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Adding inverses to diagrams II: Invertible homotopy theories are spaces, preprint available at math.AT/0710.2254
"... Abstract. In previous work, we showed that there are appropriate model ..."
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Abstract. In previous work, we showed that there are appropriate model
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
ON AN EXTENSION OF THE NOTION OF REEDY CATEGORY
"... Abstract. We extend the classical notion of a Reedy category so as to allow nontrivial automorphisms. Our extension includes many important examples ..."
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Abstract. We extend the classical notion of a Reedy category so as to allow nontrivial automorphisms. Our extension includes many important examples
HOMOTOPY FIBER PRODUCTS OF HOMOTOPY THEORIES
, 2008
"... Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is wellbehaved with respect to translating it into the setting of more gen ..."
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Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is wellbehaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have welldefined homotopy pullbacks.
SYMBOLIC DYNAMICS AND THE CATEGORY OF GRAPHS
"... Abstract. Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of nonnegative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial gro ..."
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Abstract. Symbolic dynamics is partly the study of walks in a directed graph. By a walk, here we mean a morphism to the graph from the Cayley graph of the monoid of nonnegative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, C∗algebras, etc. We
Homotopy equivalence of isospectral graphs.
, 906
"... to this model structure. We endow the categories of Nsets and Zsets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Zsets, and to sh ..."
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to this model structure. We endow the categories of Nsets and Zsets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Zsets, and to show that two finite directed graphs are almostisospectral if and only if they are homotopyequivalent in our sense. §0. Introduction. Mathematicians often study complicated categories by means of invariants (which are equal for isomorphic objects in the category). Sometimes a complicated category can be replaced by a (perhaps simpler) homotopy category which is better related to the various invariants used to study it. In topology, this was first achieved by declaring two continuous functions to be equivalent when one could be deformed into the other. But it
Preface Prerequisites Notational Conventions Acknowledgments
"... Categorical homotopy theory Emily RiehlTo my students, colleagues, friends who inspired this work.... what we are doing is finding ways for people to understand and think about mathematics. ..."
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Categorical homotopy theory Emily RiehlTo my students, colleagues, friends who inspired this work.... what we are doing is finding ways for people to understand and think about mathematics.
Contents
"... Abstract. The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to ..."
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Abstract. The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushoutproduct) constructions. The general theory is developed in parallel with examples, which we use to prove that familiar formulae for homotopy limits and colimits indeed have the desired properties.
DENDROIDAL SETS AND SIMPLICIAL OPERADS
"... Abstract. We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence ..."
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Abstract. We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence
FILTERED COLIMITS OF ∞CATEGORIES
"... 0.1. The ∞category of (small) ∞categories admits all limits and colimits. In general, colimits are difficult to describe in explicit terms. The purpose of this note is to give an explicit description of filtered colimits of ∞categories. Our conventions regarding ∞categories follow those of [DG]. ..."
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0.1. The ∞category of (small) ∞categories admits all limits and colimits. In general, colimits are difficult to describe in explicit terms. The purpose of this note is to give an explicit description of filtered colimits of ∞categories. Our conventions regarding ∞categories follow those of [DG]. In particular, whenever we