The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on ‘higher props,’ we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.     

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.     

A simplicial homotopy algorithm for computing zero points on polytopes

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On cellularization for simplicial presheaves and motivic homotopy

We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.     

Enriched simplicial presheaves and the motivic homotopy category

We construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category.     

On the homotopy of simplicial algebras over an operad

According to a result of H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra.

     

Coarse homotopy groups

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of a cone over a compact simplicial complex coincide with the usual homotopy groups of the underlying compact simplicial complex. To prove this we develop geometric triangulation techniques for cones which we expect to be of relevance also in different contexts.     

Minimal fibrations and the organizing theorem of simplicial homotopy theory

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss–Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.     

Symmetric homotopy construction

Homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system. Computations will be performed more efficiently if the symmetric structure of the system can be exploited. This paper presents the construction of a symmetric homotopy. Using this homotopy, only the paths according to the generating solutions have to be traced during continuation.     

1-Cohomology of simplicial amalgams of groups

We develop a cohomological method to classify amalgams of groups. We generalize this to simplicial amalgams in any concrete category. We compute the non-commutative 1-cohomology for several examples of amalgams defined over small simplices.     

Volume distortion in homotopy groups

Given a finite metric CW complex X and an element \({\alpha \in \pi_n(X)}\), what are the properties of a geometrically optimal representative of \({\alpha}\)? We study the optimal volume of \({k\alpha}\) as a function of k. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an invariant with respect to the rational homotopy of X. We provide a number of examples and techniques for studying this invariant, with a special focus on spaces with few rational homotopy groups. Our main theorem characterizes those X in which all non-torsion homotopy classes are undistorted, that is, their distortion functions are linear.