Unity of Opposites

From my own reading it seems that monad is more common in general but triple does seem to continue to live on in the homological algebra community. Barr and Beck both continue to use it. I found their paper "Acyclic Models and Triples" to be very enlightening - such a simple idea. I am working with a CS advisor who studies connections between category theory and programming languages. Computer scientists have found it useful to use monads to structure sequential composition of programs, so monads come up a lot there - see Moggi's "Notions of Computations and Monads". Really, a lot of the reason I am learning homological algebra is to understand the purpose and meaning of these "canonical resolutions." I am picking up French a little at a time to read the book by Godement, which is excellent. His goal in the book is to cover the homology theory of simplicial sets and simplicial complexes, briefly cover spectral sequences, and cover t...

I insist vehemently that I am a logician. My colleagues tend to believe me less and less each time we speak, because every time they see me, I seem to be carrying around a book on algebraic topology, homological algebra, or differential forms. It is true that I have not spent as much time as I should lately on logic proper. But my interest as a logician led me to topos theory, and the deeper I started to get into topos theory, the more I enjoyed it and wanted to appreciate it on its own standing - not merely as a setting for the semantics of intuitionistic logics, but as a rich branch of mathematics, invented to provide a setting for Weil's hypothesized cohomology theory, and drawing inspiration and motivation from all over algebraic topology and algebraic geometry. Since then I have begun to read broadly into classical geometry, trying to understand the historical context in which topos theory was created, and answer for myself the question "What is cohomology, really?" ...

One method of extending the definitions of homology and cohomology to more general systems of coefficients is with (cohomological) coefficient systems, as they are called in Gelfand and Manin's book Methods of Homological Algebra and Godement's book Topologie Algebrique et Theorie des Faisceau , and in more modern writing, "local systems." Given a simplicial set \(X : \Delta^{op}\to \mathbf{Sets} \) (and here, it is not essential that \(\Delta\) really be the standard simplex category; it seems that this notion would work reasonably well if we replaced it with the category of finite ordinals and strictly increasing maps between them, or, as Godement prefers, a skeleton of the category of finite sets) a cohomological coefficient system is a covariant functor from \(\mathbf{el}(X)\) to Ab (or R-mod ; or your favorite working Abelian category to take coefficients in) where by \(\mathbf{el}(X)\) we mean the category of elements of \(X\) as a presheaf, the result of...

I have begun organizing, this semester, a category theory seminar. So far we have developed the basics of Kan extensions and coends, and started to discuss simplicial sets. It has quickly become clear to me that we could easily spend the rest of the semester on the theory of simplicial sets. Since the coronavirus shut the school's operation down, we have moved online. So here is my first video lecture. More to come soon! The next one on the geometry of simplicial sets and the Eilenberg-Zilber theorem.

I suspect that the value of memorization is unfairly maligned in learning mathematics. It would be wrong to say that mathematics is "only about learning high level concepts, not about learning facts." Quite simply, every field of knowledge involves storing information in the brain. This is indisputable. Therefore, it seems rational to (upon picking up a book) decide whether one is intending to learn the material permanently, or allow it to slip away. The basic gist of my argument is that it costs many many hours to initially learn some difficult concept in mathematics (like how to solve a class of exercises with a new technique) and very little time, speaking in comparison, to occasionally practice the skill every few months or so to ensure that one is keeping it fresh. 1. When you begin learning a new subject, you are quickly overwhelmed by a mass of definitions that you must master quickly in order to make sense of the material. For example: the taxonomy of ri...

I have a lot of notes that I want to get out on the Internet. Hopefully, Google's trawlers eventually find them, and get this page catalogued by the search engine; and if someone happens to have a question about a missing detail in a proof that matches a detail I worked out here, and is able to artfully Google, then the twain shall meet. I welcome constructive criticism of the stylistic layout, as long as you can actually tell me how to improve it. I am a bit clueless with the finer points of LaTeX, but I am always looking to learn. http://www.math.wisc.edu/~nicodemus/Homological_Algebra_Weibel.pdf http://www.math.wisc.edu/~nicodemus/Homological_Algebra_Weibel.tex

Because I just started this blog, I have a lot of writing material that I've been hanging onto. I will probably end up dumping a lot of old stuff here for the time being. The following is a letter I wrote to a friend with some thoughts on monads. It's nice to look back at old writing, as it reminds you that you are making progress : I finally did end up reading that section on monads and homology in Categories for the Working Mathematician, and found it quite rewarding. I have even learned enough French to start working through the (excellent) book by Godement. The notes on monads and the Stone-Cech compactification that I mention in the email can be found here: https://www.math.wisc.edu/~nicodemus/Monads_and_Stone_Cech_Compactification.tex https://www.math.wisc.edu/~nicodemus/Monads_and_Stone_Cech_Compactification.pdf ~~~ Let me just give an update on my life and interests recently. Category theory was born from within homological algebra in an effort to keep track...

The classic textbook by Spanier is infamously dense. On one hand, his insistence on spelling out in the most precise formality the details of proofs can be a benefit for a reader who gets frequently snagged on minor technical notes and needs them spelled out precisely and in detail, without any geometric handwaving. In a book like that of Hatcher, where such handwaving is frequent, it is easy to think one understands a proof or construction, only to realize later, when you need to use some detail of the construction, that you did not understand it at all. Spanier lets the reader have no such illusions to comprehension. On the other hand, opening up such a book can be quite intimidating, especially given his tendency to collect a wall of results together and leave their proofs as exercises in the reader, and express these statements in exceeding levels of generality. I have spent some of the past few months studying Chapters 4 and 5 of his book, and taken some notes. In some places, S...