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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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 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.
   • CommentAuthorUrs
   • CommentTimeOct 29th 2019
   • (edited Oct 29th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSome naive ramblings, just thinking out loud, in case anyone feels inspired to offer a comment: I am trying to see how close to an ordered configuration space of points one can get with mapping spaces, which a priori give un-ordered configuration spaces of points. The idea I have is -- in words -- the following: An ordered configuration of points in $\mathbb{R}^3$ (say) is, up to homotopy, the same as a) An un-ordered configuration of points in $\mathbb{R}^3 \times \mathbb{R}^1$, such that this b) projects to an $\mathbb{R}^1$-labeled un-ordered configuration in $\mathbb{R}^3$; and c) projects to an $\mathbb{R}^3$-labeled un-ordered configuration in $\mathbb{R}^1$. Meaning that the points in the configuration are distinct not only as points in $\mathbb{R}^3 \times \mathbb{R}^1$, but also after projection as points in $\mathbb{R}^3$ and as points in $\mathbb{R}^1$. Here condition c) is what imposes an ordering on the "labels" in $\mathbb{R}^1$, since an arrangement of distict points on the real line puts these points into linear order. The formal statement of this idea should be that * the ordered un-labeled configuration space $\underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3)$ is a fiber product (in the 1-category of topological spaces) of * the unordered $X$-labeled configuration spaces $Conf( \mathbb{R}^k, X )$ with points disappearing when labeled by the base-point of $X$ as follows: $$ \underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3) \;\simeq_{homeo}\; Conf\big( \mathbb{R}^3, S^1 \big) \underset{ Conf\big( \mathbb{D}^{3+1} \big)_{rel} }{\times} Conf\big( \mathbb{R}^1, S^3 \big) $$ That this is the case should essentially come down to observing that this fiber product encodes the above "in words" description. First I thought that this 1-categorical fiber product is a homotopy-retraction of the corresponding homotopy fiber product, but now I think this can't be. This is because all items in the above are homotopy equivalent to based mapping spaces as $$ Maps^{\!\ast/\!}\big( S^3, S^4\big) \underset{ Maps^{\!\ast/\!}\big( S^0, S^4\big) }{\times^h} Maps^{\!\ast/\!}\big( S^1, S^4\big) $$ and of these I know the rational models, and there is no way for any homotopy fiber product of these to be equivalent to the rational model for the ordered configuration space. So now I am thinking that maybe I should regard the configuration spaces above as smooth manifolds, and as such as smooth stacks, and then think of them as differential refinements of the homotopy types of these mapping spaces (which are Cohomotopy cocycle spaces), to find that the ordered configuration space, as a smooth manifold/stack, is a homotopy fiber product of differentially refined Cohomotopy cocycle spaces. But not sure if that's a fruitful picture...

   Some naive ramblings, just thinking out loud, in case anyone feels inspired to offer a comment:

   I am trying to see how close to an ordered configuration space of points one can get with mapping spaces, which a priori give un-ordered configuration spaces of points.

   The idea I have is – in words – the following:

   An ordered configuration of points in $\mathbb{R}^3$ (say) is, up to homotopy, the same as

   a) An un-ordered configuration of points in $\mathbb{R}^3 \times \mathbb{R}^1$,

   such that this

   b) projects to an $\mathbb{R}^1$-labeled un-ordered configuration in $\mathbb{R}^3$;

   and

   c) projects to an $\mathbb{R}^3$-labeled un-ordered configuration in $\mathbb{R}^1$.

   Meaning that the points in the configuration are distinct not only as points in $\mathbb{R}^3 \times \mathbb{R}^1$, but also after projection as points in $\mathbb{R}^3$ and as points in $\mathbb{R}^1$.

   Here condition c) is what imposes an ordering on the “labels” in $\mathbb{R}^1$, since an arrangement of distict points on the real line puts these points into linear order.

   The formal statement of this idea should be that

   • the ordered un-labeled configuration space $\underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3)$

   is a fiber product (in the 1-category of topological spaces) of

   • the unordered $X$-labeled configuration spaces $Conf( \mathbb{R}^k, X )$ with points disappearing when labeled by the base-point of $X$

   as follows:

   $$\underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3) \;\simeq_{homeo}\; Conf\big( \mathbb{R}^3, S^1 \big) \underset{ Conf\big( \mathbb{D}^{3+1} \big)_{rel} }{\times} Conf\big( \mathbb{R}^1, S^3 \big)$$

   That this is the case should essentially come down to observing that this fiber product encodes the above “in words” description.

   First I thought that this 1-categorical fiber product is a homotopy-retraction of the corresponding homotopy fiber product, but now I think this can’t be.

   This is because all items in the above are homotopy equivalent to based mapping spaces as

   $$Maps^{\!\ast/\!}\big( S^3, S^4\big) \underset{ Maps^{\!\ast/\!}\big( S^0, S^4\big) }{\times^h} Maps^{\!\ast/\!}\big( S^1, S^4\big)$$

   and of these I know the rational models, and there is no way for any homotopy fiber product of these to be equivalent to the rational model for the ordered configuration space.

   So now I am thinking that maybe I should regard the configuration spaces above as smooth manifolds, and as such as smooth stacks, and then think of them as differential refinements of the homotopy types of these mapping spaces (which are Cohomotopy cocycle spaces), to find that the ordered configuration space, as a smooth manifold/stack, is a homotopy fiber product of differentially refined Cohomotopy cocycle spaces.

   But not sure if that’s a fruitful picture…

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 30th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor what it's worth, I have uploaded a graphics that illustrates this construction of the ordered unlabeled configuration space as a fiber product of unordered labeled configuration spaces: [here](https://ncatlab.org/nlab/show/configuration+space+of+points#OrderedUnlabeledConfigurationsFromUnorderedLabeledConfigurations)

   For what it’s worth, I have uploaded a graphics that illustrates this construction of the ordered unlabeled configuration space as a fiber product of unordered labeled configuration spaces: here