Kernels, lax algebras, décalage, and supercoherence
Martin Markl, Dominik Trnka
TL;DR
The paper characterizes categories with kernels as precisely the normalized lax algebras for the arrow 2-monad $\overline{\mathbb{A}}$ on the category of $\mathbb{A}$-coalgebras, where $\mathbb{A}$ is the arrow comonad. It then shows that such categories are the décalages of supercoherent structures (Jardine’s framework), thereby interpreting kernels as a weak unary operadic-type structure. A detailed analysis connects the arrow-comonad/co-monad framework with nerves, the spindle embedding, and simplicial decales, culminating in a simplicial criterion: a pointed category has kernels iff its nerve is the upper decalage of an almost-strict supercoherent structure. The work further suggests that weak unary operadic categories arise naturally as kernels, while fully lax operadic theories require a bivariant extension that handles both kernels and cokernels, pointing to future directions toward abelian and double-structure generalizations.
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 décalage of a supercoherent structure. We will then interpret categories with kernels as the sought-after weak version of unary operadic categories.