In noncommutative geometry, one typically equips an associative algebra A with further structure such that it can then be regarded as encoding a “noncommutative space”. A general axiomatic approach is via the notion of a differential calculus (aka exterior algebra), introduced by Woronowicz. Unfortunately, such an approach is, in general, insufficient to yield existence or uniqueness of a compatible notion of form, and such procedures are typically not functorial.
In this work, we aim to address this problem treating the geometry of a category E as a relative notion: it will emerge when E is equipped with an isofibration E-->Mon(V) into the category of monoids in a monoidal additive category V.
We start by considering the notion of first order differential calculi in the setting of monoids internal to a monoidal additive category V and show that the standard results concerning first order differential calculi extend to this setting. Then, we establish sufficient conditions on the faithful isofibration E-->Mon(V) such that E admits a canonical functor into the category of first order differential calculi in V.
This talk is based on joint work with Keegan Flood and Giacomo Tendas, which can be found at arXiv:2512.20742.