Smoothness in Relative Geometry

TL;DR

This paper generalizes the concept of smoothness from classical ring theory to a broader categorical context using homotopical methods and model structures, extending the notion to relative geometry in monoidal categories.

Contribution

It introduces a new notion of smoothness in the setting of monoidal categories, generalizing classical smoothness via homotopical conditions and model structures.

Findings

Defines smoothness in $sComm(C)$ using homotopical conditions.

Shows the new notion generalizes classical smooth morphisms in rings.

Provides examples of smooth morphisms in various categorical contexts.

Abstract

In \cite{tva}, Bertrand Toen and Michel Vaquie defined a scheme theory for a closed monoidal category $(C,\otimes,1)$. In this article, we define a notion of smoothness in this relative (and not necesarilly additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists practically in changing homological finiteness conditions into homotopical ones using Dold-Kahn correspondance. To do this, we provide the category $sC$ of simplicial objects in a monoidal category and all the categories $sA-mod$, $sA-alg$ ($a\in sComm(C)$) with compatible model structures using the work of Rezk in \cite{r}. We give then a general notions of smoothness in $sComm(C)$. We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and provide some examples of smooth morphisms in $N-alg$, $Comm(Set)$ and Comm(C).