Bartosz Milewski

Another way of looking at this picture is that it's a perspective view of a square rotating around a circle. The area of the projection depens only on the angle of viewing.

Modeling Identity Types Previously: Identity Types. Let me first explain why the naive categorical model of dependent types doesn’t work very well for identity types. The problem is that, in such a model, any arrow …

New post on homotopy type theory.
bartoszmilewski.com/2025/10/18/m...

Identity Types Previously: Models of (Dependent) Type Theory. There is a deep connection between mathematics and programming. Computer programs deal with such mathematical objects as numbers, vectors, monoids, fu…

A new post in the Homotopy Type Theory series.
bartoszmilewski.com/2025/09/22/i...

Models of (Dependent) Type Theory Previously: (Weak) Factorization Systems. It’s been known since Lambek that typed lambda calculus can be modeled in a cartesian closed category, CCC. Cartesian means that you can form product…

New blog post in the series on HoTT.

bartoszmilewski.com/2025/09/05/m...

It looks better when it's rotated 180 degrees, so it's lighted from above.

I bet they are marketing it to dyslexic parents

Did the fish survive?

(Weak) Factorization Systems Previously (Weak) Homotopy Equivalences. An average function between sets, $latex f \colon A \to B$ is neither surjective nor injective. We can however isolate the two “failure modes” i…

Another homotopy blog post.
bartoszmilewski.com/2025/07/26/w...

(Weak) Homotopy Equivalences Previously: Fibrations and Cofibrations. In topology, we say that two shapes are the same if there is a homeomorphism– an invertible continuous map– between them. Continuity means that …

New blog post on homotopy equivalences. Working my way towards HoTT.
bartoszmilewski.com/2025/06/20/w...

Fibrations and Cofibrations We are used to thinking of a mapping as either being invertible or not. It’s a yes or no question. A mapping between sets is invertible if it’s both injective and surjective. It means t…

New blog post about homotopies, fibers and cofibers.
bartoszmilewski.com/2025/05/30/f...

In a very Putinesque turn of events, Trump's press secretary Leavitt waved a kompromat file proving that Jeff Bezos partnered with a "Chinese propaganda arm." Soon afterwards Bezos kissed Trump's ass (TM), and the file has been put back in storage. The oligarchs have been tamed.

If I were to teach softwared design, I would use WordPress as the example of the most unusable and buggy UI design.

Subfunctor Classifier Previously: Subobject Classifier. In category theory, objects are devoid of internal structure. We’ve seen however that in certain categories we can define relationships between objects that …

Another post on categorical topology: the construction of the subobject classifier for presheafs.
bartoszmilewski.com/2025/04/21/s...

Working on a new blog post about subobject classifier for presheaves. Decoding just one page out of 600 in Mac Lane Moerdijk.

Mathstodon A Mastodon instance for maths people. We have LaTeX rendering in the web interface!

I'm unable to connect to mathstodon.xyz . PR_CONNECT_RESET_ERROR
Is this a known problem? I'm in hotel in Reims, France, and all other connections work just fine.

Not really. There are some grisly scenes of torture and death.

I thought he was just repeating the lines with no emotional inflection. It makes sense in a comedy, when you're saying something funny with dead seriousness. But in a drama?

I got tricked into watching it after watching the Tom Cruise movie, which wasn't half bad. I think an AI would do a much better job with both the script and the performance.

Embalming fluid?

Pete Hegseth: I'll drink to that 🍸🍸🍸🍸🍸

YouTube video by Veritasium The Bayesian Trap

If your prior belief is already 100% (absolute certainty), then according to Bayes' theorem, no amount of new evidence can change your belief.
www.youtube.com/watch?v=R13B...

Subobject Classifier Proviously Sieves and Sheaves. We have seen how topology can be defined by working with sets of continuous functions over coverages. Categorically speaking, a coverage is a special case of a sieve,…

New post
bartoszmilewski.com/2025/03/20/s...

School shootings 2009-2018. First US 288, next Mexico 8.

To prove that you're wrong, I would need just one counterexample: a guy who died playing Russian roulette. To counter statistical argument you need overwhelming staticstics, like this:

I assume that you're not trolling, and you genuinely believe that you're making a valid argument. Many people miss the difference between a counterexample and anectodal evidence.

If you say that Russian roulette is perfectly safe because you know a guy who survived one round, that's anecdotal.

Seriously? You're bringing kindergarten-level arguments to a discussion about children being killed?

The sea is sometimes very slippery when wet

Fixed!

Understanding Attention in LLMs There are many excellent AI papers and tutorials that explain the attention pattern in Large Language Models. But this essentially simple pattern is often obscured by implementation details and opt…

New blog post.
bartoszmilewski.com/2025/03/06/u...

You might be right. Maybe we should try something more effective.