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We study the categories governing infinity (wheeled) properads. The graphical category, which was already known to be generalized Reedy, is, in fact, an Eilenberg-Zilber category. A minor alteration to the definition of the wheeled graphical category allows us to show that it is a generalized Reedy category. Finally, we present model structures for Segal properads and Segal wheeled properads. [ABSTRACT FROM AUTHOR]

In this paper we define Martin-Löf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-Löf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-Löf complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-Löf complexes are a model of homotopy 1-types. [ABSTRACT FROM AUTHOR]

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 non-negative integers. Sets of these walks are also important in other areas, such as stochastic processes, automata, combinatorial group theory, C"-algebras, etc. We put a Quillen model structure on the category of directed graphs, for which the weak equivalences are those graph morphisms which induce bijections on the set of walks. We determine the resulting homotopy category. We also introduce a "finite-level" homotopy category which respects the natural topology on the set of walks. To each graph we associate a basal graph, well defined up to isomorphism. We show that the basal graph is a homotopy invariant for our model structure, and that it is a finer invariant than the zeta series of a finite graph. We also show that, for finite walkable graphs, if B is basal and separated then the walk spaces for X and B are topologically conjugate if and only if X and B are homotopically equivalent for our model structure [ABSTRACT FROM AUTHOR]

Abstract: We establish a connection between differential graded and simplicial categories by constructing a three-step zig-zag of Quillen adjunctions relating the homotopy theories of the two. In an intermediate step, we extend the Dold–Kan correspondence to a Quillen equivalence between categories enriched over non-negatively graded complexes and categories enriched over simplicial modules. As an application, we obtain a simple calculation of Simpson''s homotopy fiber, which is known to be a key step in the construction of a moduli stack of perfect complexes on a smooth projective variety. [Copyright &y& Elsevier]

Abstract: We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories. [Copyright &y& Elsevier]