arXiv subjects

Sergei O. Ivanov

Publications and source records attributed to Sergei O. Ivanov.

At least 19 recordsLinked to original sources

arXiv2026

A non-nilpotent localization of a finite p-group

We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations need not preserve nilpotence and that the localization map $G\to L(G)$ need not be surjective when $G$ is a finite $p$-group.

math.GR↗

arXiv2026

Pretriangulated categories failing (TR4)

We introduce gemstones in a pretriangulated category, which are used to perturb pretriangulated structures. For a triangulated category $\mathcal T$ whose stable category of coherent functors admits a gemstone, we construct pretriangulated structures on $\mathcal T$ which fail the octahedral axiom (TR4). This confirms a long-standing conjecture on triangulated categories.

math.CT↗

arXiv2026

The discrete homotopy hypothesis for directed graphs

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also known as $A$-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an $\infty$-category, denoted by ${\sf DGra}_\infty$. We prove that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.

math.AT↗

arXiv2026

On diagonal digraphs, Koszul algebras and triangulations of homology spheres

We study magnitude homology of digraphs, with a particular focus on diagonal digraphs, i.e., digraphs whose magnitude homology is concentrated on the diagonal. For any digraph $G$, we provide a complete description of the second magnitude homology ${\rm MH}_{2,k}(G)$. This allows us to define a combinatorial condition, denoted by $(\mathcal{V}_\ell)$, which is equivalent to the vanishing of ${\rm MH}_{2,k}(G,\mathbb{Z})$ for all $k>\ell$. In particular, diagonal digraphs satisfy $(\mathcal{V}_2)$. As a corollary, we obtain that the 2-dimensional CW-complex obtained from a diagonal undirected graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $σG$ is Koszul over any field, and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $Ω^\bullet(G)$ is Koszul over any field. To provide a source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$ of a pure simplicial complex $K$. For a triangulation $K$ of a topological manifold $M$, we express the non-diagonal part of the magnitude homology of $\hat G_K$ in terms of the homology of $M$. As a corollary, we obtain that if $K$ is a triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere.

math.KT↗

arXiv2026

Ring-induced localizations of nilpotent groups

For a commutative ring $R$, we study the $R$-localization functor on the category of groups, defined as localization with respect to the homomorphism $\mathbb{Z}\to R$. Our main result is that, when $R$ is a binomial ring, the $R$-localization of a nilpotent group is again nilpotent. Taking $R=\mathbb{Z}_p$, the ring of $p$-adic integers, yields a new example of a localization functor that preserves nilpotency. To prove this, we characterize $R$-local groups in terms of $R$-groups in the sense of Myasnikov-Remeslennikov. We call an $R$-group a Hall-Petresco $R$-group if it satisfies a version of the Hall-Petresco identity, and show that these form a quasivariety closed under quotients by the center. The crucial input to our main result is that every $R$-local group carries a unique Hall-Petresco $R$-group structure.

math.GR↗

arXiv2025

Bousfield-Kan completion as a codensity $\infty$-monad

Working in the setting of $\infty$-categories, we develop a general theory of the codensity monad $T_\mathcal{D}$ associated with a full subcategory $\mathcal{D}\subseteq \mathcal{C}$. We show that $T_\mathcal{D}$ has a canonical monad structure (unique up to a contractible space of choices), and characterize it as a terminal monad preserving all objects of $\mathcal{D}$. For a monad $\mathcal{M}$ on an $\infty$-category $\mathcal{C}$, we consider the $\mathcal{M}$-completion functor defined as the totalization of the cosimplicial resolution associated with $\mathcal{M}$. We show that the $\mathcal{M}$-completion functor is the codensity monad associated with the full subcategory of $\mathcal{C}$ spanned by objects that admit a structure of $\mathcal{M}$-algebra. In particular, the $\mathcal{M}$-completion functor is the terminal monad preserving all objects that admit a structure of an $\mathcal{M}$-algebra. This gives a full $\infty$-categorical characterization of the classical Bousfield-Kan $R$-completion functor as the terminal monad on the category of spaces preserving the empty space and all products of Eilenberg-MacLane spaces $K(A,n)$, where $A$ is an $R$-module.

math.AT↗

arXiv2025

On $\ell_p$-Vietoris-Rips complexes

We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.

math.AT↗

arXiv2024

A Note on the Non-Existence of Functors

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions that we consider are questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, there is no a natural non-trivial way to define an abelian subgroup of a group, or a perfect quotient group of a group. As an auxiliary result we prove that, for any non-trivial subfunctor $F$ of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of $F.$ The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the $\infty$-category of spaces.

math.CT↗

arXiv2024

Transfinite version of the Mittag-Leffler condition for the vanishing of the derived limit

We give a necessary and sufficient condition for an inverse sequence $S_0 \leftarrow S_1 \leftarrow \dots$ indexed by natural numbers to have ${\rm lim}^1S=0$. This condition can be treated as a transfinite version of the Mittag-Leffler condition. We consider inverse sequences in an arbitrary abelian category having a generator and satisfying Grothendieck axioms ${\rm (AB3)}$ and ${\rm (AB4^*)}.$ We also show that the class of inverse sequences $S$ such that ${\rm lim}\: S={\rm lim}^1 S=0$ is the least class of inverse sequences containing the trivial inverse sequence and closed with respect to small limits and a certain type of extensions.

math.KT↗

arXiv2024

Path homology of digraphs without multisquares and its comparison with homology of spaces

For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $Ω_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $Ω_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $χ^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $Ω_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $χ^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic depends on whether the characteristic of $\mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$

math.AT↗

arXiv2024

Nested homotopy models of finite metric spaces and their spectral homology

For a real $r\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the $r$-minimal model of $X$. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an $r$-homotopy invariant ${\rm SH}^r_{n,I}(X)$ of a quasimetric space $X,$ called spectral homology, that generalizes many other invariants: the pages of the magnitude-path spectral sequence, including path homology, magnitude homology, blurred magnitude homology and reachability homology.

math.AT↗

arXiv2024

Magnitude homology is a derived functor

We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of quasimetric spaces, finite quasimetric spaces, and finite digraphs. For quasimetric spaces, we define the notion of a distance module over a quasimetric space, define the functor of (co)invariants of a distance module and show that the magnitude (co)homology can be presented via its derived functors. As a corollary we obtain that the magnitude cohomology of a quasimetric space can be presented in terms of Ext functors in the category of distance modules. For finite quasimetric spaces, we show that magnitude (co)homology can be presented in terms of Tor and Ext functors over a certain graded algebra, called the distance algebra of the quasimetric space. For finite digraphs, the distance algebra is a bound quiver algebra. In addition, we show that the magnitude cohomology algebra of a finite quasimetric space can be described as a Yoneda algebra.

math.KT↗

arXiv2024

Limits via relations

In this paper, we study operations on functors in the category of abelian groups simplar to the derivation in the sense of Dold-Puppe. They are defined as derived limits of a functor applied to the relation subgroup over a category of free presentations of the group. The integral homology of the Eilenberg-Maclane space $K(\mathbb Z,3)$ appears as a part of description of these operations applied to symmetric powers.

math.KT↗

arXiv2024

On the path homology of Cayley digraphs and covering digraphs

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.

math.AT↗

arXiv2023

Simplicial approach to path homology of quivers, marked categories, groups and algebras

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.

math.AT↗

arXiv2023

HR-length of a free group via polynomial functors

We prove that for a subring $R\subseteq \mathbb Q$ and a free group $F$ of rank at least $2$ the length of the Bousfield's $HR$-localization tower for $F$ is at least $ω+ω$. The key ingredient of the proof is the theory of polynomial functors over $\mathbb Q.$

math.GR↗

arXiv2023

Completions and Terminal Monads

We consider the terminal monad among those preserving the objects of a subcategory, and in particular preserving the image of a monad. Several common monads are shown to be uniquely characterized by the property of being terminal objects in the category of co-augmented endo-functors. Once extended to infinity categories, this gives, for example, a complete characterization of the well-known Bousfield-Kan R-homology completion. In addition, we note that an idempotent pro-completion tower can be associated with any co-augmented endo functor M, whose limit is the terminal monad that preserves the closure of ImM, the image of M, under finite limits. We conclude that some basic properties of the homological completion tower of a space can be formulated and proved for general monads over any category with limits, and characterized as universal

math.CT↗

arXiv2022

Quadric Hypersurface Intersection for Manifold Learning in Feature Space

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces -- simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.

cs.LG↗