Bartosz Milewski
@bartoszmilewski.bsky.social
Physicist, mathematician, programmer.
"The Dao of Functional Programming"
Regularly updated work in progress. PDF on GitHub:
https://github.com/BartoszMilewski/DaoFP/blob/master/DaoFP.pdf
"The Dao of Functional Programming"
Regularly updated work in progress. PDF on GitHub:
https://github.com/BartoszMilewski/DaoFP/blob/master/DaoFP.pdf
New post on homotopy type theory.
bartoszmilewski.com/2025/10/18/m...
bartoszmilewski.com/2025/10/18/m...
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 …
bartoszmilewski.com
October 18, 2025 at 8:30 AM
New post on homotopy type theory.
bartoszmilewski.com/2025/10/18/m...
bartoszmilewski.com/2025/10/18/m...
A new post in the Homotopy Type Theory series.
bartoszmilewski.com/2025/09/22/i...
bartoszmilewski.com/2025/09/22/i...
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…
bartoszmilewski.com
September 22, 2025 at 1:13 PM
A new post in the Homotopy Type Theory series.
bartoszmilewski.com/2025/09/22/i...
bartoszmilewski.com/2025/09/22/i...
Another homotopy blog post.
bartoszmilewski.com/2025/07/26/w...
bartoszmilewski.com/2025/07/26/w...
(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…
bartoszmilewski.com
July 26, 2025 at 12:54 PM
Another homotopy blog post.
bartoszmilewski.com/2025/07/26/w...
bartoszmilewski.com/2025/07/26/w...
New blog post on homotopy equivalences. Working my way towards HoTT.
bartoszmilewski.com/2025/06/20/w...
bartoszmilewski.com/2025/06/20/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 …
bartoszmilewski.com
June 20, 2025 at 9:48 AM
New blog post on homotopy equivalences. Working my way towards HoTT.
bartoszmilewski.com/2025/06/20/w...
bartoszmilewski.com/2025/06/20/w...
New blog post about homotopies, fibers and cofibers.
bartoszmilewski.com/2025/05/30/f...
bartoszmilewski.com/2025/05/30/f...
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…
bartoszmilewski.com
May 30, 2025 at 9:04 AM
New blog post about homotopies, fibers and cofibers.
bartoszmilewski.com/2025/05/30/f...
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.
April 29, 2025 at 8:37 PM
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.
April 23, 2025 at 9:28 AM
If I were to teach softwared design, I would use WordPress as the example of the most unusable and buggy UI design.
Another post on categorical topology: the construction of the subobject classifier for presheafs.
bartoszmilewski.com/2025/04/21/s...
bartoszmilewski.com/2025/04/21/s...
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 …
bartoszmilewski.com
April 21, 2025 at 11:36 AM
Another post on categorical topology: the construction of the subobject classifier for presheafs.
bartoszmilewski.com/2025/04/21/s...
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.
April 17, 2025 at 3:35 PM
Working on a new blog post about subobject classifier for presheaves. Decoding just one page out of 600 in Mac Lane Moerdijk.
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.
Is this a known problem? I'm in hotel in Reims, France, and all other connections work just fine.
Mathstodon
A Mastodon instance for maths people. We have LaTeX rendering in the web interface!
mathstodon.xyz
March 29, 2025 at 9:24 AM
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.
Is this a known problem? I'm in hotel in Reims, France, and all other connections work just fine.
New blog post.
bartoszmilewski.com/2025/03/06/u...
bartoszmilewski.com/2025/03/06/u...
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…
bartoszmilewski.com
March 6, 2025 at 7:03 PM
New blog post.
bartoszmilewski.com/2025/03/06/u...
bartoszmilewski.com/2025/03/06/u...
I had a nice dinner in Nice. Creative dishes, including a bone-marrow Madelaine for dessert (not bad, considering). But I just don't dig this new fad of oxidized wines (vins jaunes). France has the best wines in the world, and they keep pushing this abomination. I like sauerkraut, but not as wine.
January 26, 2025 at 7:40 PM
I had a nice dinner in Nice. Creative dishes, including a bone-marrow Madelaine for dessert (not bad, considering). But I just don't dig this new fad of oxidized wines (vins jaunes). France has the best wines in the world, and they keep pushing this abomination. I like sauerkraut, but not as wine.
I had no idea one could use math notation and diagrams in Haskell code documentation.
hackage.haskell.org/package/mono...
hackage.haskell.org/package/mono...
Data.Functor.Monoidal
hackage.haskell.org
January 25, 2025 at 1:43 PM
I had no idea one could use math notation and diagrams in Haskell code documentation.
hackage.haskell.org/package/mono...
hackage.haskell.org/package/mono...
I decided to rearrange the Dao of FP by moving the chapter about Applicatives ahead of Monads. This forced me to separate the discussion of effects into a separate chapter. This goes against the usual approach in category theory, where monads are everything, and applicatives are barely mentioned.
January 24, 2025 at 9:10 AM
I decided to rearrange the Dao of FP by moving the chapter about Applicatives ahead of Monads. This forced me to separate the discussion of effects into a separate chapter. This goes against the usual approach in category theory, where monads are everything, and applicatives are barely mentioned.
Oh great! Now everybody knows how exciting and dangerous the work of a mathematician is. Our cover is blown. Thank you Apple TV!
January 23, 2025 at 10:46 PM
Oh great! Now everybody knows how exciting and dangerous the work of a mathematician is. Our cover is blown. Thank you Apple TV!
New blog post about Game of Life, Advent of Code, and comonad composition. bartoszmilewski.com/2025/01/04/l...
Legalizing Comonad Composition
The yearly Advent of Code is always a source of interesting coding challenges. You can often solve them the easy way, or spend days trying to solve them “the right way.” I personally pr…
bartoszmilewski.com
January 4, 2025 at 11:28 AM
New blog post about Game of Life, Advent of Code, and comonad composition. bartoszmilewski.com/2025/01/04/l...
Populations of Siberia and Saudi Arabia are almost equal at 37 mln. Siberia produces 11 mln barrels of oil per day, Saudi Arabia, 12 mln. Yet people in Siberia are dirt poor! Somebody, please, explain it to them.
December 23, 2024 at 9:23 PM
Populations of Siberia and Saudi Arabia are almost equal at 37 mln. Siberia produces 11 mln barrels of oil per day, Saudi Arabia, 12 mln. Yet people in Siberia are dirt poor! Somebody, please, explain it to them.
I don't get the vinyl snobism. I grew up at times when vinyl records and tape were the only media available, and I hated the hissing and the scratching. I dreamed of a more solid and lasting medium, so I welcomed wholeheartedly the arrival of CDs.
Wax cylinders, on the other hand, are really cool.
Wax cylinders, on the other hand, are really cool.
December 20, 2024 at 9:53 PM
I don't get the vinyl snobism. I grew up at times when vinyl records and tape were the only media available, and I hated the hissing and the scratching. I dreamed of a more solid and lasting medium, so I welcomed wholeheartedly the arrival of CDs.
Wax cylinders, on the other hand, are really cool.
Wax cylinders, on the other hand, are really cool.
Lately I've been thinking about relativistic ray tracing. Regular ray tracing assumes infinite lightspeed, so it doesn't matter if the light comes from the source to your eye or from your eye to the source. But in relativistic physics, you have to trace the ray backward in both space and time...
December 9, 2024 at 4:54 PM
Lately I've been thinking about relativistic ray tracing. Regular ray tracing assumes infinite lightspeed, so it doesn't matter if the light comes from the source to your eye or from your eye to the source. But in relativistic physics, you have to trace the ray backward in both space and time...
I have a solution to the Fermi's paradox. Every sufficiently advanced civilization solves the problem of interstellar transport and communication by harnessing the space-shrinking technology. The unfortunate side effect is that every time they shrink some space in one place, it expands in another...
December 9, 2024 at 11:41 AM
I have a solution to the Fermi's paradox. Every sufficiently advanced civilization solves the problem of interstellar transport and communication by harnessing the space-shrinking technology. The unfortunate side effect is that every time they shrink some space in one place, it expands in another...
Lies, damned lies, and AIs
November 17, 2024 at 9:45 PM
Lies, damned lies, and AIs
Continuing the series on categorical approach to topology.
bartoszmilewski.com/2024/11/16/s...
bartoszmilewski.com/2024/11/16/s...
Sieves and Sheaves
Previously: Covering Sieves. We’ve seen an intuitive description of presheaves as virtual objects. We can use the same trick to visualize natural transformations. A natural transformation can…
bartoszmilewski.com
November 16, 2024 at 5:08 PM
Continuing the series on categorical approach to topology.
bartoszmilewski.com/2024/11/16/s...
bartoszmilewski.com/2024/11/16/s...
Season 2 of Silo has just begun (I've read the books).
November 15, 2024 at 10:51 PM
Season 2 of Silo has just begun (I've read the books).