Category theory | Algebraic topology

Size functor

Given a size pair where is a manifold of dimension and is an arbitrary real continuous function definedon it, the -th size functor, with , denoted by , is the functor in , where is the category of ordered real numbers, and is the category of Abelian groups, defined in the following way. For , setting , , equal to the inclusion from into , and equal to the morphism in from to , * for each , * In other words, the size functor studies theprocess of the birth and death of homology classes as the lower level set changes.When is smooth and compact and is a Morse function, the functor can bedescribed by oriented trees, called − trees. The concept of size functor was introduced as an extension to homology theory and category theory of the idea of size function. The main motivation for introducing the size functor originated by the observation that the size function can be seen as the rankof the image of . The concept of size functor is strictly related to the concept of , studied in persistent homology. It is worth to point out that the -th persistent homology group coincides with the image of the homomorphism . (Wikipedia).

Video thumbnail

Micrometer/diameter of daily used objects.

What was the diameter? music: https://www.bensound.com/

From playlist Fine Measurements

Video thumbnail

Dimensions Chapter 5

Chapter 5 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Video thumbnail

Dimensions Chapter 1

Chapter 1 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Video thumbnail

Vernier caliper / diameter and length of daily used objects.

subscribe to my channel: https://www.youtube.com/channel/UC7HtImwPgmb1axsTJE5lxeA?sub_confirmation=1

From playlist Fine Measurements

Video thumbnail

How big are complex numbers?

How big are complex numbers? We discuss a way of measuring them via the modulus. The ideas use Pythagorus' theorem. Free ebook http://bookboon.com/en/introduction-to-complex-numbers-ebook

From playlist Intro to Complex Numbers

Video thumbnail

Dimensions Chapter 6

Chapter 6 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Video thumbnail

Micrometer / diameter of daily used objects

What was the diameter? music: https://www.bensound.com/

From playlist Fine Measurements

Video thumbnail

Volume of a sphere

Another practice problem dealing with the volume of a sphere

From playlist Middle School - Worked Examples

Video thumbnail

Higher Algebra 7: Non-abelian derived functors

In this video, we discuss the notion of non-abelian derived functors and Animation. Along the way, we discuss the Yoneda lemma. Warning: The Yoneda exercises stated at 35:00 is a bit hard given the technology we have, so I recommend simply proving the analogous statement for ordinary cat

From playlist Higher Algebra

Video thumbnail

Martina Scolamiero 9/15/21: Extracting persistence features with hierarchical stabilisation

Title: Extracting persistence features with hierarchical stabilisation Abstract: It is often complicated to understand complex correlation patterns between multiple measurements on a dataset. In multi-parameter persistence we represent them through algebraic objects called persistence mo

From playlist AATRN 2021

Video thumbnail

Serge Bouc: Correspondence functors

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Algebraic and Complex Geometry

Video thumbnail

Emily Riehl: On the ∞-topos semantics of homotopy type theory: The simplicial model of...- Lecture 2

HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 22, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual M

From playlist Topology

Video thumbnail

Justin Lynd: Control of fixed points and centric linking systems

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: Workshop "Fusion systems and equivariant algebraic topology"

From playlist HIM Lectures: Junior Trimester Program "Topology"

Video thumbnail

Spheres and Cylinders

An introduction to the process of finding the volume of a sphere

From playlist Middle School - Worked Examples

Video thumbnail

Sejong Park: Double Burnside rings and Mackey functors with applications to fusion systems

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: Workshop "Fusion systems and equivariant algebraic topology"

From playlist HIM Lectures: Junior Trimester Program "Topology"

Video thumbnail

Ivan Di Liberti - Towards higher topology

Talk at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/DiLibertiSlidesToposesOnline.pdf We categorify the adjunction between locales and topological spaces, this amounts t

From playlist Toposes online

Video thumbnail

Emily Riehl: On the ∞-topos semantics of homotopy type theory: All ∞-toposes have... - Lecture 3

HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 24, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual M

From playlist Topology

Video thumbnail

Paul Turner: A hitchhiker's guide to Khovanov homology - Part II

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Geometry

Video thumbnail

Dimensions (2 of 3: A More Flexible Definition - Scale Factor)

More resources available at www.misterwootube.com

From playlist Exploring Mathematics: Fractals

Video thumbnail

Yonatan Harpaz - New perspectives in hermitian K-theory III

For questions and discussions of the lecture please go to our discussion forum: https://www.uni-muenster.de/TopologyQA/index.php?qa=k%26l-conference This lecture is part of the event "New perspectives on K- and L-theory", 21-25 September 2020, hosted by Mathematics Münster: https://go.wwu

From playlist New perspectives on K- and L-theory

Related pages

Functor | Manifold | Persistent homology | Size homotopy group | Size function | Continuous function | Category theory | Homomorphism | Category (mathematics) | Size theory | Morphism