📖 Read pages 21-24 of Abstract and Concrete Categories: The Joy of Cats by Jirí Adámek, Horst Herrlich, George E. Strecker
📖 Read pages i-20 the front matter and Introduction of Abstract and Concrete Categories: The Joy of Cats by Jirí Adámek, Horst Herrlich, George E. Strecker
Some initial discussion of sets, classes, and conglomerates to keep us out of trouble with some of the potential foundational issues that can be found in set theory.
Highlights, Quotes, Annotations, & Marginalia
completions of partially orderedsets and of metric spaces,ˇCech-Stone compactifications of topological spaces, sym-metrizations of relations, abelianizations of groups, Bohr compactifications of topo-logical groups, minimalizations of reachable acceptors, etc. ❧
The tough part of category theory is lists of things like this right up front which will tend to scare off almost any reader but those who are working on Ph.D.s in mathematics…
November 30, 2018 at 09:29PM
I really wish more math textbooks had motivation sections like this one does.
November 30, 2018 at 09:35PM
Therefore we advise the beginner to skip from here, go directly to§3, and return to this section only when the need arises. ❧
They’ve buried the lede here apparently.
November 30, 2018 at 10:15PM
Kuratowski definition of an ordered pair ❧
I don’t think I’ve ever seen the specific name Kuratowski attached to this.
November 30, 2018 at 10:31PM
📗 Started reading Abstract and Concrete Categories: The Joy of Cats by Jirí Adámek, Horst Herrlich, George E. StreckerSyndicated copies to:
This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases.
A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
Mike Miller’s at UCLA beginning in January 2019.Syndicated copies to:
Wikipedia enforces its entries to adopt an NPOV – a neutral point of view . This is appropriate for an encyclopedia.
However, the nLab is not Wikipedia, nor is it an encyclopedia, although it does aspire to provide a useful reference in many areas (among its other purposes). In particular, the nLab has a particular point of view, which we may call the nPOV or the n- categorical point of view .
To some extent the nPOV is just the observation that category theory and higher category theory, hence in particular of homotopy theory, have a plethora of useful applications.
This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases. A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.
Mike Miller has announced in class that he’ll be using Abstract and Concrete Categories: The Joy of Cats as the textbook for hisat UCLA Extension this winter.
Naturally, he’ll be supplementing it heavily with his own notes.
A free .pdf copy of the text is also available online.Syndicated copies to:
Here are the notes from a basic course on category theory. Unlike the Fall 2015 seminar, this tries to be a systematic introduction to the subject. A good followup to this course is my Fall 2018 course. If you discover any errors in the notes please email me, and I'll add them to the list of errors. You can get all 10 weeks of notes in a single file here: here. Their typeset version was based on these handwritten versions:
I’m teaching a course on category theory at U.C. Riverside, and since my website is still suffering from reduced functionality I’ll put the course notes here for now. I taught an introductory course on category theory in 2016, but this one is a bit more advanced. The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.
This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.
Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.
January 8, 2019 - March 19, 2019
Tuesday 7:00PM - 10:00PM
Instructor: Michael Miller
The new catalog is out today and Mike Miller’s Winter class in Category Theory has been officially announced.
Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.Syndicated copies to:
Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.
For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)
We’ve covered the following topics:
- Class Introduction
- Erdős Discrepancy Problem
- Hilbert’s Cube Lemma (1892)
- Schur (1916)
- Van der Waerden (1927)
- Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
- Ramsey Theory
- Erdős (1943)
- Gallai (1944)
- Steinberg’s alternate (1944)
- DeBruijn and Erdős (1948)
- Motzkin (1951)
- Dirac (1951)
- Kelly & Moser (1958)
- Tao-Green Proof
- Homework 1 (homeworks are generally not graded)
Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.
I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.
Category Theory for Winter 2019
As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!Syndicated copies to:
As I get amped up for the start of Mike Miller’s Fall math class Gems and Astonishments of Mathematics, which is still open for registration, I’m even more excited that he’s emailed me to say that he’ll be teaching Category Theory for the Winter Quarter in 2019!!
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"Category theory is a universal modeling language."
Success is founded on information. A tight connection between success (in anything) and information. It follows that we should (if we want to be more successful) study what information is.
Grant proposals. These are several grant proposals, some funded, some in the pipeline, others not funded, that explain various facets of my research project.
This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field. [PDF]
Friends! I am so happy to share that my little booklet “What is Applied Category Theory?” is now available on the arXiv. It’s a collection of introductory, expository notes inspired by the ACT workshop that took place earlier this year. Enjoy! https://t.co/EPYP19z14x pic.twitter.com/O4uVhj401s
— Tai-Danae Bradley (@math3ma) September 18, 2018
See also Notes on Applied Category TheorySyndicated copies to:
This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory. The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems, which are mixtures of continuous dynamics---e.g. as described by a differential equation---and discrete jumps. In particular, the derivative of a continuous real-valued function is internally defined. We construct a semantics for the temporal type theory in the topos of sheaves on a translation-invariant quotient of the standard interval domain. In fact, domain theory plays a recurring role in both the semantics and the type theory.
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This morning at ACT2018, David Spivak gave a VERY cool talk on using topos theory to model how airplanes can maintain a safe distance from each other in flight. You can watch the talk here! https://t.co/pUXZhj6SXA Also check out “Temporal Type Theory” at https://t.co/6LWNOQWtqw pic.twitter.com/7x9yjBwVIA
— Tai-Danae Bradley (@math3ma) May 2, 2018