Beck torsors, formally unramified objects, and Kähler differentials

Abstract

Let $\mathcal{C}$ be a category with pullbacks. We define a $\text{Beck torsor}$ 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 $\text{formally unramified}$ 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 $\text{Ka¨hler differentials}$ 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 Kähler differentials, then an object $X$ of $\mathcal{C}$ is formally unramified if and only if ${\Omega }_X$ is a zero object in $\text{Ab}(\mathcal{C}/X)$.