# Questions tagged [duality]

Use for questions regarding duality of mathematical object, i.e. dual spaces, objects with two possible interpretations etc.

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### References: properties of $L_p$ spaces involving time and probability space

Questions are from the theory of PDEs\SPDEs
Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...

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### Gross-Hopkins duality

$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...

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### Lovasz extension and underlying matroid

Let $F \colon 2^V \rightarrow \mathbf{R}$ be a set function for some ground set $V = \{ 1, \dots, n\}$. The domain of $F$ is $\{ 0,1\}^n$ , under the usual identification of a set $S$ with its ...

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### Is there Sperner-type lemma where the colors are on the edges?

I am looking for Sperner-type lemmas where:
The colors are on the edges (1-dimensional simplices) - rather than on the vertices;
The target is a fully-colored vertex (a vertex adjacent to n edges of ...

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### How does Gabriel–Ulmer duality extend to (limit, colimit) sketches?

$\newcommand\Sketch{\mathit{Sketch}}\newcommand\Set{\mathit{Set}}
\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\Colim{Colim}\DeclareMathOperator\Mod{Mod}\newcommand\mod{\operatorname{mod}}\...

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### Heap torsors and dual objects

Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...

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### Dual space of Carleman functions

Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which
$$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...

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### Are more duality mappings for matrix norms known?

When reading A Unifying Representer Theorem for Inverse Problems and Machine Learning by Michael Unser and Duality Mapping for Schatten Matrix Norms by his PhD student
Shayan Aziznejad, I wondered if ...

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### Is $(K^*)^{**}=(K^{**})^*$ for any cone $K$?

I'm considering the dual cone $K^*$ of a non-convex cone $K$. I came up with a theory that $K^{**}$ is the closure of convex hull of $K$. Then I wonder whether $(K^*)^{**}=(K^{**})^*$ holds for any ...

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### Does Grothendieck duality hold without taking RHom?

I asked this very basic question about Grothendieck Duality on the Stack--exchange some time ago, without any replies.
I'm therefore asking the question here to test my luck.
Let $f:X\to Y$ be a ...

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### Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...

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### Unifying two definitions of $L^\infty$

Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon measure on $X$.
Definitions:
A subset $E\subseteq X$ is called locally Borel if $F \cap E$ is Borel for every Borel set $F\subseteq X$ ...

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### Is the Serre dualizing complex local in the analytic topology?

There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...

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### Why aren‘t op and co switched?

When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...

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### Simplest sensitivity analysis in semi-infinite linear programming

Consider a standard linear program of the form
\begin{align*}
\min_{x}c^{\top}x & \,\,\text{subject to}\\
Ax & =b\\
b & \geq0\,.
\end{align*}
It is well-known that if we perturb the right-...

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### Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari.
After some arguments, we get a exact sequence
$$
\mathbf{P}^1_S(k,M^{'})^* \...

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### Reference for free symmetric monoidal categories with duals on symmetric monoidal categories

The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons.
In ...

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### Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?

Let $\Omega \subset \mathbb{R}^n$ be open and bounded.
What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...

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### Order of factors for unit/counit of duals

While working through Categorical quantum mechanics by Abramsky and Coecke, I noticed that their definition of the order of factors for the unit and counit of a dualizable object disagree with the one ...

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### $L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...

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### Does coercivity/supercoercivity conjugates?

According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if,
$$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$
and it is super-coercive if
$$\...

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### Self-dual hypergraph on $\omega$

Let $H=(V,E)$ be a hypergraph. For $v\in V$ we let $v^* = \{e\in E:v\in e\}$. We define the dual of $H$ by $H^*= (E, V^*)$ where $V^* = \{v^*: v\in V\}$. We say that a $H$ is self-dual if $H \cong H^*$...

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### Reference request for a complete and formal Duality Principle in category theory

Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...

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### Does rigidity imply a unique dualizing functor?

Let $\mathcal{C}$ be a rigid, monoidal category. Can I talk about $\mathcal{C}$ as having a unique, well-defined, dualizing functor (i.e. one that maps objects and morphisms onto their respective ...

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### What are all the natural maps between iterated duals of vector spaces, and equations between these?

Fix a field. Given a vector space $V$ it has a dual $V^\ast$, which has its own dual $V^{\ast \ast}$, which has its own dual $V^{\ast \ast \ast}$, and so on ad infinitum.
There are lots of natural ...

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### Evolution PDE in dual space : Generalization of a result of Gelfand

The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand :
Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...

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### Continuum of Lagrange multipliers, duality gap, and minimax theorem

Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...

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### Effect of dualization of density

Let $D\subset X$ be a dense subset of a complete separable locally convex space $X$ over $\mathbb{R}$. Though the question seems simple enough, I can't seem to find the answer in the literature:
If $...

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### Category with binary biproducts but no zero morphism

Is there a category with binary biproducts but no zero morphism?
I'm wondering if the definition of biproducts as objects that are simultaneously products and coproducts that obey some identities on ...

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### Dual norm for weighted Sobolev space

Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm:
\begin{equation}
\|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{...

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### Reference for duality inducing bijections between subobjects and quotients?

I'm not the most category-theoretic person but I have run into the following statement in my work. Suppose we have a category duality $\mathcal F:\mathcal C\to\mathcal D$. For my needs you can ...

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### Top cohomology of profinite Poincaré duality group

The paper "Cohomology of p-adic analytic groups" by Symonds and Weigel is considered one of the main references for continuous cohomology of profinite groups. There is a passage I do not ...

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### No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...

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### Generalization of Bernstein’s inequality

I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim:
Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...

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### Confusing definition of homogeneous Sobolev norm of order -1

Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...

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### Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry

Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...

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### Dual of union of Reproducing Kernel Hilbert Spaces

I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...

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### Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ on a Polish space $X$ is given by
$$
W_p^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...

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### Origin and context of adjunctions inducing equivalences between full subcategories

The following is well-known.
Theorem. Let $F\dashv U$ be a pair of adjoint functors
$$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$
with unit $(\eta_A\colon A\to U(F(A)))_{...

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### Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...

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### Comparison: Formal Wirthmüller isomorphism of Fausk-Hu-May vs. Balmer et. al

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\tensor}{\otimes}$
$\DeclareMathOperator{\Sp}{Sp}$
This question is about comparing the approaches for a formal Wirthmüller ...

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### Can one characterize maximal antichains in terms of distributive lattices?

This is inspired by the recent question Verification of a maximal antichain
The celebrated duality between finite posets and finite distributive lattices has several nice formulations. One of them ...

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### Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?

The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of ...

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### On the dimension of the dual variety of a singular hypersurface

I was primarily interested in the following question. Let $n\geq 3$, and let $X\subset \mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is ...

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### Duality argument for elliptic regularity

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...

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### Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?

$\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$
A duality between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$)
is, by definition, a bilinear form
$$
\...

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### Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...

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### Calculating vertex weights

Vertex weights are a metaphor for a constant value $\pi_i$ that is added to the weight of every edge $e_{ij}$ that is adjacent to vertex $v_i$ in a symmetric graph $G(V,E)$ with weighted edges.
The ...

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### Weak duality sign

Let $\left\{\begin{matrix}
\operatorname{min}_xc^Tx\\Ax\leq b
\\ x\in \mathbb{R}^+
\end{matrix}\right.$ be a LP primal problem and $\left\{\begin{matrix}
\operatorname{max}_yb^Ty\\A^Ty\geq c
\\ y\in ...

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### Duality of finite signed measures and bounded continuous functions

Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $E\to\mathbb R$ (equipped with the supremum norm), $\mathcal M(E)$ denote the space of finite signed measures on ...