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Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories
We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily...
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Homotopy Spheres
This is a beautiful and illuminating example. It shows how the general surgery methods, which we have developed so far, apply to a concrete problem,...
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Weak Graph Map Homotopy and Its Applications
The authors introduce a notion of a weak graph map homotopy (they call it M -homotopy), discuss its properties and applications. They prove that the...
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Homotopy prefactorization algebras
We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras...
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Homotopy commutativity in symmetric spaces
We extend previous results of Ganea and the two of the authors with Takeda on the homotopy commutativity of the loop spaces of Hermitian symmetric...
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From samples to persistent stratified homotopy types
The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a...
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Homotopy Sheaves on Generalised Spaces
We study the homotopy right Kan extension of homotopy sheaves on a category to its free cocompletion, i.e. to its category of presheaves. Any...
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Homotopy type through homology groups
We show that for certain pairs of nonnegative integers, if the reduced integral homology groups of a space are nonzero only in that pair of degrees...
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The cube axiom and resolutions in homotopy theory
We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model...
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Nonexistence of Colimits in Naive Discrete Homotopy Theory
We show that the quasicategory defined as the localization of the category of (simple) graphs at the class of A-homotopy equivalences does not admit...
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Weight Decompositions on Algebraic Models for Map** Spaces and Homotopy Automorphisms
We obtain restrictions on the rational homotopy types of map** spaces and of classifying spaces of homotopy automorphisms by means of the theory of...
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Homotopy, homology, and persistent homology using closure spaces
We develop persistent homology in the setting of filtrations of (Čech) closure spaces. Examples of filtrations of closure spaces include metric...
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Random simple-homotopy theory
We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on...
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Data Assimilation: A Dynamic Homotopy-Based Coupling Approach
Homotopy approaches to Bayesian inference have found wide- spread use especially if the Kullback–Leibler divergence between the prior and the...
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K-flatness and orthogonality in homotopy categories
K-flatness for unbounded complexes of modules over a ring R was introduced by Spaltenstein [27], as an analogue of the classical notion of flatness...
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Homotopy
Retracts, deformation retracts, homotopy of maps and paths are all discussed here. All the material needed to define fundamental groups in the next...
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Homotopy cardinality via extrapolation of Morava–Euler characteristics
We answer a question of John Baez, on the relationship between the classical Euler characteristic and the Baez–Dolan homotopy cardinality, by...
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Homotopy and Hom Construction in the Category of Finite Hypergraphs
We define notions of a weak homotopy for finite hypergraphs and an exponential hypergraph with a right adjoint to the categorical product of finite...
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Stable homotopy groups of spheres: from dimension 0 to 90
Using techniques in motivic homotopy theory, especially the theorem of Gheorghe, the second and the third author on the isomorphism between motivic...