Related papers: Augmented Homotopical Algebraic Geometry

Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of…

Representations over diagrams of abelian categories unify quite a few notions appearing widely in literature such as representations of categories, presheaves of modules over categories, representations of species, etc. In this series of…

Many interesting classes of maps from homotopical algebra can be characterised as those maps with the right lifting property against certain sets of maps (such classes are sometimes referred to as cofibrantly generated). In a more…

Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction…

In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for…

For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical…

We introduce a new cubical model for homotopy types. More precisely, we'll define a category Qs with the following features: Qs is a PROP containing the classical box category as a subcategory, the category Qs-Set of presheaves of sets on…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

We prove that the derived categories of abelian categories have unique enhancements -- all of them, the unbounded, bounded, bounded above and bounded below derived categories. The unseparated and left completed derived categories of a…

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial…

We develop a cofibrantly generated model category structure in the category of topological spaces in which weak equivalences are A-weak equivalences and such that the generalized CW(A)-complexes are cofibrant objects. With this structure…

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological…

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…

The homotopy theory of higher categorical structures has become a relevant part of the machinery of algebraic topology and algebraic K-theory, and this paper contains contributions to the study of the relationship between B\'enabou's…

In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable…

The Andr\'e-Quillen cohomology of an algebra with coefficients in a module is defined by deriving a functor based on K\"ahler differential forms. It can be computed using a cofibrant resolution of the algebra in a model category structure…

We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.

This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold…

We make a study of ll-extensions of model category structures. We prove an existence result of ll-extensions, present some specific and some rather formal results about them and give an application of the existence result to the homotopy…