Beck torsors, formally unramified objects, and K\"ahler differentials

Abstract

Let $\mathcal{C}$ be a category with pullbacks. We define a $\textit{Beck torsor}$ in $\mathcal{C}$ as a morphism $Z\to Y$ in $\mathcal{C}$ which is a torsor for a Beck module over $Y$. We say that an object $X$ of $\mathcal{C}$ is $\textit{formally unramified}$ if, for every Beck torsor $Z\to Y$ in $\mathcal{C}$, the canonical map $\text{Hom}_{\mathcal{C}}(X, Z)\to \text{Hom}_{\mathcal{C}}(X, Y)$ is injective. If $A$ is a commutative ring with identity, then an $A$-algebra $B$ is formally unramified in the category of $A$-algebras if and only if the ring homomorphism $A\to B$ is formally unramified. Given that $A\to B$ is formally unramified if and only if $\Omega_{B/A} = 0$, we seek a similar classification for general formally unramified objects. We say that $\mathcal{C}$ has $\textit{K\"ahler differentials}$ if, for each object $X$ of $\mathcal{C}$, the forgetful functor $\text{Ab}(\mathcal{C}/X)\to \mathcal{C}/X$ from the category of Beck modules over $X$ has a left adjoint $\Omega: \mathcal{C}/X\to \text{Ab}(\mathcal{C}/X)$. Our main result is that if $\mathcal{C}$ has K\"ahler differentials, then an object $X$ of $\mathcal{C}$ is formally unramified if and only if $\Omega_X$ is a zero object in $\text{Ab}(\mathcal{C}/X)$.