Search arXiv⌕ Search

arXiv · 0704.1624

Complete Segal spaces arising from simplicial categories

Abstract

In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Julia E. Bergner. 2007-10-11. Complete Segal spaces arising from simplicial categories. https://arxiv.org/abs/0704.1624

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

On The Rational Realization of Even-dimensional Spheres and Products of Eilenberg--MacLane Spaces as Classifying Spaces

In this paper, we study the rational realization problem for the classifying space $B\mathrm{aut}_1(X)$. We prove that if $S^{2n}$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected space $X$, then $X$ is $π$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\mathbb{Q}^s,n)\times K(\mathbb{Q}^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $π$-finite space $X$. Moreover, we prove that if $r\geq 2$, $n\geq 3$ and $K(\mathbb{Q}^r,n)$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected $π$-finite space $X$, then $X$ has the rational homotopy type of $K(\mathbb{Q}^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

math.AT↗

A complete classification of the rational realization of gap-two products of Eilenberg--MacLane spaces as classifying spaces

In this paper, we study the rational realization of the gap-two product $K(\mathbb{Q},n)\times K(\mathbb{Q},n+2)$ as the classifying space $B\mathrm{aut}_1(X)$ for a simply-connected $π$-finite space $X$ with rational homology of finite type. We prove that $K(\mathbb{Q},n)\times K(\mathbb{Q},n+2)$ can be realized as $B\mathrm{aut}_1(X)$ up to rational homotopy equivalence if and only if $n=3,4$ or $4m+2$ for $m\geq 1$. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such $X$ is unique for $n=3$ or $4m+2$ for $m\geq 1$; there exist infinitely many rational homotopy types of such $X$ for $n=4$.

math.AT↗