Free decomposition spaces
Philip Hackney, Joachim Kock

TL;DR
This paper introduces free decomposition spaces, a new class of simplicial spaces generated by inert maps, and explores their properties and applications in combinatorics, including connections to quasi-symmetric functions.
Contribution
It defines free decomposition spaces, shows their relation to M"obius decomposition spaces via Kan extension, and establishes an equivalence of categories, with applications to combinatorics.
Findings
Free decomposition spaces are generated by inert maps.
Left Kan extension maps objects to M"obius decomposition spaces.
All deconcatenation type comultiplications arise from free decomposition spaces.
Abstract
We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion takes general objects to M\"obius decomposition spaces and general maps to CULF maps. We establish an equivalence of -categories . Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Intracranial Aneurysms: Treatment and Complications
