I will suggest in this article a certain marriage of the notion of a classical probability space with ideas from algebraic homotopy theory.

Recall that an algebraic probability space over $C$ is a unital algebra of random variables together with a unit preserving linear map $\iota$, called expectation, to $C$. The target $C$ of the map $\iota$ is itself a unital algebra but $\iota$ is not expected to preserve the algebra structure. In fact the deviation and higher deviations from $\iota$ being an algebra map measure strength and depth of correlations between events thought of as singletons, as pairs, as triples etc. and this is the main structure in a probability space. Such infinite hierarchy of deviations being algebra map is defined with respect to a suitable notion of independence via the notion of culmulants.

On the other hand, in algebraic homotopy theory, one is dealing with the same kind of algebraic structure but now enriched by an underlying chain complex. One expects to study algebra preserving morphisms, but now only up to chain homotopy in the underlying chain complex. The chain homotopy is part of an infinite hierarchy of homotopies, homotopies of homotopies, homotopies of homotopies of homotopies etc.

The usefulness of the two sides may be combined in the notion of homotopical probability space/probabilistic homotopy theory to be introduced in this talk. The idea is that the organizing data of successive deviations from independence -- i.e. of $\iota$ being an algebra morphism, will be replaced by a generalized morphism of homotopy probability space incorporating the notion of algebra morphisms up to infinite homotopy. The classical independence in commutative probability world correponds to the Lie world of $\infty$-homotopy theory. For free independence in the noncommutative case, there is corresponding $\infty$-homotopy theory, and associative $\infty$-homotopy theory corresponds to so called Boolean independence.

Such homotopy probability correspondences, although natural in the above unification, might not be practical for studying ordinary probability space. I will, however, argue that the very "computability" of the joint distributions (the probability law) of random variables indicates that the space has a hidden infinite homotopical structure, whose presence may be exploited to determine distributions up to finite ambiguity. Conversely, the joint distributions of random variables may also serve to provide invariants of algebraic homotopy types.