An introduction to Topos Theory (Phd course 2018/19)

This year I will teach an introduttive course on Topos Theory.

Topos theory has many different aspects. On the one hand, a topos is a generalisation of a topological space. On the other hand, every topos can be thought of as a mathematical universe in which one can do mathematics. In fact, there is a duality between Grothendieck topoi and certain first-order theories of logic, called geometric theories. Topos theory grew out of the observation that the category of sheaves over a fixed topological space forms a universe of “continuously variable sets” which obeys the laws of intuitionistic logic. After recalling some basic notions in Category Theory such as functors, natural transformations, limits and adjunctions, we will examine categories of presheaves and their fundamental properties, Grothendieck sites and sheaves and the notion of elementary topos. Applications to logic will be treated.

The (tentative) course calendar is as follows:

  • Tuesday, 7 May 2019, 15:00 (Aula P18, DipMat). Introduction to the course. Categories, functors, natural transformations, adjoint functors and equivalences. A motivation for considering sheaves: dualities.
  • Wednesday, 8 May 2019, 15:30 The category of C-sets and six examples. Representable C-sets and their computation in the examples. (Sala Riunioni, DipMat)
  • Tuesday, 14 May 2019, 15:00 Products, coproducts and other limits and colimits in the category of C-sets, with their calculation in the six examples. Yoneda lemma and Yoneda embedding. (Sala Riunioni, DipMat)
  • Wednesday, 15 May 2019, 15:30. Every C-set is a colimit of representable C-sets. Intrinsic properties of representable objects: connectivity, irreducibility and continuity. Sections, retractions and idempotents. (Sala Riunioni, DipMat)
  • Tuesday, 21 May 2019, 15:30 (Sala Riunioni, DipMat)
  • Wednesday, 22 May 2019, 16:00 (Sala Riunioni, DipMat)
  • Tuesday, 28 May 2019, 15:00 (Sala Riunioni, DipMat)
  • Wednesday, 29 May 2019, 15:30 (Sala Riunioni, DipMat)
  • Tuesday, 4 June 2019, 15:00 (Sala Riunioni, DipMat)
  • Wednesday, 5 June 2019, 15:30 (Sala Riunioni, DipMat)

The references for the course are:

  • F. William Lawvere and Steve Schanuel, Conceptual Mathematics: A First Introduction to Categories, Cambridge U. Press, Cambridge, 1997.
  • Reyes, Reyes, Zolfaghari – Generic figures and their glueings. Polimetrica, 2008.
  • MacLane, Saunders, Moerdijk, Ieke. Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer Universitext, 1994.
  • Robert Goldblatt, Topoi, the Categorial Analysis of Logic. Dover Revised edition, 2006.
  • Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium, Oxford U. Press, Oxford. Volume 1 (2002), Volume 2, (2002), Volume 3 (in preparation).

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