All Questions
Tagged with duality reference-request
31 questions
13
votes
0
answers
810
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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) ...
- 63.1k
1
vote
1
answer
120
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Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 8,945
1
vote
1
answer
65
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Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
0
votes
1
answer
111
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Kernel of restriction in étale cohomology of curves over number fields
Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \...
- 37k
10
votes
0
answers
534
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Isbell duality between algebras and sheaves
nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...
- 1
2
votes
0
answers
147
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
- 6,377
96
votes
36
answers
17k
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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....
1
vote
1
answer
214
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Homology of complements and homotopy equivalence
Albrecht Dold gave a short proof of the Jordan-Alexander complement theorem in the following form.
Given two closed sets $A$, $B \subset {\bf R}^n$ that are homeomorphic, the singular homology groups $...
- 18.7k
4
votes
0
answers
113
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
10
votes
1
answer
403
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Finite domination and Poincaré duality spaces
Here are some definitions:
A space is homotopy finite if it is homotopy equivalent to a finite CW complex.
A space finitely dominated if it is a retract of a homotopy finite space.
A space $X$ is a ...
- 18.9k
2
votes
1
answer
108
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Is the polar dual of a semi-algebraic convex body also semi-algebraic?
Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as
$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$
with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a ...
- 13.6k
2
votes
1
answer
353
views
References on duality of fractional order Sobolev spaces
I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite ...
CommunityBot
- 1
2
votes
0
answers
124
views
Heap torsors and dual objects
Suppose that $G$ is a group and $P:G\rightarrow G$ is a permutation of $G$. We would now like to apply a forgetful functor to the algebra $(G,\cdot,P,^{-1},e)$. Whenever $g\in G$, let $gP$ be the ...
1
vote
0
answers
138
views
$L_p$ estimate in mixed boundary problem for elliptic equation
Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...
- 46
3
votes
0
answers
162
views
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 ...
- 3,513
5
votes
0
answers
841
views
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 ...
- 53.3k
10
votes
1
answer
1k
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Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 18.7k
2
votes
0
answers
122
views
A new topology on the dual of a locally convex space?
Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
- 42k
5
votes
0
answers
141
views
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 \...
- 1,593
19
votes
4
answers
1k
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Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 761
13
votes
1
answer
624
views
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 ...
- 16.5k
5
votes
0
answers
64
views
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 ...
- 289
1
vote
0
answers
190
views
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 ...
- 474
3
votes
0
answers
210
views
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 ...
- 60.6k
2
votes
1
answer
144
views
Representation of the elements of $c_0^\perp$ as integrals over ultrafilters
Let
$$
X=\big\{\varphi\in\ell_\infty^{\,*}(\mathbb N) : \varphi(\{a_n\})=0\,\,\text{whenever $a_n\to 0$}\big\}.
$$
If $\varphi_{\mathscr F}(\{a_n\})$ is the limit of $\{a_n\}$ with respect to the non-...
- 4,461
2
votes
0
answers
357
views
Reference Request: Algebraic Serre's Duality Theorem for Curves
Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne (...
CommunityBot
- 1
4
votes
1
answer
923
views
Electromagnetic duality symmetry
This question arose while reading http://prd.aps.org/abstract/PRD/v13/i6/p1592_1 (Duality transformations of Abelian and non-Abelian gauge fields, by Stanley Deser and Claudio Teitelboim). It is well ...
- 4,577
5
votes
0
answers
344
views
Duality between K-theory and K-homology in the non-compact, spin$^c$ case
Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\...
- 2,998
6
votes
1
answer
670
views
Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
- 20.5k
9
votes
0
answers
322
views
Twisted duality in a symmetric monoidal category
I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?
Definition. Let $\mathcal{C}$ be a ...
- 63.1k
5
votes
2
answers
712
views
Alexander duality theorem for CW-complexes and stable homotopy theory
In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem
Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$
for finite CW-complexes with a "nice ...
- 23.5k