arXiv subjects

John Boiquaye

Publications and source records attributed to John Boiquaye.

3 recordsLinked to original sources

arXiv2026

Duplicial sets, crossed $G$-sets and descent categories

The goal of this note is to show that the left $χ$-coalgebra, which is an additional structure on one of the coefficients used in the construction of the cyclic operator for the cyclic sets that generalises the twisted nerve of a group by Loday is an instance of a general theory of left $χ$-coalgebras. It is also shown that in the general case, the resulting cyclic operator requires that the other coefficient needs to be equipped with a crossed $G$-sets structure.

math.QA↗

arXiv2025

Duplicial functors, descent categories and generalized Hopf modules

Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^χ_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $χ$ between two comonads. For the key example associated to a bialgebra $H$, right $χ$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $χ$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples.

math.CT↗

arXiv2022

Cyclic duality for slice and orbit 2-categories

The self-duality of the paracyclic category is extended to a certain class of homotopy categories of (2,1)-categories. These generalise the orbit category of a group and are associated to certain self-dual preorders equipped with a presheaf of groups and a cosieve. Slice 2-categories of equidimensional submanifolds of a compact manifold without boundary form a particular case, and for $S^1$, one recovers cyclic duality. This provides in particular a visualisation of the results of Böhm and Ştefan on the topic.

math.CT↗