Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans

  title={Locally (co)Cartesian fibrations as realisation fibrations and the classifying space of cospans},
  author={Jan Steinebrunner},
  journal={Journal of the London Mathematical Society},
  • J. Steinebrunner
  • Published 16 September 2019
  • Mathematics
  • Journal of the London Mathematical Society
We show that the conditions in Steimle's ‘additivity theorem for cobordism categories’ can be weakened to only require locally (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category. 

The classifying space of the one‐dimensional bordism category and a cobordism model for TC of spaces

The homotopy category of the bordism category hCd has as objects closed oriented (d−1) ‐manifolds and as morphisms diffeomorphism classes of d ‐dimensional bordisms. Using a new fiber sequence for

N ov 2 02 2 The equifibered approach to ∞-properads

We define a notion of ∞-properads that generalises ∞-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new

Culf maps and edgewise subdivision

We show that, for any simplicial space X, the ∞-category of culf maps over X is equivalent to the ∞-category of right fibrations over Sd(X), the edgewise subdivision of X (when X is a Rezk complete

Two-variable fibrations, factorisation systems and $\infty$-categories of spans

. We prove a universal property for ∞ -categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show



A cobordism model for Waldhausen K ‐theory

We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy

(1) Proceedings of the London Mathematical Society (2) Journal of the London Mathematical Society

THE London Mathematical Society now issues its transactions in two volumes a year, the Proceedings and the Journal. Vol. 25 of the Proceedings contains 31 technical papers on various branches of

Homotopy Relat

  • Struct., pages 703–714,
  • 2018

Semi-simplicial spaces

J. Ebert was partially supported by the SFB 878. O. Randal-Williams was supported by EPSRC grant number EP/M027783/1.

Algebraic and geometric topology

Contains sections on Structure of topological manifolds, Low dimensional manifolds, Geometry of differential manifolds and algebraic varieties, $H$-spaces, loop spaces and $CW$ complexes, Problems.

The homotopy type of the cobordism category

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main

An additivity theorem for cobordism categories

We give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries. Unlike the existing proofs, it is


The algebraic K–theory of spaces is a variant, invented by F. Waldhausen in the late 1970’s, of the standard algebraic K–theory of rings. Until that time, applications of algebraic K–theory to