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Bartosz Milewski's Programming Cafe | Category Theory, Haskell, Concurrency, C++

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In all branches of science we sooner or later encounter the global vs. local duality. Topology is no different. In topology we have the global definition of continuity: counter-images of all open sets are open. But we perceive a discontinuity as a local jump. How are the two pictures related, and can we express this topologically, that is without talking about sizes and distances? All we have at our disposal are open sets, so exactly what properties of open sets are the most relevant? They do form a (thin) category with inclusions as arrows, but so does any set of subsets. As it turns out open sets can be stitched together to create coverings. Such coverings let us zoom in on finer and finer details, thus creating the bridge between the global and the local picture. Open sets are plump–they can easily fill the bulk of space. They are also skinless, so they can’t touch each other without some overlap. That makes them perfect for constructing covers. Covering, unlike tiling, requires ove

Bartosz Milewski's Programming Cafe | Category Theory, Haskell, Concurrency, C++ Home About Bartosz Milewski's Programming Cafe Category Theory, Haskell, Concurrency, C++ June 13, 2026 Kan Extensions in Double Categories Posted by Bartosz Milewski under Category Theory , Double category , Haskell , Kan extensions , Profunctor Equipment | Tags: Category Theory , Double Category , Profunctors | Leave a Comment Previously: Kan extensions in Haskell . In a double category that is also a proarrow equipment, we have the ability to bend arrows. In particular, in the definition of the counit

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