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 (Sala Riunioni, DipMat). The category of \mathcal{C}-sets and six examples. Representable \mathcal{C}-sets and their computation in the examples.
  • Tuesday, 14 May 2019, 15:00 (Sala Riunioni, DipMat). Products, coproducts and other limits and colimits in the category of \mathcal{C}-sets, with their calculation in the six examples. Yoneda lemma and Yoneda embedding.
  • Wednesday, 15 May 2019, 15:30 (Sala Riunioni, DipMat). Every \mathcal{C}-set is a colimit of representable C-sets. Intrinsic properties of representable objects: connectivity, irreducibility and continuity. Sections, retractions and idempotents.
  • Tuesday, 21 May 2019, 15:30 (Sala Riunioni, DipMat). The equivalence between the Cauchy completion of \mathcal{C} and the full subcategory of continuous objects in Sets^{\mathcal{C}^{op}}.
  • Wednesday, 22 May 2019, 16:00 (Sala Riunioni, DipMat) Exponentials and Subobject classifiers, with examples.
  • Tuesday, 28 May 2019, 15:00 (Sala Riunioni, DipMat) There will not be lectures this week.
  • Wednesday, 29 May 2019, 15:30 (Sala Riunioni, DipMat) There will not be lectures this week.
  • Tuesday, 4 June 2019, 15:00 (Sala Riunioni, DipMat) Frames and point-free geometry. The algebraic structure of the subobject classifier.
  • Wednesday, 5 June 2019, 15:30 (Sala Riunioni, DipMat) The interpretation of geometric logic in a topos. The internal logic of a topos.
  • Tuesday, 11 June 2019, 15:00 (Sala Riunioni, DipMat) Geometric functors. Grothendieck topoi.
  • Wednesday, 12 June 2019, 15:30 (Sala Riunioni, DipMat) Classifying topoi.

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).

PhD Course, Category Theory (May-June 2018)

Contents of the page


Topics of the course

  • Categories, universal properties, functors.
  • Natural transformations, adjoint functors and categorical equivalences.
  • Concrete dualities.
  • Yoneda Lemma.
  • Sheaves and topoi.

Course material


  • Lecturer: Luca Spada
  • Duration of the course: 20 hours.

Preliminary programme

  • Friday 18 May 2018, from 11:00 to 13:00;
  • Monday 21 May 2018, from 9:00 to 11:00;
  • Wednesday 23 May 2018, from 9:00 to 11:00;
  • Friday 25 May 2018, from 9:00 to 11:00;
  • Monday 4 June 2018, from 15:00 to 17:00;
  • Wednesday, June 6, 2018, from 9:00 to 11:00;
  • Friday 8 June 2018, from 9:00 to 11:00;
  • Thursday 21 June 2018, from 9:00 to 11:00;
  • Monday 9 July 2018, from 15:00 to 17:00;
  • Wednesday 11 July 2018, from 9:00 to 11:00;

Comments, complaints, questions: write to Luca Spada

General affine adjunctions, Nullstellensätze, and dualities

At last, we have finished and submitted our paper on “General affine adjunctions, Nullstellensätze, and dualities” co-authored with Olivia Caramello and Vincenzo Marra.

Abstract. We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert’s Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.

The preprint is available on arXiv.  We made another preprint available some years ago(!), but the manuscript has changed in many respects.  The main differences between the two versions on arXiv are the following:

  1. The comparison with the existing literature is now more thorough.
  2. The categories R and D are now taken directly without passing through the quotient categories. In our opinion, this is cleaner and, as a consequence, it is now clearer what are the minimal assumption on the triplet I: T -> S.
  3. There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.