Previously (Weak) Homotopy Equivalences. An average function between sets, is neither surjective nor injective. We can however isolate the two “failure modes” if we insert a third set in between. We can, for instance, pick this set to be the subset of that is the image of under . We then define to be the […]| Bartosz Milewski's Programming Cafe
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
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
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
The Fredholm index, its robustness, and how it relates to Euler characteristic| John D. Cook
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
A paper from last week with high press visibility that makes claims to climate1 applicability is titled: Topology shapes dynamics of higher-order networks The higher-order Topological Kuramoto dyna…| GeoEnergy Math
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: Sheaves and Topology. In our quest to rewrite topology using the language of category theory we introduced the category of open sets with set inclusions as morphisms. But when we needed…| Bartosz Milewski's Programming Cafe
Here’s a fun thing: if you want to generate a random finite space, instead select a random subset from , the -fold power of the Sierpinski space , since every space embeds into some (arbitrary) product of copies of the Sierpinski space. (Recall that has underlying set , and the only open subsets are . … Continue reading Small fun observation about finite topological spaces, and some challenges| theHigherGeometer
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
What is Topology? When talking about topology, people draw cups with handles turning into donuts. When I think of topology, I see nutritious food. In mathematics, topology is defined as a family of…| Bartosz Milewski's Programming Cafe
By introducing some notation on the image and pre-image function, one could derive more elegant definitions of continuity, subspace topology and quotient topology. This notation is not new.| Toby Lam’s Blog
Recently, there’s been a great deal of excitement and interest in deep neural networks because they’ve achieved breakthrough results in areas such as computer vision. However, there remain a number…| Christopher Olah's Blog
I’ve been talking about writing a topology textbook introductory notes on topology for years. Basically since I wrote my Rethinking Topology (or a Personal Topologodicy) post 2 years ago R…| Christopher Olah's Blog
Application deadline 3 May. Quoting from the ALG-TOP mailing list: Dear Topologists There is a vacant position as postdoctoral fellow for 2 years from September 2013 within the project “Topology in…| Motivic stuff
Having been through the process of finding and applying for postdoc positions in Europe over the last couple of years, here is now the big list that I wished someone had given me a few years ago.| Motivic stuff