Chain complex | EPFL Graph Search

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but are used in several areas of mathematics,

Collapse-invariant properties of spaces equipped with signals or directions

Stefania Ebli

Collapsing cell complexes was first introduced in the 1930's as a way to deform a space into a topological-equivalent subspace with a sequence of elementary moves. Recently, discrete Morse theory techniques provided an efficient way to construct deformation retracts collapsing one space into the other while preserving global topological properties. This type of collapse, called a Morse matching, has been widely used to speed up computations in (persistent) homology by reducing the size of complexes. Unlike classical collapses, in this thesis we consider topological spaces equipped with signals or directions. The main goal is then to reduce the size of the spaces while preserving as much as possible of both the topological structure and the properties of the signals or the directions. In the first part of the thesis we explore collapsing in topological signal processing. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resultant Hodge decomposition.In Article 2.1 we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. We first prove that any deformation retract of real finite-dimensional based chain complex is equivalent to a Morse matching. We then study the interaction between the Hodge decomposition and signal compression and reconstruction. Specifically, we prove that parts of a signal's Hodge decomposition are preserved under compression and reconstruction for specific classes of discrete Morse deformation retracts of a given based chain complex. Finally, we provide an algorithm to compute Morse matchings with minimal reconstruction error.Complementary to our theoretic results in topological signal processing, we provide two applications in this field. Article 2.2 extends graph convolutional neural networks to simplicial complexes, while Article 2.3 presents a novel algorithm, inspired by the well-known spectral clustering algorithm, to embed simplices in a Euclidean space. The object of our studies in the second part of the thesis is topological spaces equipped with a sense of direction. In the directed setting, the topology of the space is characterized by directed paths between fixed initial and terminal points. Motivated by applications in concurrent programs, we focus on directed Euclidean cubical complexes and their spaces of directed paths.In Article 3.1 we define a notion of directed collapsibility for Euclidean cubical complexes using the notion of past links, a combinatorial local representations of cubical complexes. We show that this notion of collapsability preserves given properties of the directed path spaces. In particular, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex.In Article 3.2 we extend these results, providing further conditions for directed collapses to preserve the contractability or conneectdness of spaces of directed paths. Furthermore, we provide simple combinatorial conditions for preserving the topology of past links. These conditions are the first step towards developing an algorithm that checks at each iteration if a collapse preserves certain properties of the directed space.

EPFL2022

Homotopic Descent over Monoidal Model Categories

The starting point for this project is the article of Kathryn Hess [11]. In this article, a homotopic version of monadic descent is developed. In the classical setting, one constructs a category D(𝕋) of coalgebras in the Eilenberg-Moore category of algebras D𝕋 for a given monad 𝕋 on a category D. There is a canonical functor Can𝕃𝕋 from D to D(𝕋), and if Can𝕃𝕋 is fully faithful, then 𝕋 satisfies descent, while if Can𝕃𝕋 is an equivalence of categories, then 𝕋 satisfies effective descent [19]. In [11], these two conditions are replaced by a weaker one, that these hold only up to homotopy. This is achieved by working with model categories that are enriched over simplicial sets. Homotopic descent is then defined by demanding that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence of simplicial sets. A similar but stronger condition involving the path components in D(𝕋) expresses effective homotopic descent. The first goal of this project is to develop a framework of homotopic descent for model categories that are enriched over model categories other than simplicial sets. The most important examples we have in mind are chain complexes and spectra. In order to achieve this goal, we tried to determine the most general conditions that are sufficient and necessary to make the theory work. To ease the formulation, let us say that we are working with a model category D that is enriched over a monoidal model category V. The crucial constructions we need are realization, respectively totalization, of (co)simplicial objects in D. These functors have to be Quillen functors to ensure that they have the correct homotopical behaviour. This implies that there must exist a Quillen adjunction between V and simplicial sets. Furthermore, we need to be able to transfer the enrichment and (co)tensoring over V to an enrichment and (co)tensoring over simplicial sets. This forces the Quillen adjunction to be monoidal. Another main point that has to be adressed is the question, of whether the enrichment of D carries over to an enrichment of D𝕋 and D(𝕋) and how this enrichment behaves. It turns out that this works well under mild assumptions on V. This leads then to the definition of homotopic descent by requiring that each component in (Can𝕃𝕋)A,B : MapD(A,B) → MapD(𝕋) (Can𝕃𝕋(A), Can𝕃𝕋 (B)) be a weak equivalence in M and similarly for effective homotopic descent. Using this definition, the theorems in [11] carry over to this more general context. Although the conditions on V are rather constraining regarding the relation with simplicial sets, the cases of chain complexes and spectra are included. For the time being we do not see how the constraints on V could be weakened. The second goal of this project is to apply the theory of homotopic descent to concrete examples. A good source of examples is homotopic Grothendieck descent in the category of spectra, i.e., S-modules. Classical Grothendieck descent deals with the adjunction induced by a morphism φ : B → A of monoids in a monoidal category (M,Λ, S), – BΛ A : ModB ⇄ ModA : φ*, which in turn induces a monad 𝕋φ := φ*(– ΛB A) on ModB. We consider in particular the case when the morphism in question is the unit of an S-algebra E, η : S → E There is a close relationship between comodules over a Hopf algebroid and objects in D(𝕋η). Associated to η we have the canonical co-ring Wη := E ΛS E and an isomorphism between D(𝕋η) and the category of comodules over Wη in the category of S-modules. This relationship is explored in an analysis of the stable Adams spectral sequence, the construction of which heavily relies on the monadic properties of the functor η*(E ΛS –) and can therefore be expressed in terms of D(𝕋η). We construct a spectral sequence that generalizes the stable Adams spectral sequence to any stable pointed model category such as unbounded chain complexes. One can give a description of the E2-term as an Ext in D(𝕋η), E2s,t = ExtD(𝕋η) (Can(A), Can(B)). If the spectral sequences converges, it abuts to π⁎MapD(A,B η^), where Bη^ is the derived 𝕋η-completion of B, which agrees with the usual derived completion in well-known special cases. Furthermore, Bη^ := Tot B^•, and B^• is kind of a fibrant cosimplicial resolution of B. Furthermore, the language of relative homological algebra for modules and comodules generalizes to definitions for algebras in D𝕋η and coalgebras in D(𝕋η). This shows that the construction of the Adams spectral sequence works in a more general setting, where one applies a functor to an abelian category, for example π⁎, only at the end, to be able to do computations in homological algebra. This general Adams spectral sequence is closely related to the descent spectral sequence of [11], and we have clarified this relationship.

EPFL2011

Why H Z-algebra Spectra are Differential Graded Algebras?

Varvara Karpova

In homological algebra, to understand commutative rings R, one studies R-modules, chain complexes of R-modules and their monoids, the differential graded R-algebras. The category of R-modules has a rich structure, but too rigid to efficiently work with homological invariants and homotopy invariant properties. It appears more appropriate to operate in the derived category D(R), which is the homotopy category of differential graded R-modules. Algebra of symmetric spectra offers a generalization of homological algebra. In this frame, spectra are objects that take the place of abelian groups; in particular, the analogue of the initial ring Z is the sphere spectrum S. Tensoring over S endows the category of spectra with a symmetric monoidal smash product, analogous to the tensor product of abelian groups. Thus, spectra are S-modules, and ring spectra, which extend the notion of rings, are the S-algebras. To any discrete ring R, one can associate the Eilenberg-Mac Lane ring spectrum HR, which is commutative if R is.

2010

Collapse-invariant properties of spaces equipped with signals or directions

Stefania Ebli

EPFL2022

Homotopic Descent over Monoidal Model Categories

EPFL2011