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Homotopy composition of cospans

Joachim Kock, David I. Spivak
2017 Communications in Contemporary Mathematics  
It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra.  ...  In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.  ...  In light of the growing importance of homotopy viewpoints in modern mathematical sciences, we wonder whether the finer composition of cospans, provided by the homotopy pushout, may be of relevance in the  ... 
doi:10.1142/s0219199716500474 fatcat:ldhjtuy2fjaixfzjrhkybmsko4

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

Jan Steinebrunner
2022 arXiv   pre-print
As an application we compute the difference in classifying spaces between the infinity category of cospans of finite sets and its homotopy category.  ...  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  ...  Acknowledgements I would like to thank my advisor Ulrike Tillmann for her support throughout all stages of the writing process.  ... 
arXiv:1909.07133v2 fatcat:6c6oqhtflrgwfovqpp3xiw3tae

An extension of Brown functor to cospans of spaces [article]

Minkyu Kim
2020 arXiv   pre-print
In this paper, we go further by giving a sufficient condition for an A-valued Brown functor to extend to a homotopy-theoretic analogue of A-valued TQFT for arbitrary A.  ...  If A is the category of abelian groups, then the TQFTs obtained by Dijkgraaf-Witten theory of abelian groups or Turaev-Viro theory of bicommutative Hopf algebras factor through A up to a scaling.  ...  Acknowledgements The author was supported by FMSP, a JSPS Program for Leading Graduate Schools in the University of Tokyo, and JPSJ Grant-in-Aid for Scientific Research on Innovative Areas Grant Number  ... 
arXiv:2005.10621v1 fatcat:6skooobzangjlkok5dxgkqvh5i

Cubical cospans and higher cobordisms (Cospans in algebraic topology, III) [article]

Marco Grandis
2008 arXiv   pre-print
We also build a second structure, called a quasi cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has lax identities.  ...  After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological spaces.  ...  by means of homotopy pushouts.  ... 
arXiv:0806.2359v1 fatcat:id52h55kprf4vg4qcwleofdg74

Indeterminacies and models of homotopy limits [article]

Alisa Govzmann, Damjan Pištalo, Norbert Poncin
2021 arXiv   pre-print
In the case of homotopy pullbacks, we introduce the concept of full homotopy pullback by identifying the homotopy pullbacks associated with three different model structures of the category of cospan diagrams  ...  Finally, we define generalized representatives or models of homotopy limits and full homotopy pullbacks.  ...  We say that A is a generalized representative of the σ-homotopy limit of X or is a σhomotopy limit model of X, if there exists a fibrant replacement F σ X of X such that the composite of universal morphisms  ... 
arXiv:2109.12395v1 fatcat:36wflqyd6jb6hbcprav2tvsy4a

Homotopy limits in type theory

JEREMY AVIGAD, KRZYSZTOF KAPULKIN, PETER LEFANU LUMSDAINE
2015 Mathematical Structures in Computer Science  
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant.  ...  We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material.  ...  Most of this work was carried out during the Special Year on Univalent Foundations at the Institute for Advanced Study, and all three authors are grateful to the Institute for its hospitality and support  ... 
doi:10.1017/s0960129514000498 fatcat:h72zjn2fknduzls36hsw5mkvqu

Equivariant Higher Hochschild Homology and Topological Field Theories [article]

Lukas Müller, Lukas Woike
2019 arXiv   pre-print
For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the (∞,1)-category of cospans of E_∞-algebras  ...  We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group G. As coefficients, we allow E_∞-algebras with G-action.  ...  1670 "Mathematics inspired by String theory and Quantum Field Theory" and thanks the Heriot-Watt University in Edinburgh and in particular Richard Szabo for their hospitality during the time when part of  ... 
arXiv:1809.06695v2 fatcat:iqim3xgv6rggnamtnxis53hnja

Calabi-Yau structures for multiplicative preprojective algebras [article]

Tristan Bozec, Damien Calaque, Sarah Scherotzke
2021 arXiv   pre-print
In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions.  ...  Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.  ...  We refer to [19, 31] for a detailed introduction to dg-categories and their homotopy theory.  ... 
arXiv:2102.12336v2 fatcat:7cdz6jy54vawxbyabysq2z346m

A cobordism model for Waldhausen K‐theory

George Raptis, Wolfgang Steimle
2018 Journal of the London Mathematical Society  
As an example, we discuss this model for A-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausen's A(*).  ...  We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans.  ...  We warmly acknowledge the support and the hospitality of the Hausdorff Institute for Mathematics in Bonn where a preliminary outline of this work was completed.  ... 
doi:10.1112/jlms.12182 fatcat:gmssuf4g4jdydmryie6temv3tu

Simplicial Waldhausen categories and topological K-theory [article]

Yi-Sheng Wang
2020 arXiv   pre-print
Utilizing simplicial Waldhausen theory, we prove that the geometric realization of the topologized category of bounded chain complexes over complex numbers (resp. real numbers) is an infinite loop space  ...  The required natural transformation is then given by the composition, O(A · × A · × B · ) → O(Cospan(A · ) × B · ) O(θ A · )×id − −−−−−− → O(Cospan(A · ) × B · ) → O(Cospan(C · )) φ C · − − → M(Cospan(  ...  For the sake of convenience, we call the above composition F ·· . The argument so far is a straightforward generalization of the one in [19, p.372 ]. 2.  ... 
arXiv:1705.05192v3 fatcat:fq7v4alxvfh3bgtysgdepusw5m

A pair of homotopy-theoretic version of TQFT's induced by a Brown functor [article]

Minkyu Kim
2021 arXiv   pre-print
The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short).  ...  A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category.  ...  Overview of previous study In this section, we give an overview of our previous study. In subsection 2.1, we give main theorems in [10] .  ... 
arXiv:2006.10438v3 fatcat:ydchgacdlbc7lh2uioml3tyuce

Higher Equipments, Double Colimits and Homotopy Colimits [article]

Redi, Haderi
2019 arXiv   pre-print
As an application we prove that certain double colimits are naturally interpreted as homotopy colimits.  ...  However there is no double category of spaces even though notions of bimodule are conceivable.  ...  Acknowledgements I would like to thank all the members of the Mathematics Department at Bilkent University.  ... 
arXiv:1908.06201v1 fatcat:7wjx6gvx5vhvjorhmgtn5ga6da

Calabi-Yau structures on (quasi-)bisymplectic algebras [article]

Tristan Bozec, Damien Calaque, Sarah Scherotzke
2022 arXiv   pre-print
We prove along the way that the fusion process (a) corresponds to the composition of Calabi-Yau cospans with "pair-of-pants" ones, and (b) preserves the duality between non-degenerate double quasi-Poisson  ...  structures on (dg-versions of) these algebras.  ...  Since the homotopy between the 1-forms in the cospan k[z] ∐ R≥3 −→ k x 1 , x 2 ∐ R≥3 ←− R is trivial, the zero-homotopy of the composition of Calabi-Yau cospans is given by the image of ω under the map  ... 
arXiv:2203.14382v1 fatcat:7jxflnshknfnznheyzhqdpbd6q

A new approach to model categorical homotopy fiber sequences

Alisa Govzmann, Damjan Pištalo, Norbert Poncin
2022 Dissertationes Mathematicae  
This advantage arises from the understanding that the homotopy theory of the model category's arrow category contains all homotopical information about its long fibration sequences.  ...  We propose a simplified definition of Quillen's fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action.  ...  So composition of morphisms of homotopy fiber sequences is induced by the composition of M. We denote by h(M) the category of homotopy fiber sequences of M.  ... 
doi:10.4064/dm858-5-2022 fatcat:gksydqysozbtfozwmzijdconke

The AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory [article]

Damien Calaque, Rune Haugseng, Claudia Scheimbauer
2021 arXiv   pre-print
Finally, we construct forgetful functors from the unoriented bordism (∞,n)-category to cospans of spaces, and from the oriented bordism (∞,n)-category to cospans of spaces equipped with an orientation;  ...  We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable.  ...  Composing cobordisms by gluing gives in particular a homotopy pushout of topological spaces, and so corresponds to a composition of cospans.  ... 
arXiv:2108.02473v1 fatcat:gnu44ixktvg2tcv5xnknxqaogq
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