# Using the internal language of toposes in algebraic geometry

@inproceedings{Blechschmidt2021UsingTI, title={Using the internal language of toposes in algebraic geometry}, author={Ingo Blechschmidt}, year={2021} }

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…

## 15 Citations

### Syntactic presentations for glued toposes and for crystalline toposes

- Mathematics
- 2022

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

- Mathematics
- 2022

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

- Mathematics
- 2018

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…

### THE INTERNAL AB AXIOMS

- Mathematics
- 2022

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

- MathematicsProof and Computation II
- 2021

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

- Mathematics
- 2018

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

- MathematicsPaul Lorenzen -- Mathematician and Logician
- 2021

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

- MathematicsAppl. Categorical Struct.
- 2018

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

- Mathematics
- 2021

. 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

- Mathematics2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

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.

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