h/t to @michael_nielsen via Nuzzel
John Watrous's excellent quantum book just came out. It's still available free on his webpage: https://t.co/D2rr5FTly6
— michael_nielsen (@michael_nielsen) April 28, 2018
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together.
For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.
Tom Leinster has released a digital e-book copy of his textbook Basic Category Theory on arXiv. 
My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:
It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.
Huh? What’s a ‘universal property’?
In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.
A universal property amounts to a precise description of what an object does.
Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:
through representable functors (which are how you actually hand someone a universal property),
through limits (which are ways of building a new object out of a bunch of old ones),
through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).
If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.
Here’s what he wrote, over on the category theory mailing list:
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.
There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.
Instagram filter used: Clarendon
Photo taken at: UCLA Bookstore
I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.
I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.
Peter Woit has just made the final draft (dated 10/25/16) of his new textbook Quantum Theory, Groups and Representations: An Introduction freely available for download from his website. It covers quantum theory with a heavy emphasis on groups and representation theory and
“contains significant amounts of material not well-explained elsewhere.” He expects to finish up the diagrams and publish it next year some time, potentially through Springer.
Advanced Data Analysis from an Elementary Point of View
by Cosma Rohilla Shalizi
This is a draft textbook on data analysis methods, intended for a one-semester course for advance undergraduate students who have already taken classes in probability, mathematical statistics, and linear regression. It began as the lecture notes for 36-402 at Carnegie Mellon University.
By making this draft generally available, I am not promising to provide any assistance or even clarification whatsoever. Comments are, however, welcome.
The book is under contract to Cambridge University Press; it should be turned over to the press before the end of 2015. A copy of the next-to-final version will remain freely accessible here permanently.
Table of contents:
I. Regression and Its Generalizations
- Regression Basics
- The Truth about Linear Regression
- Model Evaluation
- Smoothing in Regression
- The Bootstrap
- Weighting and Variance
- Additive Models
- Testing Regression Specifications
- Logistic Regression
- Generalized Linear Models and Generalized Additive Models
- Classification and Regression Trees
II. Distributions and Latent Structure
- Density Estimation
- Relative Distributions and Smooth Tests of Goodness-of-Fit
- Principal Components Analysis
- Factor Models
- Nonlinear Dimensionality Reduction
- Mixture Models
- Graphical Models
III. Dependent Data
- Time Series
- Spatial and Network Data
- Simulation-Based Inference
IV. Causal Inference
- Graphical Causal Models
- Identifying Causal Effects
- Causal Inference from Experiments
- Estimating Causal Effects
- Discovering Causal StructureAppendices
- Data-Analysis Problem Sets
- Reminders from Linear Algebra
- Big O and Little o Notation
- Taylor Expansions
- Multivariate Distributions
- Algebra with Expectations and Variances
- Propagation of Error, and Standard Errors for Derived Quantities
- chi-squared and the Likelihood Ratio Test
- Proof of the Gauss-Markov Theorem
- Rudimentary Graph Theory
- Information Theory
- Hypothesis Testing
- Writing R Functions
- Random Variable Generation
- Unified treatment of information-theoretic topics (relative entropy / Kullback-Leibler divergence, entropy, mutual information and independence, hypothesis-testing interpretations) in an appendix, with references from chapters on density estimation, on EM, and on independence testing
- More detailed treatment of calibration and calibration-checking (part II)
- Missing data and imputation (part II)
- Move d-separation material from “causal models” chapter to graphical models chapter as no specifically causal content (parts II and IV)?
- Expand treatment of partial identification for causal inference, including partial identification of effects by looking at all data-compatible DAGs (part IV)
- Figure out how to cut at least 50 pages
- Make sure notation is consistent throughout: insist that vectors are always matrices, or use more geometric notation?
- Move simulation to an appendix
- Move variance/weights chapter to right before logistic regression
- Move some appendices online (i.e., after references)?
(Text last updated 30 March 2016; this page last updated 6 November 2015)