Kernels, lax algebras, d\'ecalage, and supercoherence

TL;DR

This paper characterizes categories with kernels as lax algebras for a specific 2-monad and relates them to supercoherent structures, providing a new perspective on weak unary operadic categories.

Contribution

It establishes an equivalence between categories with kernels, lax algebras for the arrow 2-monad, and décalage of supercoherent structures, offering a novel categorical framework.

Findings

Categories with kernels are equivalent to lax algebras for the arrow 2-monad.

Such categories are also the décalage of supercoherent structures.

Provides a new interpretation of categories with kernels as weak unary operadic categories.

Abstract

We prove that a pointed category has kernels if and only if it is a lax algebra for the arrow 2-monad, and that this holds if and only if it is the d\'ecalage of a supercoherent structure. We will then interpret categories with kernels as the sought-after weak version of unary operadic categories.