arXiv subjects
Publications and source records attributed to Ryo Horiuchi.
In this note, we show the category of $\Gamma$-sets with the sphere spectrum, the smash product, and the substitution product gives rise to an isomix linearly distributive category, which is a categorical semantics of the multiplicative fragment of linear logic. We also show Lydakis' assembly map between these two monoidal products corresponds to the mixor.
In this note, we investigate the Witt vectors in the min-plus algebra of extended non-negative real numbers, and consider categories enriched over them viewed as a monoidal category.
In this note, we investigate a mixture of combinatorial spectra and stratified simplicial sets, which would be thought of as a model of the spectrum objects of $(\infty, \infty)$-categories.
An analogous construction of simplicial homotopy group for Kan complex can be applied to saturated complicial sets to give monoids. In this paper, we investigate how the construction of loop spaces of Kan complexes lifts to the complicial setting and relates to such monoids.
For a Kan complex with a vertex, we have the notion of its simplicial homotopy groups. In this paper, for a weak complicial set in the sense of Verity with a vertex, we construct monoids which are a generalization of simplicial homotopy groups.
It is well known that the underlying simplicial set of any simplicial group is a Kan complex. Roughly speaking, Kan complex is an infinite-dimensional analogue of groupoid, and the relation between groupoids and categories resembles that between groups and monoids. Thus one may ask if the underlying simplicial set of each simplicial monoid is a quasi-category. In this short note, we construct a simplicial monoid whose underlying simplicial set is not a quasi-category.
Hesselholt defined a spectrum $\operatorname{TP}(X)$, the periodic topological cyclic homology of a scheme $X$, using topological Hochschild homology and the Tate construction, which is a topological analogue of Connes-Tsygan periodic cyclic homology $\operatorname{HP}$ defined by Hochschild homology and the Tate construction. Goodwillie proved that for $R$ an algebra over a field of characteristic 0 and $I$ a nilpotent ideal of $R$, the quotient map $R\to R/I$ induces an isomorphism on $\operatorname{HP}$. In this paper, we show that the analogous result for $\operatorname{TP}$ does not hold, that is to say, there is an algebra of positive characteristic and a nilpotent ideal such that the quotient map does not induce an isomorphism on $\operatorname{TP}$, even rationally.
Let $p$ be a prime number, and let $A$ be a ring in which $p$ is nilpotent. In this paper, we consider the maps $$K_{q+1}(A[x]/(x^m), (x))\to K_{q+1}(A[x]/(x^{mn}), (x)),$$induced by the ring homomorphism $A[x]/(x^{m})\to A[x]/(x^{mn})$, $x\mapsto x^n$. We evaluate these maps, up to extension, for general $A$ in terms of topological Hochschild homology, and for regular $\mathbb{F}_p$-algebras $A$, in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative $K$-group of $\mathcal{O}_{K}/p\mathcal{O}_{K}$ for certain perfectoid fields $K$.