# Connections in terms of tangent ($\infty$-)categories?

Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, the nLab page), which could be adapted to $\infty$-categorical settings: there is a rich theory of the cotangent complex formalism in Lurie's Higher Algebra, section 7.3.

My question is that, whether we have an analogue for connections. Given a commutative ring $k$, a commutative $k$-algebra $A$ and a right $A$-module $M$, we define a connection $\nabla$ to be a $k$-linear map $M\to M\otimes_A\Omega_{A/k}^1$ satisfying the Leibniz rule: $\nabla(ma)=(\nabla m)a+m\otimes da$ for $m\in M$ and $a\in A$. I wonder whether we can also describe this through category theory and imagine a link between this and the deformation theory, so that we can generalize the connection into $\infty$-categorical setting?

EDIT: Following Jason Starr's comment, we can find a description of connections in Illusie's Complexe Cotangent et Déformations, Remarque 2.3.7.5: A connection $\nabla\colon M\to M\otimes_A\Omega_{A/k}^1$ one-to-one corresponds to a splitting $A$-linear map $M\to M\otimes_AP_{A/k}^1$ of the short exact sequence $$0\to M\otimes_A\Omega_{A/k}^1\to M\otimes_AP_{A/k}^1\to M\to0$$ where $P_{A/k}^1=A\oplus\Omega_{A/k}^1$ is the trivial square-zero extension of $A$ with respect to $\Omega_{A/k}^1$ also endowed with a $A$-algebra structure by $A\to P_{A/k}^1,a\mapsto a\oplus da$. This could be naturally generalized to the $\infty$-categorical setting.

• Isn't a connection a splitting of the Atiyah exact sequence? If so, this is discussed at length near the beginning of volume I of Illusie's "Complexe Cotangent et Deformations" where he discusses properties of the functor $P$. – Jason Starr Mar 16 '18 at 11:26
• The thing I find confusing about this is that I like to think of a connection as a path-lifting operation. From this perspective, any (co)cartesian fibration comes equipped with a unique "connection" given by taking (co)cartesian lifts. This is analogous to the fact that any Serre fibration has a path-lifting operation which is unique up to coherent homotopy. A connection in terms of a Leibniz rule is supposed to just be an "infinitesimal path-lifting operation", but if the space of connections is contractible, this leaves me puzzled as to what information could be stored in the connection. – Tim Campion Mar 16 '18 at 17:32
• Of course, it's also true in differential geometry that the space of connections on a vector bundle is contractible (in fact, it's an affine space), but that's okay because differential geometry is not homotopy-invariant, so it makes sense for the connection to store information which is not homotopy-invariant. On the other hand, infinity categories are "homotopy-invariant" in the sense that any meaningful notion is invariant under equivalence of infinity categories. – Tim Campion Mar 16 '18 at 17:37
• @TimCampion In differential geometry, a useful fact is that holonomy is a equivariant under gauge transformations. Perhaps a similar equivariance plays a role here. – Mike Miller Mar 16 '18 at 19:41
• @JasonStarr Thanks. Are you referring to (1.2.6.3) by Atiyah exact sequence, or stacks.math.columbia.edu/tag/09CH? – Yai0Phah Mar 18 '18 at 8:38