arXiv subjects

Kimball Strong

Publications and source records attributed to Kimball Strong.

5 recordsLinked to original sources

arXiv2026

An Enriched Approach to the Strictification of $(\infty,1)$-Categories

We define a functor which takes in an $(\infty,1)$-category and outputs an $(ω,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in $\infty$-groupoids, and then "locally strictifying" (applying the strictification of $\infty$-groupoids to each hom space) to obtain a category enriched in $ω$-groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in $ω$-groupoids, which is equivalently an $(ω,1)$-category. We prove that this functor is conservative by proving a slightly stronger statement on systems of chain complexes parameterized by the homotopy $(2,1)$-category of an $(\infty,1)$-category, and explain how this generalizes the Homological Whitehead Theorem from spaces to $(\infty,1)$-categories.

math.CT↗

arXiv2026

Change of Enrichment for Monoidal Model Categories

A classical theorem of category theory says that given an adjunction between monoidal categories with lax monoidal right adjoint, there is an induced adjunction between categories of enriched categories. We extend this result to monoidal model categories (with some model categorical assumptions), improving on previous theorems which required the derived left adjoint to be strong monoidal. As an application, we consider the category of chain complexes over a varying groupoid of operators, equipped with two different monoidal structures: the standard tensor product, and the cartesian product. We show that both of these products yield monoidal model categories, and use our theorems to reproduce the square-zero extensions adjunction for augmented dg-categories.

math.CT↗

arXiv2025

Strictification of $\infty$-Groupoids is Comonadic

We investigate the universal strictification adjunction from weak $\infty$-groupoids (modeled as simplicial sets) to strict $\infty$-groupoids (modeled as simplicial T-complexes). We prove that any simplicial set can be recovered up to weak homotopy equivalence as the totalization of its canonical cosimplicial resolution induced by this adjunction. This generalizes the fact due to Bousfield and Kan that the homotopy type of a simply connected space can be recovered as the totalization of its canonical cosimplicial resolution induced by the free simplicial abelian group adjunction. Furthermore, we leverage this result to show that this strictification adjunction induces a comonadic adjunction between the quasicategories of simplicial sets and strict $\infty$-groupoids.

math.AT↗

arXiv2019

Cardinalities of Prime Spectra of Precompletions

Given a complete local (Noetherian) ring $T$, we find necessary and sufficient conditions on $T$ such that there exists a local domain $A$ with $|A| < |T|$ and $\widehat{A} = T$, where $\widehat{A}$ denotes the completion of $A$ with respect to its maximal ideal. We then find necessary and sufficient conditions on $T$ such that there exists a domain $A$ with $\widehat{A} = T$ and $|\mbox{Spec}(A)| < |\mbox{Spec}(T)|$. Finally, we use "partial completions" to create local rings $A$ with $\widehat{A} = T$ such that $\mbox{Spec}(A)$ has varying cardinality in different varieties.

math.AC↗

arXiv2019

Structure of Spectra of Precompletions

Let T be a complete local (Noetherian) ring and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T. We investigate the possible structures of the partially ordered set Spec(A). Specifically, we explore the minimal prime ideals of A and their formal fibers, the maximal chains of prime ideals in A, and the number of prime ideals in A containing combinations of minimal prime ideals of A.

math.AC↗