Related papers: Simplicial approach to derived differential manifo…
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold…
We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore,…
We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a…
We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of…
Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the…
We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting…
We construct differential equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct…
Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic…
In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…
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.
On one hand, together with Pelle Steffens, we recently characterized the infinity category of derived manifolds up to equivalence by a universal property. On the other hand, it is shown in recent work of Behrend-Liao-Xu that the category of…
We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the…
The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…
In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log \'etale maps and use this to define derived log stacks.
Derived geometry can be defined as the universal way to adjoin finite homotopical limits to a given category of manifolds compatibly with products and glueing. The point of this paper is to show that a construction closely resembling…
We construct a complex of differential forms on a local $C^\infty$-ringed space. The two main classes of spaces we have in mind are differential spaces in the sense of Sikorski and $C^\infty$-schemes. Just as in the case of manifolds the…
By Rickard's work, two rings are derived equivalent if there is a tilting complex, constructed from projective modules over the first ring such that the second ring is the endomorphism ring of this tilting complex. In this work I describe,…
In this paper, we study some relationships existing between some particular mathematical structures: discrete surfaces coming from discrete topology and mathematical morphology, poset-based connected manifolds coming from discrete topology,…
An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical…
Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work…