## 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.

## MV-algebras, infinite dimensional polyhedra, and natural dualities

Leo and I have just finished our paper on the connection between natural dualities and the duality between semisimple MV-algebras and compact Hausdorff spaces with definable maps. Actually, we provide a description of definable maps that is intrinsically geometric. In addition, we give some applications to semisimple tensor products, strongly semisimple and polyhedral MV-algebras.

The paper can be downloaded here.

## A(nother) duality for the whole variety of MV-algebras

This is the abstract of a talk I gave in Florence at Beyond 2014.

Given a category \(C\) one can form its

ind-completionby taking all formal directed colimits of objects in \(C\). The “correct” arrows to consider are then families of some special equivalence classes of arrows in \(C\) (Johnstone 1986, V.1.2, pag. 225). Thepro-completionis formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category \(C\) is dually equivalent to thepro-completion of the dual category \(C^{\rm op}\).$$\textrm{ind}\mbox{-}C\simeq (\textrm{pro}\mbox{-}(C^{\rm{op}}))^{\rm{op}}. \qquad\qquad (1)$$

Ind- and pro- completions are very useful objects (as they are closed under directed (co)limits) but cumbersome to use, because of the involved definitions of arrows between objects. We prove that if \(C\) is an algebraic category, then the situation considerably simplifies.

If \(V\) is any variety of algebras, one can think of any algebra \(A\) in \(V\) as colimit of finitely presented algebras as follows.

Consider a presentation of \(A\) i.e., a cardinal \(\mu\) and a congruence [/latex]\theta[/latex] on the free \(\mu\)-generated algebra \(\mathcal{F}(\mu)\) such that \(A\cong \mathcal{F}(\mu)/\theta\). Now, consider the set \(F(\theta)\) of all finitely generated congruences contained in \(\theta\), this gives a directed diagram in which the objects are the finitely presented algebras of the form \(\mathcal{F}(n)/\theta_{i}\) where \(\theta_{i}\in F(\theta)\) and \(X_{1},…,X_{n}\) are the free generators occurring in \(\theta_{i}\). It is straightforward to see that this diagram is directed, for if \(\mathcal{F}(m)/\theta_{1}\) and \(\mathcal{F}(n)/\theta_{2}\) are in the diagram, then both map into \(\mathcal{F}(m+n)/\langle\theta_{1}\uplus\theta_{2}\rangle\), where \(\langle\theta_{1}\uplus\theta_{2}\rangle\) is the congruence generated by the disjoint union of \(\theta_{1}\) and \(\theta_{2}\). Now, the colimit of such a diagram is exactly \(A\).

Denoting by \(V_{\textrm{fp}}\) the full subcategory of \(V\) of finitely presented objects, the above reasoning entails

$$V\simeq\textrm{ind}\mbox{-}V_{\textrm{fp}}. \qquad\qquad (2)$$

We apply our result to the special case where \(V\) is the class of MV-algebras. One can then combine the duality between finitely presented MV-algebras and the category \(P_{\mathbb{Z}}\) of rational polyhedra with \(\mathbb{Z}\)-maps (see here), with (1) and (2) to obtain,

$$MV\simeq\textrm{ind}\mbox{-}MV_{\textrm{fp}}\simeq \textrm{pro}\mbox{-}(P_{\mathbb{Z}})^{\rm{op}}. \qquad\qquad (3)$$

This gives a categorical duality for the whole class of MV-algebras whose geometric content may be more transparent than other dualities in literature. In increasing order of complexity one has that any MV-algebra:

- is dual to a polyhedron (Finitely presented case);
- is dual to an intersection of polyhedra (Semisimple case);
- is dual to a countable nested sequence of polyhedra (Finitely generated case);
- is dual to the directed limit of a family of polyhedra. (General case).

Here are the slides of this talk

## Algebra|Coalgebra seminar

Starting from October 2013 I am organising the Algebra|Coalgebra seminar at the ILLC. The webpage of the seminar are here. See you there!