# homology Published articles for homology. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## The Čech Complex and the Vietoris-Rips Complex DevFeed: [The Čech Complex and the Vietoris-Rips Complex]() Original publisher: [Read original article]() Published: 2015-08-06T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [data](), [datasets](), [Point cloud](), [Computing]() Tags: [approximation](), [cech-complex](), [complex](), [computational-topology](), [data-mining](), [data-science](), [dataset](), [homology](), [linear-algebra](), [mathematics](), [persistent-homology](), [point](), [points](), [simplicial-complex](), [vietoris-rips-complex]() ### AI overview This article introduces computational topology for analyzing the shape of data. It explains how point clouds can be converted into simplicial complexes so homology and persistent homology can identify qualitative features such as connected components and holes, with some resistance to noise. ### Source excerpt It's about time we got back to computational topology. Previously in this series we endured a lightning tour of the fundamental group and homology, then we saw how to compute the homology of a simplicial complex using linear algebra. What we really want to do is talk about the inherent shape of data. Homology allows us to compute some qualitative features of a given shape, i.e., find and count the number of connected components or a given shape, or the number of "2-dimensional holes" it has. ## Fixing Bugs in "Computing Homology" DevFeed: [Fixing Bugs in "Computing Homology"]() Original publisher: [Read original article]() Published: 2014-01-23T23:05:16Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Mathematics](), [Code](), [function](), [Matrix]() Tags: [bugs](), [code](), [computing](), [function](), [homology](), [linear-algebra](), [mathematics](), [matrices](), [persistent-homology](), [programming](), [python](), [quotients](), [row-reduction](), [test]() ### AI overview This article corrects bugs in code for computing homology. It identifies an indexing error, explains a mathematical mistake in simultaneous row and column reduction, and shows how further row reduction is needed to obtain the correct rank for a triangulation of the Möbius band. ### Source excerpt A few awesome readers have posted comments in Computing Homology to the effect of, "Your code is not quite correct!" And they're right! Despite the almost year since that post's publication, I haven't bothered to test it for more complicated simplicial complexes, or even the basic edge cases! When I posted it the mathematics just felt so solid to me that it had to be right (the irony is rich, I know). ## Functoriality in Category Theory: Mappings That Preserve Morphisms DevFeed: [Functoriality in Category Theory: Mappings That Preserve Morphisms]() Original publisher: [Read original article]() Published: 2013-07-14T10:03:29Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Category Theory](), [Mathematics](), [Math and Logic]() Tags: [categories](), [category-theory](), [coproducts](), [functor](), [homology](), [ml](), [morphisms](), [universal-properties]() ### AI overview This tutorial introduces functoriality in category theory. It explains functors as mappings between categories that assign objects and morphisms while preserving identity morphisms and composition, with homology as an example of a functorial construction. ### Source excerpt Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we've repeatedly stressed, the meat of category theory is in the morphisms. One natural question one might ask is, what notion of morphism is there between categories themselves? ## Computing Homology DevFeed: [Computing Homology]() Original publisher: [Read original article]() Published: 2013-04-10T16:38:25Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Code](), [Algorithm](), [Collections](), [.NET 11 Preview 7]() Tags: [algebraic-topology](), [algorithm](), [code](), [collections](), [computing](), [homology](), [linear-maps](), [matrices](), [matrix](), [ordering](), [row-reduction](), [structure]() ### AI overview This tutorial explains an algorithmic approach to computing homology for a simplicial complex. It represents chain groups and boundary homomorphisms as matrices, uses row reduction, and works through an extended example for computing H₁. ### Source excerpt Update: the mistakes made in the code posted here are fixed and explained in a subsequent post (one minor code bug was fixed here, and a less minor conceptual bug is fixed in the linked post). In our last post in this series on topology, we defined the homology group. Specifically, we built up a topological space as a simplicial complex (a mess of triangles glued together), we defined an algebraic way to represent collections of simplices called chains as vectors in a vector space, we defined the boundary homomorphism $ \partial_k$ as a linear map on chains, and finally defined the homology groups as the quotient vector spaces