Searcharxiv⌕ Search

arXiv subjects

Viktoriya Ozornova

Publications and source records attributed to Viktoriya Ozornova.

At least 19 recordsLinked to original sources
arXiv2026

About the contractibility of the walking coinductive equivalence

We study the marked simplicial set obtained as the Roberts-Street nerve of the walking coinductive equivalence. We show that it is a non-contractible saturated complicial set for which all of its finite truncations are contractible. When regarding saturated complicial sets as a model for right $(\infty,\infty)$-categories, it represents a concrete and explicit example of a right $(\infty,\infty)$-category that is itself non-contractible, but whose reflection to a left $(\infty,\infty)$-category is contractible.

math.AT↗
arXiv2026

Cores and localizations of $(\infty,\infty)$-categories

We consider $(\infty,d)$-categories in the limit $d\to \infty$ via the core or localization functors that forget or invert higher non-invertible arrows, respectively. We compare the two resulting $(\infty,1)$-categories of $(\infty,\infty)$-categories and exhibit the localization-limit as a reflective localization of the core-limit. On the side, we study intermediate localizations that arise from notions of invertibility that only emerge at $d=\infty$ such as the one defined by coinduction.

math.AT↗
arXiv2025

$(\infty,n)$-categories in context

This note is a contribution written for the second volume of the Encyclopedia of mathematical physics. We give an informal introduction to the notions of an $(\infty,n)$-category and $(\infty,n)$-functor, discussing some of the different models that implement them. We also discuss the notions of a symmetric monoidal $(\infty,n)$-category and symmetric monoidal $(\infty,n)$-functor, recalling some important results whose statements employ the language of $(\infty,n)$-categories.

math.AT↗
arXiv2024

Waldhausen's $S_\bullet$-construction

This note is an expository contribution for a proceedings volume of the workshop "Higher Segal Spaces and their Applications to Algebraic K-theory, Hall Algebras, and Combinatorics". We survey various versions of Waldhausen's S-construction and the role they play in defining K-theory, and we discuss their 2-Segality properties.

math.AT↗
arXiv2024

A model for the coherent walking $ω$-equivalence

We prove that a certain $ω$-category, which was constructed in previous work by the third and fourth author, is a model for the fully coherent walking $ω$-equivalence. Further, appropriate truncations of it give models for the fully coherent walking $n$-equivalence for each $n\geq1$.

math.CT↗
arXiv2023

A categorical characterization of strong Steiner $ω$-categories

Strong Steiner $ω$-categories are a class of $ω$-categories that admit algebraic models in the form of chain complexes, whose formalism allows for several explicit computations. The conditions defining strong Steiner $ω$-categories are traditionally expressed in terms of the associated chain complex, making them somewhat disconnected from the $ω$-categorical intuition. The purpose of this paper is to characterize this class as the class of polygraphs that satisfy a loop-freeness condition that does not make explicit use of the associated chain complex and instead relies on the categorical features of $ω$-categories.

math.CT↗
arXiv2023

Pushouts of Dwyer maps are $(\infty,1)$-categorical

The inclusion of 1-categories into $(\infty,1)$-categories fails to preserve colimits in general, and pushouts in particular. In this note, we observe that if one functor in a span of categories belongs to a certain previously-identified class of functors, then the 1-categorical pushout is preserved under this inclusion. Dwyer maps, a kind of neighborhood deformation retract of categories, were used by Thomason in the construction of his model structure on 1-categories. Thomason previously observed that the nerves of such pushouts have the correct weak homotopy type. We refine this result and show that the weak homotopical equivalence is a weak categorical equivalence. We also identify a more general class of functors along which 1-categorical pushouts are $(\infty,1)$-categorical.

math.AT↗
arXiv2023

What is an equivalence in a higher category?

The purpose of this survey is to present in a uniform way the notion of equivalence between strict $n$-categories or $(\infty,n)$-categories, and inside a strict $(n+1)$-category or $(\infty,n+1)$-category.

math.AT↗
arXiv2022

An $(\infty,2)$-categorical pasting theorem

We show that any pasting diagram in any $(\infty,2)$-category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a pasting scheme is the homotopy colimit of its cells as an $(\infty,2)$-category. We prove this explicitly in the simplicial categories model and then explain how to deduce the model-independent statement from that calculation.

math.AT↗
arXiv2022

Model independence of $(\infty,2)$-categorical nerves

For most models of $(\infty,2)$-categories an embedding of the $\infty$-category of 2-categories into that of $(\infty,2)$-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve embeddings induce equivalent functors, modulo change of model. We also show that all the nerve embeddings realize the $\infty$-category of 2-categories as the sub-$\infty$-category of $(\infty,2)$-categories that are local with respect to a certain class of maps.

math.AT↗
arXiv2021

Nerves and cones of free loop-free ω-categories

We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $ω$-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard $m$-simplex is equivalent to the complicial nerve of the $m$-oriental.

math.AT↗
arXiv2020

Gray tensor product and saturated $N$-complicial sets

We show that the pretensor and tensor products of simplicial sets with marking are compatible with the homotopy theory of saturated $N$-complicial sets (which are a proposed model of $(\infty,N)$-categories), in the form of a Quillen bifunctor and a homotopical bifunctor, respectively.

math.AT↗
arXiv2020

Stable homotopy hypothesis in the Tamsamani model

We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.

math.AT↗
arXiv2020

Comparison of Waldhausen constructions

In previous work, we develop a generalized Waldhausen $S_{\bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{\bullet}$-constructions for exact categories and for stable and exact $(\infty,1)$-categories, as well as the relative $S_{\bullet}$-construction for exact functors.

math.AT↗
arXiv2020

The unit of the total décalage adjunction

We consider the décalage construction $\operatorname{Dec}$ and its right adjoint $T$. These functors are induced on the category of simplicial objects valued in any bicomplete category $\mathcal{C}$ by the ordinal sum. We identify $T \operatorname{Dec}X$ with the path object $X^{Δ[1]}$ for any simplicial object $X$. We then use this formula to produce an explicit retracting homotopy for the unit $X\to T\operatorname{Dec}X$ of the adjunction $(\operatorname{Dec},T)$. When $\mathcal{C}$ is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in $\mathcal{C}$.

math.AT↗