Simplicial Sets Research Articles

Action of Crossed Modules and Bar Construction

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.

The discrete flow category: structure and computation

AbstractIn 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.

On Dealing with Minima at the Border of a Simplicial Feasible Area in Simplicial Branch and Bound

AbstractWe 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.

The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

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 ${\mathbb{F}}$.

Non‐accessible localizations

AbstractIn a 2005 paper, Casacuberta, Scevenels, and Smith construct a homotopy idempotent functor on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe . When specialized to an appropriate family, this produces a localization which when interpreted in the ‐topos of spaces agrees with the localization corresponding to . Our approach generalizes the approach of Casacuberta et al. (Adv. Math. 197 (2005), no. 1, 120–139) in two ways. First, by working in homotopy type theory, our construction can be interpreted in any ‐topos. Second, while the local objects produced by Casacuberta et al. are always 1‐types, our construction can produce ‐types, for any . This is new, even in the ‐topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about “small” types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice that implies that sets cover and that the law of excluded middle holds.

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.

Cubical Models of (∞,1)-Categories

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences.

Simplicial techniques for operator solutions of linear constraint systems

A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Zd.

The successive dimension, without elegance

Experience shows that the poset of levels (or dimensions) of the topos of presheaves on some elegant Reedy categories may be equipped with a monotone increasing ‘successor’ function which, as the case of simplicial sets illustrates, is different from Lawvere’s Aufhebung in general. We prove that a similar result holds for the topos of presheaves on a small category with split-epi/mono factorizations; a typical feature of categories that are Reedy elegant, or skeletal, or graphic (von Neumann-)regular, but more general. In fact, we show that the more general ‘successor’ may be described as a function on the poset of full subcategories of the site that are closed under subobjects.

Decalf: A Directed, Effectful Cost-Aware Logical Framework

We present decalf , a d irected, e ffectful c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs with effects. Like calf , the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike calf , the present language takes account of effects , such as probabilistic choice and mutable state. This extension requires a reformulation of calf ’s approach to cost accounting: rather than rely on a ”separable” notion of cost, here a cost bound is simply another program . To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a cost recurrence and readily extends to higher-order, effectful programs. The development proceeds by first introducing the decalf type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify decalf via a model in the topos of augmented simplicial sets.

Are two H-spaces homotopy equivalent? An algorithmic view point

This paper proposes an algorithm that decides if two simply connected spaces represented by finite simplicial sets of finite [Formula: see text]-type and finite dimension [Formula: see text] are homotopy equivalent. If the spaces are homotopy equivalent, the algorithm finds a homotopy equivalence between their Postnikov stages in dimension [Formula: see text]. As a consequence, we get an algorithm deciding if two spaces represented by finite simplicial sets are stably homotopy equivalent.

Weak cartesian properties of simplicial sets

Weak cartesian properties of simplicial sets

On loop spaces of simplicial sets with marking

On loop spaces of simplicial sets with marking

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

AbstractWe 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.

Cubical and Simplicial Sets in the Category of Quivers

Cubical and Simplicial Sets in the Category of Quivers

Martin-Löf identity types in C-systems

This paper continues a series of papers that develop a new approach to syntax and semantics of dependent type theories. Here we study the interpretation of the rules of the identity types in the intensional Martin-Löf type theories on the C-systems that arise from universe categories. In the first part of the paper we develop constructions that produce interpretations of these rules from certain structures on universe categories while in the second we study the functoriality of these constructions with respect to functors of universe categories. The results of the first part of the paper play a crucial role in the construction of the univalent model of type theory in simplicial sets.

Computing Homotopy Classes for Diagrams

We present an algorithm that, given finite diagrams of simplicial sets X, A, Y, i.e., functors {mathcal {I}}^textrm{op}rightarrow {textsf {s}} {textsf {Set}}, such that (X, A) is a cellular pair, dim Xle 2cdot {text {conn}}Y, {text {conn}}Yge 1, computes the set [X,Y]^A of homotopy classes of maps of diagrams ell :Xrightarrow Y extending a given f:Arightarrow Y. For fixed n=dim X, the running time of the algorithm is polynomial. When the stability condition is dropped, the problem is known to be undecidable. Using Elmendorf’s theorem, we deduce an algorithm that, given finite simplicial sets X, A, Y with an action of a finite group G, computes the set [X,Y]^A_G of homotopy classes of equivariant maps ell :Xrightarrow Y extending a given equivariant map f:Arightarrow Y under the stability assumption dim X^Hle 2cdot {text {conn}}Y^H and {text {conn}}Y^Hge 1, for all subgroups Hle G. Again, for fixed n=dim X, the algorithm runs in polynomial time. We further apply our results to Tverberg-type problem in computational topology: Given a k-dimensional simplicial complex K, is there a map Krightarrow {mathbb {R}}^d without r-tuple intersection points? In the metastable range of dimensions, rdge (r+1) k+3, the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.

Local search versus linear programming to detect monotonicity in simplicial branch and bound

AbstractThis study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial partition set or to reduce its simplicial dimension to a relative border (at the boundary of the feasible set) facet (or face) by removing one (or more) vertices. A monotonicity test is more complicated for a simplicial sub-set than for a box, because its orientation does not coincide with the components of the gradient. However, one can focus on directional derivatives (DD). In a previous study, we focused on either basic directions, such as centroid to vertex or vertex to vertex directions, or finding the best directional derivative by solving an LP or MIP. The research question of this paper refers to using local search (LS) based sampling of directions from vertex to facet. Results show that most of the monotonic DD found by LP are also found by LS, but with much less computational cost. Notice that finding a monotone direction does not require to find the direction in which a derivative bound is the steepest.

On the Goldman-Millson theorem for A∞-algebras in arbitrary characteristic

Complete filtered A∞-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the A∞ analogs of several key theorems from the Maurer-Cartan theory for L∞-algebras. In contrast with the L∞ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the A∞-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set N•(A) to an A∞-algebra A, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of N•(A) in terms of the cohomology algebra H(A), and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of A is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L∞-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in [26].

Quasi-2-Segal sets

We show that the 2-Segal spaces (also called decomposition spaces) of Dyckerhoff-Kapranov and G\'alvez-Kock-Tonks have a natural analogue within simplicial sets, which we call quasi-2-Segal sets, and that the two ideas enjoy a similar relationship as the one Segal spaces have with quasi-categories. In particular, we construct a model structure on the category of simplicial sets whose fibrant objects are the quasi-2-Segal sets which is Quillen equivalent to a model structure for complete 2-Segal spaces (where our notion of completeness comes from one of the equivalent characterizations of completeness for Segal spaces). We also prove a path space criterion, which says that a simplicial set is a quasi-2-Segal set if and only if its path spaces (also called d\'ecalage) are quasi-categories, as well as an edgewise subdivision criterion.