It is not easy for me to think of CatAlg 2026 only as a conference that took place from 1 to 4 July. For many months it was made of emails, programmes, hotel reservations, lists of participants, changes, and many small practical problems that Andrea Montoli and myself had to deal with. Then, suddenly, people started arriving in Edolo, the lecture room was full, and the conference really began.

This was the twentieth-anniversary edition of CatAlg. The first meeting was held in Milan in 2006, and later there were the editions in Gargnano in 2011, 2015, 2018, and 2022. This time we moved to Edolo, at the Mountain University of the University of Milan. It was a new place for CatAlg, but at the same time the continuation of a long story.

Wednesday, 1 July — The beginning

The first morning of a conference always has a particular atmosphere. There are colleagues and friends that one has not seen for some time, new participants, people arriving with luggage, coffee, greetings, and many last-minute things to check.

At 9:30, after the greeting message of Luca Giupponi on behalf of the UniMont community, CatAlg 2026 officially started.

Indeed, it was a great pleasure for me to see the room full. At that moment the conference was no longer only something organised by us. It became something made by all the speakers and participants.

We wanted CatAlg to be a real workshop, not only a sequence of talks. For this reason, we planned long lectures, with enough time to explain ideas, give motivations, and discuss questions.

Already on the first day there was a moment that was very special to me. In his lecture, George Janelidze also spoke about coherent actions and ideals. These notions are part of a work that I am doing with Federica Piazza, my PhD student, and Manuel Mancini, a former PhD student.

At a certain point, and completely unexpectedly for me, George asked me to explain briefly these concepts to the audience. I was very pleased by this invitation. I had not planned to speak, and perhaps for this reason I enjoyed the moment even more. It was also nice to see that a work developed with Federica and Manuel entered naturally into the discussion of the conference.

From the first day, the discussions continued during coffee breaks and lunch. People started talking in small groups, sometimes with a piece of paper, sometimes at the blackboard, and sometimes only with gestures.

This was exactly the kind of conference we wanted.

Thursday, 2 July — The conference finds its rhythm - poster session

On the second day, everybody already knew the place and the conference had found its rhythm. Discussions started again from where they had stopped the day before, and mathematics continued inside and outside the lecture room.

The talks touched many different aspects of categorical algebra: cohomology, exactness, higher-dimensional structures, protomodularity, Hopf structures, metrics, norms, and other related topics.

For me, Alan Cigoli’s lectures, given on the second and third days, had a special importance. Alan presented the present state of a project that started many years ago with Sandra Mantovani and Enrico Vitale, and later involved Alan himself and me.

It was interesting, and also a little emotional, to see this long project presented as a whole. Usually, when one works on a problem for many years, one sees only one part at a time. Listening to Alan, I could see more clearly the complete development: the first questions, the results obtained in different periods, the changes of point of view, and the present state of the work.

Research does not always follow a straight line. Some projects continue for many years, stop for some time, and then start again in a new form. Alan’s lectures showed very well this kind of mathematical history.

The programme also included open problems, informal discussions, and poster presentations. The poster session was especially important because many young researchers presented their work.

The topics were very different: internal structures, central series, simplicial sets, groupoids, algebraic coherence, bialgebras, Lawvere conditions, operads, enriched groups, and other subjects.

I like poster sessions because they change the structure of the conference. There is not one speaker and one audience. There are many small discussions at the same time. People move from one poster to another, ask questions, come back later, and sometimes discover that two different problems are connected.

These are often the moments in which new ideas begin.

In the evening we had the conference dinner at La Corte di Bacco. The dinner was also an important part of the conference. People had more time to talk, not only about mathematics, but also about previous conferences, travels, work, and many other things. And the dinner itself, was outstanding!

Friday, 3 July — Talks and discussions

The third day was perhaps the day in which the workshop character of CatAlg was most clear.

Alan Cigoli continued his presentation of the project started many years ago with Sandra Mantovani and Enrico Vitale. For me, it was very satisfying to see how old ideas had developed into new results and new questions.

This is one of the best aspects of mathematics: a project is not only a collection of papers. It is also a long collaboration between people, with discussions, different ideas, changes, and sometimes long pauses.

During the day I was moving continuously between different roles: organiser, listener, colleague, and sometimes person solving practical problems. This means that I could not always follow every talk with full concentration, but at the same time I could see the conference from many different points of view.

By Friday, many mathematical discussions were also taking place outside UniMont: during walks in Edolo, at the bar, during lunch, and in the evening. The mountains were always present. They were not only the background of the conference. They also helped to create a slower and more relaxed atmosphere.

At the same time, we all started to feel that the end of the conference was near. There were still many people to speak with and many questions to discuss.

Saturday, 4 July — The last morning

The last morning of a conference is always a little strange. People arrive with suitcases, check train times, and start saying goodbye.

Enrico Vitale gave the final lecture, on the Grothendieck 2-group. It was a very good way to close the scientific programme, with a topic that connected algebra and category theory, as many talks of the conference had done.

Then came the final thanks.

As chair of the organising committee, I was of course very relieved that everything had gone well. But more than relief, I felt gratitude.

I am grateful to all the invited speakers, to all the participants, and especially to the younger researchers who presented posters. Their questions, talks, and discussions made the conference alive.

I am also very grateful to the staff of UniMont. They helped us in many practical matters and welcomed us with great kindness. Thanks to them, we felt at home from the first day.

I also want to thank the other organisers. CatAlg 2026 was possible because many people worked together for several months.

When the last participants left Edolo, the rooms became quiet again. But the conference did not feel completely finished. Some discussions had only started, some questions were still open, and some possible collaborations had appeared.

For me, this is one of the best possible results of a workshop.

The complete programme, abstracts, slides, list of participants, and photographs are available on the CatAlg 2026 conference website.

Comments and discussion are welcome on Mathstodon.