Most of my research focuses on category theory and categorical logic. This webpage gathers a bit of material and is designed for bachelor and master students that want to have a first encounter with these topics.
The biggest advantage of this webpage with respect to a much more ambitious project like the nlab is its incompleteness. Especially for a very young student, a lot of information is very often the same of too much information. In this sense, the point of view that I try to offer is both a bias and an opportunity.
The sections above are fairly independent with the exception of the one on Categorical Logic, which should not be browsed without a basic knowledge of Category Theory. The meaning of the word basic is clarified at the beginning of the CT section.
Sometimes links might be broken, please contact me if they stop working. Also, this page will never find its final version, and will be a never ending work in progress. If you like any of these topics, and you want to get in touch with me, just drop me a line!
There exists a plethora of introductions to category theory. Some of them are very famous both for the clarity of their exposition and for their historical importance. Among those one should definitely list the evergreen Categories for the working mathematician by Saunders MacLane.
Yet I do not consider that book to be the best introduction to the subject for a bachelor student. Instead, the following two references cover the basics of category theory, offering incredible clarity and an enormous amount of examples.
I suggest these two books with the same motto that characterize this webpage, they go directly to the point, covering only the essential and giving the student a good foundation on which to build future knowledge.
Once the basic material above is well understood, one can enter in the zoo of category theory at work and meet its most different contextualizations. The following two books offer a very classical perspective on several incarnations of category theory.
From the point of view of the most recent trends in category theory, both the two books will seem a bit old fashioned, and indeed they are. This does not make them less interesting, or relevant. Also, starting from the most recent trends, ignoring completely the more established ones, does not come without risks.
Notice that I am neglecting on purpose Homotopy theory because it is a very special flavor of category theory that would not fit with the ambition of generality of this list nor with the special taste of his author. That is not the case of locally presentable and accessible categories: in a sense, those offer a theory of tame categories and play a relevant role in many fields, ranging from homotopy theory (combinatorial model categories) to abstract model theory (abstract elementary classes).
In my personal experience, Category theory has proven to be a seductive and dangerous field, especially for youngsters. Please avoid the Scuttle paradox.
Formal Category Theory
We are all indebted with the Australian school of category theory for the invention of formal category theory. By this term I intend anything related to the topic of enriched categories, 2-categories and bicategories. Unfortunately it is very hard to find good and complete texts on this topic. Luckily, since this page is dedicated to youngsters, I have no ambition of providing a complete reference, and I will try to stay on the basics.
The Australian tradition has various applications, the one that is closest to my heart is the intense study of the properties of the 2-category Cat of locally small categories and functors.
In this fashion, my favorite topic in formal category theory is that of Yoneda structures, that play a relevant role in an abstract treatment of semantic representation of logic. It is almost impossible to provide a concise reference for this topic, I can recommend to start from the following link.
Depending on their background, people tend to identify categorical logic with some of its parts. People from computer science think that categorical logic is a framework that offers a semantic for type theory and λ-calculus, pure mathematicians are mostly aware of topos theory, because of its connection with the rest of mathematics, and finally just not many people are interested in the categorical approach to universal algebra and model theory.
This sad fragmentation of the field is encouraged by the available literature. Most of the books on categorical logic concentrate just on one of the aspects above-mentioned, and a complete source does not exist to my knowledge. While we wait for such a bible to appear, I suggest everyone to start the journey trough categorical logic via the historical book below.
While the book leaves uncovered almost 40 years of beautiful mathematics, it shows quite strongly that categorical logic is a unite field, where the subjects above melt into a unique discourse. Please, read it slowly and carefully, history is important and we should guard it. Once one has some understanding of what categorical logic is about, these are the books that I suggest to start with.
Theoretical Computer Science
Categorical Model Theory
A lot of very good mathematics does not lie in a book yet, this very short subsection lists some very good papers.
Logic and foundational aspects of Type Theory
Defining (mathematical) logic is too ambitious and very far for the general aim of this webpage. Students are generally introduced to the fragment of mathematical logic that is mostly studied in their department. As already stated above, this is both a limitation and an opportunity. Yet a mature researcher, or a student that wants to become a mature researcher, should be able to navigate the map of logic, being aware of its countries and their leaders. In order to restrain (which is very different from avoid) the consequences of the no true Scotsman fallacy, a first approach to this understanding should always be historical. For this purposes I suggest to give a look to the book series below.
It is absolutely impossible to read entirely such an encyclopedia, its value is to relativize the areas of classical logic and to introduce the reader to some topics. Once one has some vague idea of the historical development of logic, it's time to study. The book below is a soft but extremely comprehensive introduction to many fields and subfields of logic.
Finally, let me focus on those fields that are better known to myself, namely: Model theory and Set theory.
Foundational aspects of Type Theory
Philosophy of Language
Unfortunately I am far from being an expert in this field.