In mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by .
In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets.
Another model category is the category of chain complexes of R-modules for a commutative ring R. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and R-algebras, one of the first major applications of the theory. Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.
The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors (e.g., Mark Hovey and Philip Hirschhorn) work with closed model categories and simply drop the adjective 'closed'.
The definition has been separated to that of a model structure on a category and then further categorical conditions on that category, the necessity of which may seem unmotivated at first but becomes important later. The following definition follows that given by Hovey.
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This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous
Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a
We propose an introduction to homotopy theory for topological spaces. We define higher homotopy groups and relate them to homology groups. We introduce (co)fibration sequences, loop spaces, and suspen
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
In mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
In mathematics, more specifically , a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a . The study of such generalizations is known as . Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic and some of the advanced notions and theorems have their analogues for quasi-categories.
Explores the concept of (co)limits in homotopical algebra, discussing functor relations, special cases, and the universal properties of colimits and limits.
Explores the homotopy theory of chain complexes, focusing on retractions and model category structures.
Explores the homotopy theory of chain complexes, focusing on model categories, weak equivalences, and the retraction axiom.
We apply the Acyclicity Theorem of Hess, Kedziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of c
INT PRESS BOSTON, INC2019
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Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids i
We prove that the category of rational SO(2)-equivariant spectra has a simple algebraic model. Furthermore, all of our model categories and Quillen equivalences are monoidal, so we can use this classi