General 2-Dimensional Adjunctions, Universal Monads and Simplicial Structures

TL;DR

This paper develops a framework for 2-dimensional adjunctions and monads, introducing lax-Gray monads and a simplicial structure analogue called Delta LG, extending classical categorical concepts to higher dimensions.

Contribution

It introduces the notion of lax-Gray monads with coherence equations and constructs a 2-dimensional simplicial analogue, Delta LG, as a higher-dimensional extension of classical structures.

Findings

Derived coherence equations for lax-Gray 2-monads

Constructed the free lax-Gray 2-monad on one object

Defined the simplicial analogue Delta LG for higher-dimensional structures

Abstract

We use the general notion of 2-dimensional adjunction with given coherence equations as introduced by MacDonald-Stone, building on earlier work by Gray, to derive coherence equations for a general 2-monad, which we refer to as a lax-Gray monad. The free lax-Gray 2-monad on one object may be regarded as the suspension of a lax 2-dimensional analogue of the simplicial category Delta. We call this analogue Delta LG for lax-Gray Delta. This is analogous to the way that the free 1-monad Mnd (as presented in Schanuel-Street) is a concrete example of the suspension of the simplicial category Delta, described by Mac Lane.