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https://github.com/iblech/internal-methods

Notes on how to use the internal language of toposes in algebraic geometry
https://github.com/iblech/internal-methods

algebraic-geometry categorical-logic mathematics sheaf-theory topos-theory

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Notes on how to use the internal language of toposes in algebraic geometry

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# Using the internal language of toposes in algebraic geometry



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 translation of internal statements and

proofs is facilitated by an easy mechanical procedure.



We investigate how the internal language of the little Zariski topos

can be exploited to give simpler definitions and more conceptual

proofs of the basic notions and observations in algebraic geometry.

To this end, we build a dictionary relating internal and external notions and

demonstrate its utility by giving a simple proof of Grothendieck's generic

freeness lemma in full generality. We also employ this framework to state a

general transfer principle which relates modules with their induced quasicoherent

sheaves, to study the phenomenon that some

properties spread from points to open neighborhoods, and to compare general

notions of spectra.



We employ the big Zariski topos to set up the foundations of a synthetic account

of scheme theory. This account is similar to the synthetic account of

differential geometry, but has a distinct algebraic flavor. Central to the

theory is the notion of synthetic quasicoherence, which has no analog in

synthetic differential geometry. We also discuss how various common subtoposes

of the big Zariski topos can be described from the internal point of view and

derive explicit descriptions of the geometric theories which are

classified by the fppf and by the surjective topology.



No prior knowledge about topos theory or formal logic is assumed.



* **[Expository notes (PDF)](https://rawgit.com/iblech/internal-methods/master/notes.pdf)**

* [Slides (PDF)](https://www.speicherleck.de/iblech/stuff/gael2013-topos.pdf)

  for [GAeL 2013](https://www.mimuw.edu.pl/~gael/xxi/)

* [Poster (PDF)](https://rawgit.com/iblech/internal-methods/master/poster.pdf)

  for [GAeL 2014](https://www.mimuw.edu.pl/~gael/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-ihes2015.pdf)

  and [video](https://www.youtube.com/watch?v=7S8--bIKaWQ)

  for [Topos à l'IHÉS 2015](https://indico.math.cnrs.fr/event/747/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-pssl99.pdf)

  for [PSSL 99](https://www.iti.cs.tu-bs.de/~koslowj/PSSL99)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-cambridge2016.pdf)

  for the [Category Theory Seminar](https://talks.cam.ac.uk/talk/index/66318) at

  the Centre for Mathematical Sciences in Cambridge

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-warwick2017.pdf)

  for the [Algebraic Geometry Seminar](https://homepages.warwick.ac.uk/staff/A.Thompson.8/seminar.html#Blechschmidt)

  in Warwick

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-duesseldorf2017.pdf)

  for the Oberseminar Algebra/Geometrie/Topologie in Düsseldorf

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-pssl101.pdf)

  for [PSSL 101](https://www1.maths.leeds.ac.uk/~pmtng/pssl101.html)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-pssl103.pdf)

  for [PSSL 103](https://www.math.muni.cz/~loregianf/PSSL103/PSSL103.php)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-munich2018.pdf)

  for the [Oberseminar Mathematische Logik in München](https://www.mathematik.uni-muenchen.de/~schwicht/seminars/osem/indexosem.php)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-filmat2018.pdf)

  for [FilMat 2018](https://filmatnetwork.com/filmat-2018-program/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-unilog2018.pdf)

  for [UNILOG 2018](https://www.uni-log.org/wk6-CAT.html)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-como2018.pdf)

  for [Toposes in Como 2018](http://tcsc.lakecomoschool.org/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-leipzig2018.pdf)

  for the [Group Seminar](https://www.mis.mpg.de/calendar/lectures/2018/abstract-26140.html)

  of Jürgen Jost's group in Leipzig

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-bayreuth2018.pdf)

  for the [Colloquium Logicum 2018](https://www.cl2018.uni-bayreuth.de/en/index.html) in Bayreuth

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-pssl104.pdf)

  for [PSSL 104](https://mysite.science.uottawa.ca/phofstra/PSSL18.html) in Amsterdam

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-muenchenwiler2018.pdf)

  for the [Münchenwiler Meeting Autumn 2018](http://mw.inf.unibe.ch/) in Münchenwiler

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-padova2018.pdf)

  for the Logic Seminar in Padova*

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-birmingham2019.pdf)

  for the [6th Workshop on Formal Topology](https://www.cs.bham.ac.uk/~sjv/6WFTop/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-edinburgh2019.pdf)

  for [Category Theory 2019](http://conferences.inf.ed.ac.uk/ct2019/)

* [Slides (PDF)](https://rawgit.com/iblech/internal-methods/master/slides-herrsching2019.pdf)

  for [Proof and Computation 2019](http://www.mathematik.uni-muenchen.de/~schwicht/pc19.php)



![Sheaves of rings look like ordinary rings from the internal point of view.](images/external-internal-small.png)