Skew Monoidales in $Span$

TL;DR

This paper characterizes skew monoidales within the monoidal bicategory $Span$ by relating them to categories equipped with a specific functor satisfying certain conditions, expanding understanding of their structure.

Contribution

It provides a novel characterization of skew monoidales in $Span$ using the shift functor on simplicial sets, including an example with a non-restricted unit.

Findings

Skew monoidales in $Span$ correspond to categories with a functor satisfying specific conditions.

The paper introduces a characterization involving the decalage functor on simplicial sets.

An example demonstrates a skew monoidale with a non-restricted unit.

Abstract

This article consists of an interesting characterisation of a skew monoidale in the monoidal bicategory $Span$. After discussing the shift or decalage functor on simplicial sets we characterise these skew monoidales as categories $\mathbb C$ together with a functor $R \colon Dec(\mathbb C) \longrightarrow \mathbb C$ which satisfies two conditions and give an example where the unit of the skew monoidale is not of a restricted type.