A small blog about Rust, type theory and mathematics.

I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology: A Categorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MIT Press and will be released on August 18, 2020. But you can pre-order on Amazon now!

Slides for the course on category theory. Contribute to TallCats/CategoryTheory development by creating an account on GitHub.

An adjunction is a way to relate two objects a and c, but not directly: C(a,c) Instead, we get two maps, (L)eft and (R)ight, which allow to relate them: C(L(a),c) A(a,R(c)) 1/n The means that there's a pair of arrows, one in each direction, that these two hom-sets are isomorphic. The first example given is currying: a function in two arguments is equivalent to a function of one argument returning another function in the remaining argument. 2/n We can write this like so: C(a✕b,c) C(a,c^b) where L = _✕b, and R = _^b.

These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific...

adding equalizers and coequalizers to a cartesian closed category amounts to enriching the corresponding type system with refinement types and quotient types respectively.

In my previous blog post, Programming with Universal Constructions, I mentioned in passing that one-to-one mappings between sets of morphisms are often a manifestation of adjunctions between functo…

I love functional programming, […] and developing intuition behind complex topics. Thus G [some group] is a category with one object, in which every arrow is an isomorphism.

relaxing a notion of “sameness” gives us extra currency with which to explore new phenomena

Keynote by Dr. Emily Riehl C◦mp◦se :: Conference http://www.composeconference.org/ May 18, 2017 Slides: http://www.math.jhu.edu/~eriehl/compose.pdf Monads have famously been used to model computational effects, although, curiously, the computer science literature presents them in a form that is scarcely recognizable to a category theorist — I’d say instead that a monad is just a monoid in the category of endofunctors, what’s the problem? ;) To a categorical eye, computational effects are modeled using the Kleisli category of a monad, a perspective which suggests another categorical tool that might be used to reason about computation. The Kleisli category is closely related to another device for categorical universal algebra called a Lawvere theory, which may be a more natural framework to model computation (an idea suggested by Gibbons, Hinze, Hyland, Plotkin, Power and certainly others). This talk will survey monads, Lawvere theories, and the relationships between them and illustrate the advantages and disadvantages of each framework through a variety of examples: lists, exceptions, side effects, input-output, probabilistic non-determinism, and continuations.

Contribute to cohomolo-gy/cats-in-context development by creating an account on GitHub.

Slides for Scalaworld 2019. Contribute to cohomolo-gy/Type-Arithmetic-and-the-Yoneda-Perspective development by creating an account on GitHub.

With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor ...

Welcome back to our mini-series on categorical limits and colimits! In Part 1 we gave an intuitive answer to the question, "What are limits and colimits?" As we saw then, there are two main ways that mathematicians construct new objects from a collection of given objects: 1) take a "sub-collection," contingent on some condition or 2) "glue" things together. The first construction is usually a limit, the second is usually a colimit. Of course, this might've left the reader wondering, "Okay... but what are we taking the (co)limit of ?" The answer? A diagram. And as we saw a couple of weeks ago, a diagram is really a functor.

I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere--in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

This post made the concept of “indexing,” used across mathematics, click for me.

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like, "For every [bla bla] there existsa [yadda yadda] such thatthe following diagram commutes." and perhaps it left you wondering what it all meant.

an often fruitful way to discover properties of an object is not to investigate the object itself, but rather to study the collection of maps to or from the object.

Unlike monads, which came into programming straight from category theory, applicative functors have their origins in programming. McBride and Paterson introduced applicative functors as a programmi…

In July 2019, ICMS hosted a workshop on Category Theory.  During the workshop, Eugenia Cheng (School of the Art Institute of Chicago) gave a public lecture entitled Inclusion-Exclusion in mathematics and beyond: who stays in, who falls out, why it happens and what we could do about it. This is a recording of that talkThis talk has captions.  To turn the captions off, press CC on the bottom toolbar.

once we see the formal definition below, it will become clear that mathematical (say, first-order logical) statements, together with proofs of implication, form a category. Even though a “proof” isn’t strictly a structure-preserving map, it still fits with the roughly stated axioms above. One can compose proofs by laying the implications out one after another, this composition is trivially associative, and there is an identity proof. Thus, proofs provide a way to “transform” true statements into true statements, preserving the structure of boolean-valued truth. The section on diagram categeories was fantatsic.

Moreover, a universal property jumps right to the heart of why a construction is important. ​ I want to make this point very clear, because most newcomers to category theory are never told this. Category theory exists because it fills a need. Even if that need is a need for better organization and a refocusing of existing definitions. ​ l One hopes, then, that very general theorems proved within category theory can apply to a wide breadth of practical areas. ​ Could it be that there is some (non-categorical) theorem that can’t be proved unless you resort to category-theoretical arguments? In my optimistic mind the answer must certainly be no. Moreover, it appears that most proofs that “rely” on category theory only really do so because they’re so deeply embedded in the abstraction that unraveling them to find non-category-theoretical proofs would be a tiresome and fruitless process.

Introduction Terminal and Initial objects Terminal and initial objects 1 Terminal and initial objects 2 Terminal and initial objects 3 Products and Coproducts Products and coproducts 1 Products and…

CAUTION: Monoid fever is contagious.

Grothendieck views the mathematician and the problem as complimenting each other, the mathematician using the problem’s natural structure in its solution, rather than striking it with a foreign, invasive method. My view is that any problem that has resisted repeated direct attack from problem solvers, should naturally be of interest to theory builders. If you can’t solve a problem directly, then grow a crystal of theory around the problem and then hope that the solution you are looking for can be located somewhere inside the crystal. This is hard because the same person needs to know about both category theory and the problem domain, which is quite a heavy demand on a human brain.

I think we should levy a fee for category theorists to use the word “just”