Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 27th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea An alternative to [[complete topological vector spaces]] in the framework of [[condensed mathematics]]. Roughly, completeness is expressed as ability to integrate with respect to [[Radon measures]]. This doesn't quite work as stated, and to make this rigorous one has to bring [[L^p-spaces]] for $0\lt p\le 1$ (i.e., the non-convex case) into the picture. ## Definition A [[condensed abelian group]] $V$ is __$p$-liquid__ ($0\lt p\le 1$) if for every [[compact Hausdorff topological space]] $S$ and every morphism of [[condensed sets]] $f\colon S\to V$ there is a unique morphism of [[condensed abelian groups]] $M_{<p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{<p}(S)$. Here for a [[compact Hausdorff topological space]] $S$ and for any $p$ such that $0<p\le 1$ we have $$M_{<p}(S)=\bigcup_{q<p}M_q(S),$$ where $$M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},$$ where $$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$ where $S_i$ are finite sets such that $$S = \lim_i S_i$$ and $$M(F)_{\ell^p\le C}$$ for a [[finite set]] $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with [[l^p-norm]] at most $C$. ## References See [[condensed mathematics]]. <a href="https://ncatlab.org/nlab/revision/liquid+vector+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/liquid+vector+space">current</a>

   Created:

   Idea

   An alternative to complete topological vector spaces in the framework of condensed mathematics.

   Roughly, completeness is expressed as ability to integrate with respect to Radon measures.

   This doesn’t quite work as stated, and to make this rigorous one has to bring L^p-spaces for $0\lt p\le 1$ (i.e., the non-convex case) into the picture.

   Definition

   A condensed abelian group $V$ is $p$-liquid ($0\lt p\le 1$) if for every compact Hausdorff topological space $S$ and every morphism of condensed sets $f\colon S\to V$ there is a unique morphism of condensed abelian groups $M_{<p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{<p}(S)$.

   Here for a compact Hausdorff topological space $S$ and for any $p$ such that $0<p\le 1$ we have

   $$M_{<p}(S)=\bigcup_{q<p}M_q(S),$$

   where

   $$M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},$$

   where

   $$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$

   where $S_i$ are finite sets such that

   $$S = \lim_i S_i$$

   and

   $$M(F)_{\ell^p\le C}$$

   for a finite set $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with l^p-norm at most $C$.

   References

   See condensed mathematics.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2022
   • (edited May 28th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave fixed the link to [[p-norm|$\ell^p$-norm]]: [[p-norm|$\ell^p$-norm]] <a href="https://ncatlab.org/nlab/revision/diff/liquid+vector+space/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/liquid+vector+space/3">v3</a>, <a href="https://ncatlab.org/nlab/show/liquid+vector+space">current</a>

   have fixed the link to $\ell^p$-norm:

     [[p-norm|$\ell^p$-norm]]
   

   diff, v3, current