In 20], Yetter makes the following deenition: Fix a nite group, G. For any space X and a triangulation, T, a G-coloring of T is a map : T (1) ! G such that given any 2 T (2) , (e 1) " 1 (e 2) " 2 (e 3) " 3 = 1, whenever @ = e " 1 1 e " 2 2 e " 3 3 , for " i = 1 denoting in the rst expression non-inversion or inversion in the group G and in the second preservation or reversal of orientation. We ...

The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category CatS of S-enriched categories then has a model structure with characteriza...

The Stone dualities for accessible categories and the subclass of Diers categories provide limit and colimit structuring principles for query languages and the associated database. As a motivating example, we consider relational databases. Relational databases are given by database schemata, the syntax, and the class of sets of relations satisfying the syntactic constraints, called instances. R...

Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics which reflect global geometric properties of data sets. In order to be useful in practice, for example for feature generation in machine learning, it needs to be effectively computable. Classical homology is a computable topological invariant because of the Mayer-Vietoris exact and spectral...

We propose a definition of the Chow group of a rigid tensor triangulated category. The basic idea is to allow “generalized” cycles, with nonintegral coefficients. The precise choice of relations is open to some fine-tuning. Hypothesis 1. Let K be an essentially small tensor triangulated category. Let us assume that its triangular spectrum in the sense of [1], Spc(K) = { P ⊂ K ∣∣P is prime}, is ...

Introduction. The Freudenthal suspension theorem implies that the set of homotopy classes of continuous maps from one finite complex to another is eventually invariant under iterated suspension of the complexes. In this "stable range" the set of homotopy classes is also well behaved in other ways. For example, it has in a natural way the structure of an abelian group. These stable groups of hom...

The main characters of this paper are the moduli spaces TMg,n of rational tropical curves of genus g with n marked points, with g ≥ 2. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces Xg,n, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes ...

The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2-category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this definition. The structure present on the class C of rigged surfaces...

We first describe Krull-Schmidt theorems decomposing H spaces and simply-connected co-H spaces into atomic factors in the category of pointed nilpotent p-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic H spaces and homotopy types of atomic co-H spaces, and construct a split fibration which connects them and illuminates the decomposi...

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility. In universal algebra we deal with varieties – classes o...