Combinatorial isomorphisms and combinatorial homotopy equivalences

Combinatorial Categorical Equivalences

In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X ; (b) the observation of...

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...

Considering Stochastic and Combinatorial Optimization

Here, issues connected with characteristic stochastic practices are considered. In the first part, the plausibility of covering the arrangements of an improvement issue on subjective subgraphs is studied. The impulse for this strategy is a state where an advancement issue must be settled as often as possible for discretionary illustrations. Then, a preprocessing stage is considered that would q...