# mathematics

## ACT2020 conference notes: Main sessions

In the previous four posts I organized my notes from the ACT2020 Tutorial Day. In this post I’ll continue to note down things I’ve learned from the main sess...

## ACT2020 conference notes: Tutorial Day Lecture 1 (David Spivak)

So I’m participating in this year’s Applied Category Theory conference (with a virtual poster—check it out here!), and the preconference Tutorial Day has jus...

## A new application of category theory in linguistics (part 2)

In the previous post I laid out the disciplinary background of my application of category theory in linguistics in more detail (than I had done in my explain...

## A new application of category theory in linguistics (part 1)

This is the second part of my “portfolio” prepared for the virtual poster session at ACT2020. It introduces my category-theoretic modeling of the human langu...

## Category theory notes 20: Category theory for posets (Part 2)

In my previous post “Category theory notes 19: Category theory for posets (Part 1)” I began writing about the “poset versions” of some category-theoretic res...

## Category theory notes 19: Category theory for posets (Part 1)

Throughout this blog series I’ve been writing about category-theoretic results in their fully general forms, which are applicable to all categories from all ...

## Category theory notes 18: Reflective subcategory (Part 2)

In my previous post “Category theory notes 17: Reflective subcategory (Part 1)” I discussed the similarity and distinction between linguistic and mathematica...

## Category theory notes 17: Reflective subcategory (Part 1)

So far in this series I’ve viewed a category as an individual and independent entity. Two categories may be related by functors or even better connected by a...

## Category theory notes 16: Yoneda lemma (Part 3)

In the previous two posts, “Category theory notes 14: Yoneda lemma (Part 1)” and “Category theory notes 15: Yoneda lemma (Part 2),” I started a task of decip...

## Category theory notes 15: Yoneda lemma (Part 2)

In the previous post “Category theory notes 14: Yoneda lemma (Part 1)” I began writing about IMHO the most challenging part in basic category theory, the Yon...

## Category theory notes 14: Yoneda lemma (Part 1)

Awodey calls it “the single most used result” of category theory (Category Theory, p. 185), Crole regards it as “an indispensable tool which every category t...

## Category theory notes 13: Adjunction (Part 2)

In the previous post “Category theory notes 12: Adjunction (Part 1)” I wrote about my thoughts on adjunction, an extremely important component of category th...

## Category theory notes 12: Adjunction (Part 1)

So, after reading many textbooks, watching many video tutorials, and attending two guided courses, I finally understood adjunction. That was a real “Eureka!”...

## Category theory notes 11: Composite naturality (Part 2)

In my previous post “Category theory notes 10: Composite naturality (Part 1)” I illustrated the vertical and horizontal compositions of natural transformatio...

## Category theory notes 10: Composite naturality (Part 1)

The notion of natural transformation is surprisingly easy to follow. If you know what an arrow is and what a functor is, then you automatically know what a n...

## Category theory notes 9: Full and half inverses

Category theory textbooks often warn learners that categorical objects and arrows shouldn’t be tied to sets and functions. A poset category, for example, has...

## Category theory notes 8: Functoriality

Fong & Spivak refer to category, functor, and natural transformation as the “big three” of category theory in their newly published textbook An Invitatio...

## Category theory notes 7: Categorical idioms

Idioms and slangs are an important part of human language. They are short, expressive, and vividly reflect regional/historical mind-sets. And they are usuall...

## Category theory notes 6: Think big

Category theory is spectacularly big. But exactly how big is it? Consider a set $A$. It can hold a huge number of elements, say, all grains of sand on Earth....

## Category theory notes 5: Arrows and diagrams

Arrows are so vital to category theory that Awodey jokingly refers to the theory as “archery” (Category Theory, p. 2). Given two objects in a category, an ar...

## Category theory notes 4: Monoid

Monoid is one of those concepts that are extremely simple, extremely useful, and can at the same time be extremely confusing. It was one of the first concept...

## Category theory notes 3: Categorial or categorical?

Category in category theory is a noun, but we often need to use the term in other parts of speech (notably adjective and verb). Try the following quiz for ex...