In mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.
The mathematical field that studies topoi is called topos theory.
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of the capability of Grothendieck toposes to incarnate the “essence” of different mathematical situations is given by their use as bridges for connecting theories which, albeit written in possibly very different languages, share a common mathematical content.
A Grothendieck topos is a C which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.)
There is a D and an inclusion C ↪ Presh(D) that admits a finite-limit-preserving left adjoint.
C is the category of sheaves on a Grothendieck site.
C satisfies Giraud's axioms, below.
Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a .
Giraud's axioms for a C are:
C has a small set of s, and admits all small colimits. Furthermore, fiber products distribute over coproducts.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, and topological algebraic geometry.
The theme of the working group varies from year to year. Examples of recent topics studied include: Galois theory of ring spectra, duality in algebra and topology, topological algebraic geometry and t
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
In mathematics, specifically in , an exponential object or map object is the generalization of a function space in set theory. with all and exponential objects are called . Categories (such as of ) without adjoined products may still have an exponential law. Let be a category, let and be of , and let have all with .
In , a is Cartesian closed if, roughly speaking, any morphism defined on a of two can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by , whose internal language, linear type systems, are suitable for both quantum and classical computation.
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F.
Let X /S be a flat algebraic stack of finite presentation. We define a new & eacute;tale fundamental pro-groupoid pi(1)(X /S), generalizing Grothendieck's enlarged & eacute;tale fundamental group from SGA 3 to the relative situation. When S is of equal pos ...
We define filter quotients of -categories and prove that filter quotients preserve the structure of an elementary -topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of -ca ...
2021
,
A computer-implemented inverse rendering method is provided. The method comprises: computing an adjoint image by differentiating an objective function that evaluates the quality of a rendered image, image elements of the adjoint image encoding the sensitiv ...