Vegard Fjellbo, John Rognes
A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$-simplices given by the simplicial maps $Δ[n] \times K \to X$. We prove that $X^K$ is non-singular whenever $X$ is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it is not a topos.
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The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak)_proper_ homotopy type of X. Moreover, our techniques strictly generalize the classical simplicial situation: e.g., one obtains, in a unified way, singular homology with compact supports and (Borel-Moore) singular homology with locally finite s…
A simplicial set is said to be non-singular if its non-degenerate simplices are embedded. Let $sSet$ denote the category of simplicial sets. We prove that the full subcategory $nsSet$ whose objects are the non-singular simplicial sets admits a model structure such that $nsSet$ becomes is Quillen equivalent to $sSet$ equipped with the standard model structure due to Quillen. The model structure on $nsSet$ is right-induced from $sSet$ and it makes $nsSet$ a proper cofibrantly generated model category. Together with Thomason's model structure on small categories (1980) and Raptis' model structure…
We prove that a simplicial 2-sphere satisfying a certain condition is the underlying simplicial complex of a 3-dimensional non-singular complete fan. In particular, this implies that any simplicial 2-sphere with $\leq 18$ vertices is the underlying simplicial complex of such a fan.
Computing singular simplicial homologies of digraphs and quivers
本文为有向图与箭图发展同伦理论，构造函子实现奇异单纯同调与空间同调的同构，并提供计算持久同调的高效算法，适用于大规模网络分析。
We show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary $\ell^1$-semi-norm on singular homology in all dimensions.
Simplicial intersection homology revisited
本文直接证明了单纯与奇异交链复形间的典范映射是拟同构，并将该平行性扩展至交爆破上同调，证明了伪流形与 PL 空间中该上同调的同构性质及可计算性。
A simplicial set is said to be non-singular if the representing map of each non-degenerate simplex is degreewise injective. The inclusion into the category of simplicial sets, of the full subcategory whose objects are the non-singular simplicial sets, admits a left adjoint functor called desingularization. In this paper, we provide an iterative description of desingularization that is useful for theoretical purposes as well as for doing calculations.
The transverse singular complex
本文研究了无边界的光滑流形M的奇异单纯集Sing(M)，证明其可变形退缩到由与T中每个元素横截的光滑奇异单纯组成的子集。
Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the l^1-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology -- as already claimed by Thurst…
The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen's iterated integrals are considered in a diffeological framework. As a consequ…
The Barratt nerve, denoted $B$, is the endofunctor that takes a simplicial set to the nerve of the poset of its non-degenerate simplices. The ordered simplicial complex $BSd\, X$, namely the Barratt nerve of the Kan subdivision $Sd\, X$, is a triangulation of the original simplicial set $X$ in the sense that there is a natural map $BSd\, X\to X$ whose geometric realization is homotopic to some homeomorphism. This is a refinement to the result that any simplicial set can be triangulated. A simplicial set is said to be regular if each of its non-degenerate simplices is embedded along its $n$-th …
We show that Sullivan's model of rational differential forms on a simplicial set $X$ may be interpreted as a (kind of) $0|1$-dimensional supersymmetric quantum field theory over $X$, and, as a consequence, concordance classes of such theories represent the rational cohomology of $X$. We introduce the notion of superalgebraic cartesian sets, a concept of space which should roughly be thought of as a blend of simplicial sets and generalized supermanifolds, but valid over an arbitrary base ring. Every simplicial set gives rise to a superalgebraic cartesian set and so we can formulate the notion o…
We prove the following generalization of a classical result of Adams: for any pointed and connected topological space $(X,b)$, that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in $X$ with vertices at $b$ is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of $X$ at $b$. We deduce this statement from several more general categorical results of independent interest. We construct a functor $\mathfrak{C}_{\square_c}$ from simplicial sets t…
Natural operations in Intersection Cohomology
本研究解决了Goresky与MacPherson提出的公开问题，构造出相交上同调的分类单纯集与分类空间，证明其构成无穷loop空间，还研究了深度1分层空间的分类空间性质并提出相关猜想。
Weighted Homology and Cohomology of Weighted Polyhedra
该研究定义了加权多面体的概念，构建了其加权奇异同调理论并证明其与加权单纯同调同构，还推广了相关上同调乘积并阐释了orbifold理论。
We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semi-algebraic subset $S \subset \mathbb{R}^k$ given by a quantifier-free first order formula $ϕ$ in the language of the reals, and produces as output a simplicial complex $Δ$, whose geometric realization, $|Δ|$ is $\ell$-equivalent to $S$. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell)}}$, where $s$ is the number of polynomials appearing in the formula $ϕ$, and $d$ a bound on their degrees. For fixed $\ell$, this bound is singly exponential in $k$. In particular, since $\ell…
In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $sπ_n(X)$. They are diagrams of groups built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered h…
Projective model structures on diffeological spaces and smooth sets and the smooth Oka principle
本文否定微分拓扑空间上经光滑奇异复形转移模型结构的猜想，转而在光滑集层范畴建立适当组合、笛卡尔且与单纯集 Quillen 等价的模型结构，并推广光滑 Oka 原理以分类微分几何对象。
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain non-trivial properties. This contrasts with the ordinary, unstructured notion of a Kan fibration. We show that fundamental properties of Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, we …
Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loeh's result plays a fundamental…
The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually s…
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