Combinatorial homotopy. I

Combinatorial Homotopy. I

Combinatorial Descriptions of Homotopy Groups

Let X be a cofunctor from O to sets. Let Xn = X({0, 1, · · · , n}). The face function dj : Xn → Xn−1 is induced by the inclusion d : {0, · · · , n − 1} → {0, · · ·n} with d(i) = i for i < j, d(i) = i + 1 for i ≥ j for 0 ≤ j ≤ n. The degeneracy function sj : Xn → Xn+1 is induced by s : {0, · · · , n+ 1} → {0, · · · , n} with s(i) = i for i ≤ j and s(i) = i− 1 for i > j for 0 ≤ j ≤ n. The simplic...

Introduction to Combinatorial Homotopy Theory

1 Introduction. Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to distinguish two maps which can be continuously deformed into each other; such a wea...

Homotopy Groups of the Combinatorial Grassmannian

We prove that the homotopy groups of the oriented matroid Grass-mannian MacP(k; n) are stable as n ! 1, that 1 ((MacP(k; n)) = 1 (G(k; R n)), and that there is a surjection 2 (G(k; R n)) ! 2 ((MacP(k; n)). The theory of oriented matroids gives rise to a combinatorial analog to the Grassmannian G(k; R n). By thinking of an oriented matroid as a \combinatorial vector space", one is led to deene t...

Homotopy Probability Theory I

This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of deri...