Abstract We give the construction of the universal, natural up to homotopy Chern–Weil differential graded algebra homomorphism: for infinite‐dimensional Milnor regular Lie groups , where is a certain de Rham algebra of (Milnor up to a natural weak homotopy equivalence) and where is the algebra of continuous invariant multilinear functionals on the Lie algebra. In particular, this applies to the group of compactly generated Hamiltonian symplectomorphisms, using which we verify a conjecture of Reznikov. For the construction of , we introduce a basic geometric‐categorical notion of a smooth simplicial set. Loosely, this is to Chen spaces as simplicial sets are to spaces. We then give a new construction of the classifying space of as a smooth Kan complex, with the geometric realization weakly equivalent to the Milnor .

In this paper, we take a simplicial complex of dimension [Formula: see text] and we introduce on it oriented [Formula: see text]-simplexes and oriented [Formula: see text]-simplexes. As a result, we create a weighted simplicial set of [Formula: see text]-simplexes and [Formula: see text]-simplexes [Formula: see text], on which we introduce the cochains of dimension [Formula: see text] and we use them to construct a weighted normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text] and its weighted Dirichlet normalized Laplacian associated with [Formula: see text]-simplexes [Formula: see text]. Next, we study the spectrum of [Formula: see text] and the spectrum of [Formula: see text]. Finally, we develop the weighted isoperimetric inequalities at infinity associated to [Formula: see text] and we use them to analyze the essential spectrum of [Formula: see text] and the essential spectrum of [Formula: see text].

We study four types of (co)cartesian fibrations of \infty -bicategories over a given base \mathcal{B} , and prove that they encode the four variance flavors of \mathcal{B} -indexed diagrams of \infty -categories. We then use this machinery to setup a general theory of marked (co)limits for diagrams valued in an \infty -bicategory, capable of expressing lax, weighted and pseudo limits. When the \infty -bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of marked (co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these marked (co)limits in a wide range of examples. We finish by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of marked (co)limits, provided they exist.

A partial group with n + 1 n+1 elements is, when regarded as a symmetric simplicial set, of dimension at most n n . This dimension is n n if and only if the partial group is a group. As a consequence of the first statement, finite partial groups are genuinely finite, despite being seemingly specified by infinitely much data. In particular, finite partial groups have only finitely many im-partial subgroups. We also consider dimension of partial groupoids.

Abstract In this work, we conclude our study of fibred $\infty $ -bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $ -bicategory of 2-Cartesian fibrations over S and the $\infty $ -bicategory of contravariant functors with values in the $\infty $ -bicategory of $\infty $ -bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

This paper studies the notion of complete connectedness of a Grothendieck topos, defined as the existence of a left adjoint to a left adjoint to a left adjoint to the global sections functor, and provides many examples. Typical examples include presheaf topoi over a category with an initial object, such as the topos of sets, the Sierpiński topos, the topos of trees, the object classifier, the topos of augmented simplicial sets, and the classifying topoi of many algebraic theories, such as groups, rings, and vector spaces. We first develop a general theory on the length of adjunctions between a Grothendieck topos and the topos of sets. We provide a site characterisation of complete connectedness, which turns out to be dual to that of local topoi. We also prove that every Grothendieck topos is a closed subtopos of a completely connected Grothendieck topos.

Uniform Manifold Approximation and Projection (UMAP) is a dimensionality-reduction and clustering algorithm designed to transform high-dimensional datasets into optimized low-dimensional embeddings. While UMAP is widely used for efficient and reliable dimensionality reduction, there is a growing demand for improved scalability across heterogeneous datasets. A UMAP implementation in Haskell would boast several advantages over other languages due to Haskell's functional programming properties such as lazy evaluation and static typing, which can provide improved concurrency, parallelization, and scalability, resulting in a more user-friendly experience while enhancing reliability. Despite this, no distributed version of UMAP implemented in Haskell is currently available to the public. To address this gap, this project developed a UMAP implementation that processes fuzzy simplicial sets concurrently, specifically focusing on the parallelization of the K-Nearest-Neighbors (KNN) algorithm, a core component of UMAP. This allows for the distribution of the UMAP algorithm across multiple GPUs as each GPU only keeps a portion of the dataset in VRAM. To accomplish this, a purely functional UMAP was implemented using Haskell (GHC2021) and wrapped in a concurrency monad using the parallel programming library. Typed parsing of datasets was found to be compatible with a functional UMAP implementation, and the fuzzy simplicial sets in UMAP allowed for concurrent processing and the implementation of a scalable UMAP.

Abstract In this article, we expand upon the concepts introduced in Spivak (Metric realization of fuzzy simplicial sets, 2009) about the relationship between the category $$\textbf{UM}$$ of uber metric spaces and the category $$\textbf{sFuz}$$ of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP (McInnes et al. 2018) to apply the theory to manifold learning, dimension reduction and data visualization, while refining some of their constructions to put the corresponding theory on a more solid footing. A generalization of the adjunction between $$\textbf{UM}$$ and $$\textbf{sFuz}$$ allows us to view the adjunctions used in both publications as special cases. Moreover, we derive an explicit description of colimits in $$\textbf{UM}$$ and the realization functor $$\text {Re}:\textbf{sFuz}\rightarrow \textbf{UM}$$ , and show that $$\textbf{UM}$$ can be embedded into $$\textbf{sFuz}$$ . Furthermore, we prove analogous results for the category of extended-pseudo metric spaces $$\textbf{EPMet}$$ . We also provide rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and $$\textbf{sFuz}$$ , which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap (Tenenbaum et al. 2000) and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods. Additionally, we developed a new embedding method that allows to preserve clusters detected in the original metric space that we construct from the data. The visualization of the optimization process gives the user information, both about the inner-cluster distributions in the original metric space and their inter-cluster relations. We compare our new method with UMAP, Isomap and t-SNE on a series of low- and high-dimensional datasets and provide explanations for observed differences and improvements.

Given a commutative ring, R R , a π 1 \pi _1 - R R -equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an R R -homology equivalence between universal covers. When R R is an algebraically closed field, Raptis and Rivera [Int. Math. Res. Not. IMRN 16 (2024), pp. 11766–11811] described a full and faithful model for the homotopy theory of spaces up to π 1 \pi _1 - R R -equivalence. They use simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the Cobar functor. In this article, we prove a G G -equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf [Trans. Amer. Math. Soc. 277 (1983), pp. 275–284]. We also prove a more general result about modeling G G -simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras.

If a group $N$ acts on a set $X$, a simplicial set $Bar(X,N)$ using the usual bar construction has been provided. In this construction, if the group $N$ acts on a group $G$ via a homomorphism $f:N\rightarrow G$, then $Bar(G,N)$ has a simplicial set structure. In the case of $f$ has a crossed module structure, $Bar(G,N)$ has a normal simplicial group structure. In this work, by defining an action of a crossed module $\partial: N_1 \longrightarrow X_1$ on a homomorphism of groups $f: N_2 \longrightarrow X_2 $ via a double map $\alpha: \partial\rightarrow f$, we will construct a bisimplicial set, using the 2-dimensional version of the usual Bar construction.

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly at topological data analysis. The proposed set-up applies to any parafinite simplicial set and proceeds by associating with it a finite dimensional simplicial Hilbert space, whose simplicial operator structure is studied in some depth. It is shown in particular how the problem of determining the simplicial set’s homology can be solved within the simplicial Hilbert framework. Further, the conditions under which simplicial set theoretic algorithms can be implemented in a quantum computational setting with finite resources are examined. Finally a quantum algorithmic scheme capable of computing the simplicial homology spaces and Betti numbers of a simplicial set combining a number of basic quantum algorithms is outlined.

Abstract In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric functions over arbitrary rings to extend this result to any ring. In particular, we study the discrete isoperimetric function and show that its linearity is equivalent to hyperbolicity, which is also equivalent to it being subquadratic. We further use these ideas to obtain conditions for subgroups of higher rank hyperbolic groups to be again higher rank hyperbolic of the same rank. The appendix discusses the equivalence between isoperimetric functions and coning inequalities in the simplicial setting and the general setting, leading to combinatorial definitions of higher rank hyperbolicity in the setting of simplicial complexes and allowing us to give elementary definitions of higher rank hyperbolic groups.

We prove that for any bisimplicial set X, the natural comparison map between the diagonal dX and the total simplicial set TX is a categorical equivalence in the sense of Joyal and Lurie.

As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which Gajer associates to a couple (X,\overline{p}) of a filtered space and a perversity. We first establish some basic properties of the intersection fundamental groups, such as a Van Kampen theorem. For general intersection homotopy groups on Siebenmann CS sets, we prove a Hurewicz theorem between them and the Goresky and MacPherson intersection homology. If the CS set and its intrinsic stratification have the same regular part, we establish the topological invariance of the \overline{p} -intersection homotopy groups. Several examples justify the hypotheses made in the statements. Finally, intersection homotopy groups also coincide with the homotopy groups of the topological space itself, for the top perversity on a connected normal Thom–Mather space.

Generalizing quasicategories via model structures on simplicial sets

Abstract In this note we show that in the simplicial setting, the classifying space construction converts short exact sequences of groups not just to homotopy fibrations, but in fact to fibre bundles.

Partial groups as symmetric simplicial sets

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.

Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $R$. The weak equivalences are given by: (1) an $R$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers, and (3) $R$-homology equivalences. Analogously, for any field ${\mathbb{F}}$, we construct three model structures on the category of connected simplicial cocommutative ${\mathbb{F}}$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when ${\mathbb{F}}$ is algebraically closed, the simplicial ${\mathbb{F}}$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when ${\mathbb{F}}$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of ${\mathbb{F}}$.