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 …| Bartosz Milewski's Programming Cafe
This is the second part in a series about diagrammatic reasoning, inspired by e-graphs. Last time, we reviewed the concept of initial functor and showed by example how to calculate with diagrams and initial functors. This time, we make that calculus more systematic and we reconceive e-graphs in terms of initial functors. 1 Weak equivalence of diagrams We’ve been deriving equations by chaining together initial functors between diagrams, going in either direction. Let’s give a name to this ...| Topos Institute
An e-graph, short for “equality graph,” is a data structure that maintains a congruence relation on expression trees: an equivalence relation stable under forming new expressions. First devised by Nelson and Oppen in 1980 (Nelson 1980; Nelson and Oppen 1980), e-graphs received a surge of new attention when Willsey et al demonstrated, via their software package egg, that e-graphs combined with equality saturation can be a fast, powerful, and adaptable tool for equational reasoning (Willsey...| Topos Institute
At last week’s Topos Colloquium, Rory Lucyshyn-Wright told us about categories graded by a monoidal category, following his recent preprint (Lucyshyn-Wright 2025). Graded categories, short for locally graded categories, were first introduced by Richard Wood under a different name (Wood 1976, 1978). Graded categories are of mathematical interest because they simultaneously generalize actions of a monoidal category (“actegories”) and, via a Yoneda-type embedding, enriched categories, whil...| Topos Institute
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…| Bartosz Milewski's Programming Cafe
What really is the color GREEN? What does it mean to different people? We actually have no idea. Now, consider The Dress on its 10 year anniversary. One can either argue with others whether it is BLUE and BLACK or WHITE and GOLD, or objectively look at it’s composition from a technical standpoint. Below is … Continue reading Subjectivity and Perception| GeoEnergy Math
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 …| Bartosz Milewski's Programming Cafe
Kernel and cokernel are dual concepts, but the former is far more likely to appear in textbooks. The cokernel is often in the background without a name.| John D. Cook
A personal blog about functional programming, category theory, chess, physics and linux topics| beuke.org
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,…| Bartosz Milewski's Programming Cafe
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…| Bartosz Milewski's Programming Cafe
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 be drawn as a virtual arrow between two virtual objects corresponding to two presheaves and . Indeed, for every , seen as an arrow , we get an […]| Bartosz Milewski's Programming Cafe
Previously: Sheaves as Virtual Objects. In order to define a sheaf, we have to start with coverage. A coverage defines, for every object , a family of covers that satisfy the sub-coverage conditions. Granted, we can express coverage using objects and arrows, but it would be much nicer if we could use the language of […]| Bartosz Milewski's Programming Cafe
Previously: Coverages and Sites The definition of a sheaf is rather complex and involves several layers of abstraction. To help us navigate this maze we can use some useful intuitions. One such intuition is to view objects in our category as some kind of sets (in particular, open sets, when we talk about topology), and […]| Bartosz Milewski's Programming Cafe
Previously: Presheaves and Topology. 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 con…| Bartosz Milewski's Programming Cafe
Previously: Topology as a Dietary Choice. Category theory lets us change the focus from individual objects to relationships between them. Since topology is defined using open sets, we’d start…| Bartosz Milewski's Programming Cafe
About two years ago I wrote about a category-theoretic treatment of collaborative text editing. That post is unique in the history of Bosker Blog in having been cited – twice so far that I kno…| Bosker Blog
A personal blog about functional programming, category theory, chess, physics and linux topics| beuke.org
A personal blog about functional programming, category theory, chess, physics and linux topics| beuke.org
A personal blog about functional programming, category theory, chess, physics and linux topics| beuke.org
