Tangent $\mathrm{\infty }$-Categories and Goodwillie Calculus

Following Rosický [Ros84], Cockett and Cruttwell introduced in [CC14] a notion of tangent category to axiomatize certain categorical properties of the tangent bundle construction on a smooth manifold. Examples of tangent categories appear in many contexts including commutative algebra and algebraic geometry [CL23], operad theory [ILL24], logic, category theory, and, of course, differential geometry (ordinary and synthetic). With Bauer and Burke [BBC21], we recently added homotopy theory to this list by extending Cockett and Cruttwell’s definition to $\mathrm{\infty }$-categories in the sense of Lurie, constructing an example of a tangent $\mathrm{\infty }$-category that encodes Goodwillie’s functor calculus [Goo03].

Our definition depends on a certain symmetric monoidal $\mathrm{\infty }$-category $𝕎{\mathrm{eil}}_{\mathrm{\infty }}$ of $\mathrm{\infty }$-Weil-algebras. That $\mathrm{\infty }$-category is related to, though not the same as, the category of Weil-algebras introduced by Leung in [Leu17] to give an alternative characterization of tangent structure. In this talk, I will focus on the $\mathrm{\infty }$-category $𝕎{\mathrm{eil}}_{\mathrm{\infty }}$, giving two different descriptions, one in terms of partial commutative monoids and one in terms of $E_{\mathrm{\infty }}$-semirings. The part that is new is the proof that these two descriptions are equivalent, which provides us with additional examples of tangent $\mathrm{\infty }$-categories of $E_{\mathrm{\infty }}$-ring spectra. Our work also determines a notion of tangent ($\mathrm{\infty }$-)bicategory, which we propose as a natural way to extend Cockett and Cruttwell’s work to a bicategorical setting.