Model categories and simplicial methods (original) (raw)

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Contemporary Mathematics Model Categories and Simplicial Methods

2006

There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We’re going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subje...

Simplicial structures on model categories and functors

American Journal of Mathematics, 2001

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or 'continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

Simplicial model category structures on the category of chain functors

Homology, Homotopy and Applications, 2007

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in Ch. After an appropriate tensor and cotensor structure in Ch is established (§1, §3), Quillen's axiom SM7 is verified in §5 and §6. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing Ch with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) (§7). In §8 the model structure on Ch is shown to be proper.

Overview on Models in Homotopical Algebra

2004

A covariant functor ∆ →A is called a model object of A. Model objects produce in A a subject matter very much as algebraic topology when A is the category of topological spaces. Here we describe the settings on which such concepts are developed and describe the main features developed by the author about model objects.

The Boardman–Vogt resolution of operads in monoidal model categories

Topology, 2006

We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ -cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.

Left-determined model categories and universal homotopy theories

Transactions of the American Mathematical Society, 2003

We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. r B v / / D there exists a diagonal d : B → C.

Erratum to ``Left-determined model categories and universal homotopy theories

Transactions of the American Mathematical Society, 2008

We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories.

Projective Model Structures for Exact Categories

arXiv: Category Theory, 2016

In this article we provide sufficient conditions on weakly idempotent complete exact categories EEE which admit an abelian embedding, such that various categories of chain complexes in EEE are equipped with projective model structures. In particular we show that as soon as EEE has enough projectives, the category textbfCh+(E)\textbf{Ch}_{+}(E)textbfCh+(E) of bounded below complexes is equipped with a projective model structure. In the case that EEE also admits all kernels we show that it is also true of textbfChge0(E)\textbf{Ch}_{\ge0}(E)textbfChge0(E), and that a generalisation of the Dold-Kan correspondence holds. Supplementing the existence of kernels with a condition on the existence and exactness of certain direct limit functors guarantees that the category of unbounded chain complexes textbfCh(E)\textbf{Ch}(E)textbfCh(E) also admits a projective model structure. When EEE is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. Along the way we also discuss genera...