Simplicial set
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Weighted growing simplicial complexes Open
Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial comp…
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Homotopy theory of complete Lie algebras and Lie models of simplicial sets Open
In a previous work, by extending the classical Quillen construction to the non‐simply connected case, we have built a pair of adjoint functors, model and realization, between the categories of simplicial sets and complete differential grad…
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Yoneda Lemma for Simplicial Spaces Open
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition …
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Discrete topological complexity Open
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex and replacing the concept of homotopy by that of contiguous simplicial maps. We study th…
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Semi-simplicial spaces Open
This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The top…
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Towards a constructive simplicial model of Univalent Foundations Open
We provide a partial solution to the problem of defining a constructive\nversion of Voevodsky's simplicial model of univalent foundations. For this, we\nprove constructive counterparts of the necessary results of simplicial homotopy\ntheor…
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Strong Collapse for Persistence Open
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by …
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Simplicial Complexity: piecewise linear motion planning in robotics Open
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexit…
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The amazing world of simplicial complexes Open
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
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Principled Simplicial Neural Networks for Trajectory Prediction Open
We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture:…
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Systolic geometry and simplicial complexity for groups Open
Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial co…
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Transcendental simplicial volumes Open
We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifol…
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The fiber of persistent homology for simplicial complexes Open
We study the inverse problem for persistent homology: For a fixed simplicial complex K, we analyze the fiber of the continuous map PH on the space of filters that assigns to a filter f : K → R the total barcode of its associated sublevel s…
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Combinatorics and Representation Theory for Generalized Permutohedra I: Simplicial Plates Open
In this paper, we announce results from our thesis, which studies for the first time the categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained by certain continuity re…
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Extending Homotopy Type Theory with Strict Equality Open
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is …
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Classification of Constructible Cosheaves Open
In this paper we prove an equivalence theorem originally observed by Robert MacPherson. On one side of the equivalence is the category of cosheaves that are constructible with respect to a locally cone-like stratification. Our constructibi…
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Simplicial quantum contextuality Open
We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rath…
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Model Structures for Correspondences and Bifibrations Open
We study the notion of a bifibration in simplicial sets which generalizes the classical notion of two-sided discrete fibration studied in category theory. If $A$ and $B$ are simplicial sets we equip the category of simplicial sets over $A\…
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Dist2Cycle: A Simplicial Neural Network for Homology Localization Open
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
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Minimal volume entropy of simplicial complexes Open
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complemen…
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Computing simplicial representatives of homotopy group elements Open
A central problem of algebraic topology is to understand the homotopy groups of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group of a g…
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Semi-Simplicial Set Models for Distributed Knowledge Open
International audience
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Extendability of simplicial maps is undecidable Open
We present a short proof of the Čadek-Krčál-Matoušek-Vokřínek-Wagner result from the title (in the following form due to Filakovský-Wagner-Zhechev). For any fixed even $l$ there is no algorithm recognizing the extendability of the identity…
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Left fibrations and homotopy colimits II Open
For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence betwee…
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Hodge Decompositions for Weighted Hypergraphs Open
Weighted hypergraphs are generalizations of weighted simplicial complexes. In recent years, weighted Laplacians of weighted simplicial complexes have been studied. In 2016, as a generalization of the homology of simplicial complexes, the e…
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Dwyer–Kan homotopy theory for cyclic operads Open
We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each o…
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Lie models of homotopy automorphism monoids and classifying fibrations Open
Given X a finite nilpotent simplicial set, consider the classifying fibrations X → B aut∗ G(X) → B autG(X) and X → Z → B aut∗ π (X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed …
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Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation Open
As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for…
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Isolated factorizations and their applications in simplicial affine semigroups Open
We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. …
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k-simplex2vec: a simplicial extension of node2vec Open
We present a novel method of associating Euclidean features to simplicial complexes, providing a way to use them as input to statistical and machine learning tools. This method extends the node2vec algorithm to simplices of higher dimensio…