• Corpus ID: 127395338

Using the internal language of toposes in algebraic geometry

  title={Using the internal language of toposes in algebraic geometry},
  author={Ingo Blechschmidt},
Any scheme has its associated little and big Zariski toposes. These toposes support an internal mathematical language which closely resembles the usual formal language of mathematics, but is “local on the base scheme”: For example, from the internal perspective, the structure sheaf looks like an ordinary local ring (instead of a sheaf of rings with local stalks) and vector bundles look like ordinary free modules (instead of sheaves of modules satisfying a local triviality condition). The… 

Figures and Tables from this paper

Syntactic presentations for glued toposes and for crystalline toposes

We regard a geometric theory classified by a topos as a syntactic presentation for the topos and develop tools for finding such presentations. Extensions (or expansions) of geometric theories, which

Limits, Colimits, and Spectra of Modelled Spaces

It is well-known that the construction of Zariski spectra of (commutative) rings yields a dual adjunction between the category of rings and the category of locally ringed spaces. There are many

Flabby and injective objects in toposes

We introduce a general notion of flabby objects in elementary toposes and study their basic properties. In the special case of localic toposes, this notion reduces to the common notion of flabby


We define and study the Grothendieck AB axioms [4] for abelian (univalent) categories in Homotopy Type Theory. Our main result is that categories of modules over a ring satisfy the internal versions

Generalized Spaces for Constructive Algebra

The purpose of this contribution is to give a coherent account of a particular narrative which links locales, geometric theories, sheaf semantics and constructive commutative algebra. We are hoping

An elementary and constructive proof of Grothendieck's generic freeness lemma

We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully

Syntax for Semantics: Krull’s Maximal Ideal Theorem

Krull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations of the Axiom of Choice (AC) in ring theory. For many a consequence of AC, constructive counterparts are well within

Operations on Categories of Modules are Given by Schur Functors

Families of functors between categories of finitely generated modules which are defined for all commutative k-algebras simultaneously and are compatible with base changes are studied.

Yoneda's lemma for internal higher categories

. We develop some basic concepts in the theory of higher categories internal to an arbitrary ∞ topos. We define internal left and right fibrations and prove a version of the Grothendieck construction

Normalization for Cubical Type Theory

The normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms.



Sheaves in geometry and logic: a first introduction to topos theory

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various

Universal projective geometry via topos theory


In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian


The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the

Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves

In his fundamental paper [5], Serre initiated the study of the Cech cohomology of coherent sheaves on (separated) algebraic varieties with their Zariski topology. He proved that coherent sheaves on

Points in the fppf topology

Using methods from commutative algebra and topos-theory, we construct topos-theoretical points for the fppf topology of a scheme. These points are indexed by both a geometric point and a limit

Applications of constructive logic to sheaf constructions in toposes

The internal logic of a topos is exploited to give an easy proof of the fact that topologies in a topos form a locale and to give simple internal reformulations of two well-known variants of

Notions of Anonymous Existence in Martin-Löf Type Theory

A new notion of anonymous existence in type theory is defined and compared carefully to compare different forms of existence carefully and show possibly surprising consequences of the judgmental computation rule of the truncation.

Categories of spaces built from local models

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces.

Classifying Topoi and Preservation of Higher Order Logic by Geometric Morphisms.

Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the