348 research outputs found
The topological realization
In this paper, I argue that the newly developed network approach in neu- roscience and biology provides a basis for formulating a unique type of realization, which I call topological realization. Some of its features and its relation to one of the dominant paradigms of realization and explanation in sciences, i.e. the mecha- nistic one, are already being discussed in the literature. But the detailed features of topological realization, its explanatory power and its relation to another prominent view of realization, namely the semantic one, have not yet been discussed. I argue that topological realization is distinct from mechanistic and semantic ones because the realization base in this framework is not based on local realisers, regardless of the scale (because the local vs global distinction can be applied at any scale) but on global realizers. In mechanistic approach, the realization base is always at the local level, in both ontic (Craver 2007, 2014) and epistemic accounts (Bechtel and Richardson 2010). The explanatory power of realization relation in mechanistic approach comes directly from the realization relation-either by showing how a model is mapped onto a mechanism, or by describing some ontic relations that are explanatory in themselves. Similarly, the semantic approach requires that concepts at different scales logically satisfy microphysical descriptions, which are at the local level. In topological frame- work the realization base can be found at different scales, but whatever the scale the realization base is global, within that scale, and not local. Furthermore, topological realization enables us to answer the “why” questions, which according to Polger 2010 make it explanatory. The explanatoriness of topological realization stems from under- standing mathematical consequences of different topologies, not from the mere fact that a system realizes them
The simplicial volume of 3-manifolds with boundary
We provide sharp lower bounds for the simplicial volume of compact
-manifolds in terms of the simplicial volume of their boundaries. As an
application, we compute the simplicial volume of several classes of
-manifolds, including handlebodies and products of surfaces with the
interval. Our results provide the first exact computation of the simplicial
volume of a compact manifold whose boundary has positive simplicial volume. We
also compute the minimal number of tetrahedra in a (loose) triangulation of the
product of a surface with the interval.Comment: 24 pages, 5 figures. Section 6 has been removed, and will appear in a
separate paper by the same authors. This version has been accepted for
publication by the Journal of Topolog
Pro-categories in homotopy theory
The goal of this paper is to prove an equivalence between the model
categorical approach to pro-categories, as studied by Isaksen, Schlank and the
first author, and the -categorical approach, as developed by Lurie.
Three applications of our main result are described. In the first application
we use (a dual version of) our main result to give sufficient conditions on an
-combinatorial model category, which insure that its underlying
-category is -presentable. In the second application we
consider the pro-category of simplicial \'etale sheaves and use it to show that
the topological realization of any Grothendieck topos coincides with the shape
of the hyper-completion of the associated -topos. In the third
application we show that several model categories arising in profinite homotopy
theory are indeed models for the -category of profinite spaces. As a
byproduct we obtain new Quillen equivalences between these models, and also
obtain an example which settles negatively a question raised by Raptis
Topological comparison theorems for Bredon motivic cohomology
We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for
Bredon motivic cohomology of smooth complex and real varieties with an action
of the group of order two. This identifies equivariant motivic and topological
invariants in a large range of degrees.Comment: Corrected indices in main theorem and a few minor changes. To appear,
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