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Decomposition spaces and restriction species

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journal contribution
posted on 15.08.2018, 13:08 by Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks
We show that Schmitt's restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital 2-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce the notion of directed restriction species that subsume Schmitt's restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher-Connes-Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.

Funding

The first author was partially supported by grants MTM2012-38122-C03-01, 2014-SGR-634, MTM2013-42178-P, MTM2015-69135-P, and MTM2016-76453-C2-2-P, the second author by MTM2013- 42293-P and MTM2016-80439-P and the third author by MTM2013-42178-P and MTM2016-76453-C2- 2-P.

History

Citation

International Mathematics Research Notices, 2018, rny089

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

International Mathematics Research Notices

Publisher

Oxford University Press (OUP)

issn

1073-7928

eissn

1687-0247

Acceptance date

17/04/2018

Copyright date

2018

Available date

12/09/2019

Publisher version

https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny089/5095270

Notes

This is the last of six papers that formerly constituted the long manuscript arXiv:1404.3202;The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.

Language

en

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