Homotopy Theory Research Articles

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

We consider a geodesic problem in a manifold endowed with a Randers-Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik-Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution.

Sensitivity-dependent dynamic searching approach coupling multi-intelligent surrogates in homotopy mechanism for groundwater DNAPL-source inversion

Equifinality and premature convergence can result in considerable errors when simultaneously characterizing groundwater contamination sources and estimating contaminant transport parameters. To resolve this problem, we design a sensitivity-dependent progressive optimization system embedding ensemble-learning technique. To avoid repetitive CPU-demanding model evaluations in Sobol’ global sensitivity analysis and swarm intelligence optimization inverse modeling, Kriging, support vector regression (SVR), kernel extreme learning machine (KELM), and deep convolutional neural network (DCNN) are compared and ensembled to build an accurate surrogate of the numerical model. In addition, the sensitivities of different source characteristics and contaminant transport parameters are set as important indicators to adjust the displacement vectors of the swarm in each iteration during the optimization process to achieve a balanced identification of sensitivity-varied elements. Moreover, a homotopy-based progressive searching mechanism approach to the global optimum in large areas is developed, with the aim of preventing premature convergence for multimodal search problems. The results indicate that the ensemble learning model efficiently captures the complex input-output relationship of the numerical model with an increased determination coefficient (R2 = 0.9988), while the mean relative error is limited to 0.9314%. Although the contribution of source characteristics and contaminant transport parameters to the spatial-temporal distribution of contaminants vary dramatically, the combined application of sensitivity analysis, homotopy theory, and swarm intelligence optimization provides a more stable and accurate estimation of all the elements. The mean relative error of the identification results significantly reduced from 7.2184% to 3.2718%, whereas the maximum relative error is limited to 9.9501%.

Categorical models for path spaces

We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by considering localizations of the rigidification functor, unraveling a construction of Hinich, and using a sequence of operators originally introduced by Szczarba in 1961. As a result, we obtain several combinatorial models for the path category of a simplicial set. We then pass to the chain-level and describe a model for the path category, now considered as a category enriched over differential graded (dg) coalgebras, in terms of a suitable algebraic chain model for the underlying simplicial set. This is achieved through a version of the cobar functor inspired by Lazarev and Holstein's categorical Koszul duality. As a consequence, we obtain a conceptual explanation of a result of Franz stating that there is a natural dg bialgebra quasi-isomorphism from the extended cobar construction on the chains of a reduced simplicial set to the chains on its Kan loop group.

The digital Hopf construction

Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration that mimics the smooth setting.

Homotopy theory of net representations

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell [Formula: see text]-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.

Framed motivic $\Gamma$-spaces

We combine several mini miracles to achieve an elementary understanding of infinite loop spaces and very effective spectra in the algebro-geometric setting of motivic homotopy theory. Our approach combines $\Gamma$-spaces and Voevodsky's framed correspondences into the concept of framed motivic $\Gamma$-spaces; these are continuous or enriched functors of two variables that take values in framed motivic spaces. We craft proofs of our main results by imposing further axioms on framed motivic $\Gamma$-spaces such as a Segal condition for simplicial Nisnevich sheaves, cancellation, $\mathbb{A}^1$- and $\sigma$-invariance, Nisnevich excision, Suslin contractibility, and grouplikeness. This adds to the discussion in the literature on coexisting points of view on the $\mathbb{A}^1$-homotopy theory of algebraic varieties.

Some common fixed point theorems in bipolar metric spaces and applications

<abstract><p>In this article, we prove some common fixed point theorems for generalized rational type contractions in bipolar metric spaces. These theorems also generalize and extend several interesting results of metric fixed point theory to the bipolar metric context. In addition, we provide some examples to illustrate our theorems, and applications are obtained in areas of homotopy theory and integral equations by using iterative methods for mathematical operators on a bipolar metric space.</p></abstract>

Multicategories model all connective spectra

There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.

E-infinity structure in hyperoctahedral homology

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.

Fuzzy homotopy theory with some result

This paper aims to introduce the concept of the fuzzy homotopy and give some results. It is shown that the fuzzy homotopy is an equivalence relation between two fuzzy functions. In addition, we proved the fuzzy fundamental group, some theorems and properties.

Topological finiteness properties of monoids, I: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.

The universal six-functor formalism

We prove that Morel-Voevodsky's stable $\mathbb{A}^1$-homotopy theory affords the universal six-functor formalism.

A Pullback Diagram in the Coarse Category

This paper studies the asymptotic product of two metric spaces. It is well defined if one of the spaces is visual or if both spaces are geodesic. In this case the asymptotic product is the pullback of a limit diagram in the coarse category. Using this product construction we can define a homotopy theory on coarse metric spaces in a natural way. We prove that all finite colimits exist in the coarse category.

Homotopy theory of monoid actions via group actions and an Elmendorf style theorem

Homotopy theory of monoid actions via group actions and an Elmendorf style theorem

Synthetic spectra and the cellular motivic category

To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E-homology. We show that the $$\infty $$ -category of synthetic spectra based on E is in a precise sense a deformation of the $$\infty $$ -category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $$E_{*}$$ -based Adams spectral sequence. We describe a symmetric monoidal functor from the $$\infty $$ -category of cellular motivic spectra over $$\text {Spec}({\mathbb {C}})$$ into an even variant of synthetic spectra based on $$\textrm{MU}$$ and show that it induces an equivalence between the $$\infty $$ -categories of p-complete objects for all primes p. In particular, it follows that the p-complete cellular motivic category can be described purely in terms of chromatic homotopy theory.

Existence Suzuki Type Fixed Point Results in Ab-Metric Spaces With Application

In this paper, we give some applications to integral equations as well as homotopy theory via Suzuki contractive type common coupled fixed point results in complete Ab-metric space. We also furnish an example which supports our main result.

Absorbing homogeneous polynomials arising from rational homotopy theory and graph theory

Absorbing homogeneous polynomials arising from rational homotopy theory and graph theory

On the homotopy theory of equivariant colored operads

We build model structures on the category of equivariant simplicial operads with weak equivalences determined by families of subgroups, in the context of operads with a varying set of colors (and building on the fixed color model structures in the prequel). In particular, by specifying to the family of graph subgroups (or, more generally, one of the indexing systems of Blumberg-Hill), we obtain model structures on the category of equivariant simplicial operads whose weak equivalences are determined by norm map data.

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that C^* -algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C^* -algebras. We focus on C^* -algebras which are noncommutative CW-complexes in the sense of Eilers et al. (1998). We construct the stable \infty -category of noncommutative CW-spectra, which we denote by \mathtt{NSp} . Let \mathcal{M} be the full spectral subcategory of \mathtt{NSp} spanned by “noncommutative suspension spectra” of matrix algebras. Our main result is that \mathtt{NSp} is equivalent to the \infty -category of spectral presheaves on \mathcal{M} . To prove this, we first prove a general result which states that any compactly generated stable \infty -category is naturally equivalent to the \infty -category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an \infty -categorical version of a result by Schwede and Shipley (2003). In proving this, we use the language of enriched 1-categories as developed recently by Hinich. We end by presenting a “strict” model for \mathcal{M} . That is, we define a category \mathcal{M}_s strictly enriched in a certain monoidal model category of spectra \mathtt{Sp}^\mathtt{M} . We give a direct proof that the category of \mathtt{Sp}^\mathtt{M} -enriched presheaves \mathcal{M}_s^{\mathtt{op}}\to\mathtt{Sp}^{\mathtt{M}} with the projective model structure models \mathtt{NSp} and conclude that \mathcal{M}_s is a strict model for \mathcal{M} .