Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents

Abstract. In this paper and its follow-up [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L∞-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.

Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =

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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

Axiomatic homotopy theory for operads

Abstract. This paper is the follow-up of [MV08].

Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give also a unified axiomatic treatment of operads as functors on labeled graphs. Finally, we extend internal cohomomorphism constructions to more general categorical contexts. 0.1. Internal cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and

Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids by Zhi-ming Luo

Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids by Zhi-Ming Luo