Combinatorial homotopy. II

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 Ii

This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework can assist in the development of homotopy probability theory, where a vector space of random variables is replaced by a chain complex of random variables. This allows the principles of derived mathe...