Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

An algebraic representation of globular sets

Pages: 135 – 150

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a8


Anibal M. Medina-Mardones (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)


We describe a fully faithful embedding of the category of (reflexive) globular sets into the category of counital cosymmetric $R$-coalgebras when $R$ is an integral domain. This embedding is a lift of the usual functor of $R$-chains and the extra structure consists of a derived form of cup coproduct. Additionally, we construct a functor from group-like counital cosymmetric $R$-coalgebras to $\omega$-categories and use it to connect two fundamental constructions associated to oriented simplices: Steenrod’s cup‑$i$ coproducts and Street’s orientals. The first defines the square operations in the cohomology of spaces, the second, the nerve of higher-dimensional categories.


globular sets, higher categories, $E_{\infty}$-structures, Steenrod cup‑$i$ products

2010 Mathematics Subject Classification

18D05, 55S05

Received 24 June 2019

Received revised 7 August 2019

Accepted 26 August 2019

Published 15 April 2020