## PhD Course, Category Theory (May-June 2018)

## Contents of the page

## Announcements

## Topics of the course

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

## Course material

- Harold Simmons. An Introduction to Category Theory. Cambridge University Press, 2011.
- Robert Goldblatt Topoi: The Categorial Analysis of Logic, Dover Publications 2006.
- Saunders Mac Lane. Categories for the Working Mathematician (Second edition). Springer. 1988.

## Practicalities

- 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’sNullstellensatz, 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:

- The comparison with the existing literature is now more thorough.
- 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.
- There is now a section studying the issue of concreteness of the adjunction and comparing with the theory of concrete adjunction.