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Presentations of configuration categories

Pedro Boavida de Brito, Michael S. Weiss

Abstract

The configuration category of a manifold is a topological category which we view as a Segal space, via the nerve construction. Our main result is that the unordered configuration category, suitably truncated, admits a finite presentation as a complete Segal space if the manifold in question is the interior of a compact manifold.

Presentations of configuration categories

Abstract

The configuration category of a manifold is a topological category which we view as a Segal space, via the nerve construction. Our main result is that the unordered configuration category, suitably truncated, admits a finite presentation as a complete Segal space if the manifold in question is the interior of a compact manifold.
Paper Structure (8 sections, 21 theorems, 26 equations)

This paper contains 8 sections, 21 theorems, 26 equations.

Table of Contents

  1. Finite presentation properties of complete Segal spaces
  2. Generators and relations for configuration categories
  3. Epi-mono factorizations
  4. Attaching generators and relations
  5. Products of simplices
  6. Decompositions of $\mathsf{ucon}(M|D)$ and the proof of proposition \ref{['prop-addrel']}
  7. Defects
  8. Products of representable simplicial sets

Key Result

Lemma 1.3

Let $Y$ be a complete Segal space which is finitely presentable. Let $f\colon\! W\to Y$ be a map of simplicial spaces, where $W$ is homotopically compact as a simplicial space. Then $f$ admits a factorization \xymatrix{ W \ar[r]^-{f_1} & W' \ar[r]^-{f_2} & Y }in which $f_1$ is a cofibration, $W'$ is

Theorems & Definitions (35)

  • Definition 1.1
  • Example 1.2
  • Lemma 1.3
  • Corollary 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • ...and 25 more