Motivated by the philosophy that $C^$-algebras reflect noncommutative topology, we investigate stable homotopy theory of (opposite) category $C^$-algebras. We focus on $C^\*$-algebras which are CW-complexes in sense Eilers et al. (1998). construct $\infty$-category CW-spectra, denote $\mathtt{NSp}$. Let $\mathcal{M}$ be full spectral subcategory $\mathtt{NSp}$ spanned “noncommutative suspension...

It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, ...

Let M = (L , ∗) be a GL-monoid. An M-valued preordered set is an L-subset endowed with a reflexive and M-transitive L-relation, it is essentially a category enriched in a quantaloid generated by M. This paper presents a study of M-valued preordered sets with emphasis on symmetrization and the Cauchy completion. The main result states that symmetrization and the Cauchy completion of M-valued pre...

Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, concerning the relationship of Kan functor extensions to closed structures on functor categories, made in "Enriched functor categories" | 1] §9. It is assumed that the reader is familiar with the basic results of closed category theory, including the representation theorem. Apart fr...

We employ the notions of ‘sequential function’ and ‘interrogation’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results i...

Abstract: Allegories are enriched categories generalizing a category of sets and binary relations. Accordingly, relational products in an allegory can be viewed as a generalization of Cartesian products. There are several definitions of relational products currently in the literature. Interestingly, definitions for binary products do not generalize easily to n-ary ones. In this paper, we provid...

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

We have extended an existing lemmatizer, which relies on a lexicon of about 1.2 millions form, where lemmas are indexed by rich PoS tags, with a sequence of cascading filters, each one in charge of dealing with specific issues related to out-of-dictionary words. The last two filters are devoted to resolve semantic ambiguities between words of the same syntactic category, by querying external re...

Colony development, which includes hyphal extension, branching, anastomosis, and asexual sporulation, is a fundamental aspect of the life cycle of filamentous fungi; genetic mechanisms underlying these phenomena are poorly understood. We conducted transcriptional profiling during colony development of the model filamentous fungus Neurospora crassa, using 70-mer oligonucleotide microarrays. Rela...