countable discrete spaces?
I am wondering if the category of abstract $\sigma$-algebras is contravariantly equivalent to the pro-category of the category of countable sets.
Note: I have tweaked this question a little. Now it ...
user30211
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Point-free topology, but with $\sigma$-algebras instead of spaces
I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had:
If abstract $\sigma$-algebras (i.e. certain boolean ...
user30211
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Nests on Banach spaces and their duals
Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$.
Take $f\in X^{*}$ and suppose:
$N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$
$N=\bigcap_{M>N}M$
Is there ...
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Dual Steenrod squares
Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
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Cassels Pairing for Fine Selmer groups
Let $S_n$ be the Selmer group of $E/K_n$ where $K_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $K$ and $C_n$ be the torsion part of $S_n$.
By Cassels pairing, we know that ...
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383
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Serre functors for non-proper categories
One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
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Monge-Kantorovich duality with a $\{0,1\}$ cost function
Consider the usual Monge-Kantorovich transportation problem where $X$ and $Y$ are Polish spaces, $\mu$ and $\nu$ are probability measures on $X$ and $Y$, and $c:X\times Y \to \mathbb{R}^+ \cup \{+\...
- 4,049
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Poincare duality in families of smooth, projective curves
Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
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The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
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Generalized Gelfand triples
Normally, when we talk about Gelfand triple we have three Hilbert spaces
$$\newcommand{\X}{\mathcal{X}}
\X_+ \subset \X_0 \subset \X_-
$$
such that the subsets are dense, the embedding mappings are ...
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I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module
I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
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If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?
Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am ...
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How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?
$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
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What is the dual of generating Boolean subalgebra by subexpressions of a modal formula?
I am supposed to be answering this question rather than asking it but I really cannot figure out.
There is a variation on Stone duality linking algebraic and (descriptive) Kripke semantics for (...
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Divided power algebra is artinian as a module over the polynomial ring
I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper Homological algebra on a complete intersection, with ...
- 1,286
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Legendre transform on signed measure space
Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
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Examples of isomorphic W* algebra with non-homeomorphic weak topology
Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
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Reflection-invariant monomial ideals and Alexander duality
First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
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Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
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Dualizable presheaves with respect to Day convolution
This question was posted on MSE and got very little attention, so I'm also posting it here.
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...
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Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
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Support of cohomology of a dualizing complex
Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the ...
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Weak$^*$ topology on Bochner $L^p$-spaces
Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
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Duality of Bochner $L^{\infty}$ space
Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space
$$
E:=L^{\infty}([0,1],X),
$$
i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
1
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1
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118
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For univariate polynomial is non-negativity on an interval equivalent to having a nonnegative scalar product with non-negative polynomials
Let $\mathbb R_d[t]$ be the set of univariate polynomials in the variable $t$ of degree $d$, and $S$ be the set of elements of $\mathbb R_d[t]$ that are nonnegative on $[0, 1]$. Does the following ...
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Access to a classic reference of Dold-Puppe
There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as:
A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
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Discrete primal-duality in optimization
I would like to inquire about the existence of something perhaps similar to the duality theorem in the convex analysis or convex programming in the discrete setting. Here is a concrete example.
Let $...
- 2,251
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Why the circle for Pontryagin duality? [duplicate]
For a locally compact group $G$, we define the Pontryagin dual as $\hat G = Hom(G,\mathbb T)$ where $\mathbb T$ is the circle group and the homomorphisms are continuous group maps. This duality has a ...
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What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$?
This is different from $C$ being dualizable ($[C,D] = C^\vee \otimes D$). (EDIT: It turns out to be the same -- see Mike Shulman's answer!) But for example, if $C$ is a locally free sheaf of finite ...
- 64k
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Gabriel-Ulmer duality for $\infty$-categories
Gabriel-Ulmer duality states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable ...
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Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
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Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf?
Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded
(respectively, absolutely summable) real sequences. I need to find out if
$l^{\infty}$ equipped with the Mackey topology $\tau(l^{\...
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Uniqueness of dualizing objects
One definition of (symmetric) star-autonomous category is as a closed symmetric monoidal category $(C,\otimes,I,\multimap)$ equipped with an object $\bot$ such that all double-dualization maps $A \to (...
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Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function
We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by
$$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$
...
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Topologies of pairs and closed bounded convex sets
[I have posted this question on MSE some time ago, but received no answer.]
It is known, that if two locally convex topologies on a vector space determine the same collection of continuous linear ...
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Criterion for weak compactness
Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous ...
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Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
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If a $\otimes$-idempotent object has a dual, must it be self-dual?
Let $C$ be a symmetric monoidal category.
Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...
- 64k
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Strong duality for a particular moment problem
Reading the paper in this Link (see pag 13) with the objective of understanding a topic related to stochastic optimization I came across a problem in demonstrating one of the theorems. The situation ...
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Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
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Is the category of profunctors $Prof(A,B)$ equivalent to $Prof(B,A)^{op}$?
$\def\Prof{\mathsf{Prof}}\def\Set{\mathsf{Set}}\def\tobar{\mathrel{\mkern3mu \vcenter{\hbox{$\scriptscriptstyle+$}}\mkern-12mu{\to}}}$Let $A$ and $B$ be categories. Define a profunctor $A\tobar B$ to ...
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Are there other dualities on finite vector spaces besides the canonical one?
Let $\text{FinVec}$ denote the category of finite dimensional vector spaces over some field $k$, and let $F:\text{FinVec}\to \text{FinVec}$ be a contravariant functor such that $F^2$ is naturally ...
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Is there any dual relationship between quasi-completeness and barrelledness?
In the theory of stereotype spaces, it is known that for a locally convex space $X$,
If $X$ is pseudocomplete, then $X^{\star}$ is pseudosaturated, and
If $X$ is pseudosaturated, then $X^{\star}$ ...
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512
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Grothendieck duality for resolution of singularities
I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that ...
- 577
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Quantum subgroups of Locally Compact Groups and Parabolic Induction
In the classical theory , parabolic induction is used to construct the (reduced) dual of a (semi-simple) Lie Group. However, for this we need subgruops. Given that the theory of "quantum subgroups" of ...
- 561
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606
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Dualizing sheaf on a Cohen-Macaulay scheme
I would like to fill a detail in the answer given by Francesco Polizzi to the question "Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?".
Let $X$ be a normal, Cohen-Macaulay scheme of ...
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Arithmetic projective duality
Projective duality is a duality that associates to a (smooth) subvariety X of $\mathbb{P}^n$ the dual variety $X^*\subset\mathbb{P}^{n*}$ of tangent hyperplanes.
What makes the duality interesting ...
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210
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On Radon-Nikodym property of a dual of a Bochner space
Let ${\rm S}^{n-1}$ be the unit sphere of ${\bf R}^n$ and let us consider the dual of the space $L^p\left({\bf R}^n; C^1({\rm S}^{n-1})\right)$, for some $p\in\langle1,\infty\rangle$: it is the space ...
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607
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GOE/GSE duality and Bott periodicity
Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is https://arxiv.org/pdf/math-...
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Characterization of the weak completion of $L^2(\mathbb{R}^d)$
The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is ...
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