Constructing Canonical Calculi

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 (also sometimes called exterior algebra), introduced by Woronowicz [1].

Given an associative algebra $A$, there is a canonical choice of first order differential calculus, namely the universal first order differential calculus, obtained as the kernel of the multiplication on $A$. It is widely studied although it is often regarded to be of considerably less geometric interest than some classical commutative examples (e.g. de Rham forms for $C^{\mathrm{\infty }}$-rings or Kähler differentials for commutative algebras). Therefore, a standard practice in the noncommutative setting is to consider additional structures (bimodule relations, covariance conditions, etc.) in order to obtain more geometrically interesting notions of differential calculus. 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. The foundational work on differential calculi [1, p. 126] specifically remarks upon the “unpleasant contrast” with the classical case which this lack of functoriality results in (there Woronowicz is specifically referring to the case of quantum groups).

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 viewed in relation to a category of monoids $\mathrm{Mon}(𝒱)$ via a faithful isofibration ${(-)}_0:ℰ\to \mathrm{Mon}(𝒱)$, where $𝒱$ is a monoidal additive category. In this setting, we can define a notion of bimodule category for objects of $ℰ$, which generalises the case of bimodules over a monoid in $𝒱$, through the following pullback.

We start by generalising the notion of first order differential calculi to 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 broader setting. Then, we establish sufficient conditions on the faithful isofibration ${(-)}_0$ such that $ℰ$ admits a canonical functor ${\mathrm{Calc}}_{ℰ}^1$ to the category ${\mathrm{Calc}}^1(𝒱)$ of first order differential calculi in $𝒱$. Generalising the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical de Rham functor from $ℰ$ to the category ${\mathrm{Calc}}^{\bullet }(𝒱)$ of differential calculi in $𝒱$.

This yields a simultaneous generalisation of the de Rham complex on $C^{\mathrm{\infty }}$-rings, the Kähler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $ℰ$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\mathrm{Mon}(𝒱)$ factor suitably, their corresponding de Rham functors are related via a comparison map.