Citation: Bakalov, Bojko, and Alexander A. Kirillov. Lectures on tensor categories and modular functors. Vol. 21. American Mathematical Soc., 2001.
Web: https://www.math.stonybrook.edu/~kirillov/tensor/tensor.html
Tags: Expository, Mathematical, Monoidal-categories, Modular-tensor-categories, TQFT, Quantum-groups
This is the go-to "Modular Tensor Categories" book. I don't find it to be particularly pedagogical, but it covers most of the theory and does quite a good job. Note that it "covers most of the theory" to the extent that there was a theory to cover in 2000 - there's been a lot of progress since then. Nowadays there are lots of things that are out of date about this book.