6 ms·

Sheaf Theory Through Examples

73 points by soist 2y ago

xanderlewis 2y ago
This is a very fun book, and an unusual one too. Especially for a concept that even many pure mathematicians find abstract.
- soloist11 2y ago
  Category theory is being applied in all sorts of domains and recently has been making some inroads in ML as well, e.g. https://arxiv.org/abs/2106.07032 https://arxiv.org/abs/2106.07032
asplake 2y ago
> Sheaves are mathematical constructions concerned with passages from local properties to global ones Sounds interesting. Could someone elaborate on that?
- tristramb 2y ago
  Think of a cryptic crossword puzzle. It consists of a grid of overlapping slots and for each slot there is a clue. The question that sheaf theory addresses is what constraints do you have to put on the clues to ensure that the overall puzzle has a single solution.
  - calf 2y ago
    Is that related to #P complexity at all, the complexity class of counting the number of solutions to an NP-complete instance?
    - tristramb 2y ago
      No
      - calf 2y ago
        Why not, they sound like similar problems and the latter would motivate computer scientists to learn about sheafs as per the approach taken in OP's book which is applied/example driven.
- ibotty 2y ago
  If you know (smooth) manifolds you know a basic example of sheaves (and its cohomology group).
- QuesnayJr 2y ago
  Sheaves capture two properties: if you have a solution to a problem on a big piece of a space, you can shrink it to a smaller piece, and if you have solutions on small pieces of a space that agree with each other on overlaps, you can glue them together to get a solution on a bigger piece. An easy example is a function on a set. If you have function defined on the whole set, you can shrink it to give you a function defined on a subset. If you have functions defined on several subsets, and those functions agree on the overlaps of the subsets, then you can use that to define a function on the union of the subsets. More interesting examples arise in topology and related fields.
- 6gvONxR4sf7o 2y ago
  I don’t know sheaves (except that they are a generalization of differential geometry or something?), but a great example of local to global is the fundamental theorem of calculus. You take this property of a function that’s only defined in an arbitrarily small neighborhood of a point, and from it you can determine the function’s value anywhere else. That is, you take infinitesimally small changes (e.g. velocity) and add them up in the right way and get finite changes (e.g. distance). It’s more interesting than it sounds because you aren’t computing a sum or something with numbers when you add up infinitesimal change. Local/infinitesimal change is in some ways a different beast than finite/global change.
hackandthink 2y ago
An amazing book that takes a clear and descriptive path to topos theory. If you want to take it a little slower, you can start with Lawvere's "Conceptual Mathematics" https://api.pageplace.de/preview/DT0400.9780511590092_A23569066/preview-9780511590092_A23569066.pdf https://api.pageplace.de/preview/DT0400.9780511590092_A23569...
- auggierose 2y ago
  I've seen toposes declared as some fundamental notion, and I'd very much like to understand them. Is there a short definition somewhere out there of what a topos is in terms of first-order predicate logic? Something I can understand without reading through 200 pages of preliminary material first? I've seen statements that such a formulation in first-order logic would be misguided, because category theorists have their own notion of logic, but I'd like to understand it using my own notion of logic first.
  - hackandthink 2y ago
    Makkai's work my fit: "First Order Logic with Dependent Sorts,with Applications to Category Theory" "For instance, the definition of elementary topos (with operations defined by universal properties up to isomorphism, not specified as univalued operations) can be given as a finite set of sentences in FOLDS." https://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf https://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf
    - auggierose 2y ago
      Interesting find, but again an example of where you first need to learn some new logic FOLDS ("FOLDS has the first two of these, contexts and types (although the latter are called 'sorts'), but it does not have the third, terms (except in the rudimentary form of mere variables), and it has equality in a greatly restricted form only."). I wonder if it is impossible to describe a topos as a normal axiom system of first-order logic, or if people are just unwilling to do it.
  - soist 2y ago
auggierose 2y ago
There is something about category theory that just puts me to sleep. I cannot count the number of times I picked up a category theory text, full of best intentions, started reading, and ... woke up a few hours later. Dozens of diagrams pointing here and there don't help. It is somehow as if the abstractness of category theory is abstract in the wrong way for me.
- chongli 2y ago
  It’s not just you. Category theory has been called “abstract nonsense” for a very long time and even referred to as such by Saunders Mac Lane [1], cofounder of the discipline itself! The subject is just very difficult to motivate because it’s so abstract that it’s hard to see the relevance of its results. [1] https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
  - auggierose 2y ago
    Yet its origin seems to be something quite concrete and practical. For example, just recently I came across this text: "Foundations of Algebraic Topology", by Eilenberg and Steenrod. Its preamble is highly readable and engaging, see below. We have a topology and compute some algebraic structure from it. Sounds easy! ------------------------------ The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest and most extensively developed portion of algebraic topology, and may be regarded as the main body of the subject. The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized. It is assumed that the reader is familiar with the basic concepts of algebra and of point set topology. No attempt is made to axiomatize these subjects. This has been done extensively in the literature. Our achievement is different in kind. Homology theory is a transition (or function) from topology to algebra. It is this transition which is axiomatized. Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems. In this respect, homology theory parallels analytic geometry. How ever, unlike analytic geometry, it is not reversible. The derived algebraic system represents only an aspect of the given topological system, and is usually much simpler. This has the advantage that the geometric problem is stripped of inessential features and replaced by a familiar type of problem which one can hope to solve. It has the disadvantage that some essential feature may be lost. In spite of this, the subject has proved its value by a great variety of successful applications. Our axioms are statements of the fundamental properties of this assignment of an algebraic system to a topological system. The axioms are categorical in the sense that two such assignments give isomorphic algebraic systems.

73 points by soist 2y ago

This is a very fun book, and an unusual one too. Especially for a concept that even many pure mathematicians find abstract.

soloist11 2y ago
Category theory is being applied in all sorts of domains and recently has been making some inroads in ML as well, e.g. https://arxiv.org/abs/2106.07032 https://arxiv.org/abs/2106.07032

asplake 2y ago

> Sheaves are mathematical constructions concerned with passages from local properties to global ones Sounds interesting. Could someone elaborate on that?

tristramb 2y ago
Think of a cryptic crossword puzzle. It consists of a grid of overlapping slots and for each slot there is a clue. The question that sheaf theory addresses is what constraints do you have to put on the clues to ensure that the overall puzzle has a single solution.
- calf 2y ago
  Is that related to #P complexity at all, the complexity class of counting the number of solutions to an NP-complete instance?
  - tristramb 2y ago
    No
    - calf 2y ago
      Why not, they sound like similar problems and the latter would motivate computer scientists to learn about sheafs as per the approach taken in OP's book which is applied/example driven.
ibotty 2y ago
If you know (smooth) manifolds you know a basic example of sheaves (and its cohomology group).
QuesnayJr 2y ago
Sheaves capture two properties: if you have a solution to a problem on a big piece of a space, you can shrink it to a smaller piece, and if you have solutions on small pieces of a space that agree with each other on overlaps, you can glue them together to get a solution on a bigger piece. An easy example is a function on a set. If you have function defined on the whole set, you can shrink it to give you a function defined on a subset. If you have functions defined on several subsets, and those functions agree on the overlaps of the subsets, then you can use that to define a function on the union of the subsets. More interesting examples arise in topology and related fields.
6gvONxR4sf7o 2y ago
I don’t know sheaves (except that they are a generalization of differential geometry or something?), but a great example of local to global is the fundamental theorem of calculus. You take this property of a function that’s only defined in an arbitrarily small neighborhood of a point, and from it you can determine the function’s value anywhere else. That is, you take infinitesimally small changes (e.g. velocity) and add them up in the right way and get finite changes (e.g. distance). It’s more interesting than it sounds because you aren’t computing a sum or something with numbers when you add up infinitesimal change. Local/infinitesimal change is in some ways a different beast than finite/global change.

hackandthink 2y ago

An amazing book that takes a clear and descriptive path to topos theory. If you want to take it a little slower, you can start with Lawvere's "Conceptual Mathematics" https://api.pageplace.de/preview/DT0400.9780511590092_A23569066/preview-9780511590092_A23569066.pdf https://api.pageplace.de/preview/DT0400.9780511590092_A23569...

auggierose 2y ago
I've seen toposes declared as some fundamental notion, and I'd very much like to understand them. Is there a short definition somewhere out there of what a topos is in terms of first-order predicate logic? Something I can understand without reading through 200 pages of preliminary material first? I've seen statements that such a formulation in first-order logic would be misguided, because category theorists have their own notion of logic, but I'd like to understand it using my own notion of logic first.
- hackandthink 2y ago
  Makkai's work my fit: "First Order Logic with Dependent Sorts,with Applications to Category Theory" "For instance, the definition of elementary topos (with operations defined by universal properties up to isomorphism, not specified as univalued operations) can be given as a finite set of sentences in FOLDS." https://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf https://www.math.mcgill.ca/makkai/folds/foldsinpdf/FOLDS.pdf
  - auggierose 2y ago
    Interesting find, but again an example of where you first need to learn some new logic FOLDS ("FOLDS has the first two of these, contexts and types (although the latter are called 'sorts'), but it does not have the third, terms (except in the rudimentary form of mere variables), and it has equality in a greatly restricted form only."). I wonder if it is impossible to describe a topos as a normal axiom system of first-order logic, or if people are just unwilling to do it.
- soist 2y ago

auggierose 2y ago

There is something about category theory that just puts me to sleep. I cannot count the number of times I picked up a category theory text, full of best intentions, started reading, and ... woke up a few hours later. Dozens of diagrams pointing here and there don't help. It is somehow as if the abstractness of category theory is abstract in the wrong way for me.

chongli 2y ago
It’s not just you. Category theory has been called “abstract nonsense” for a very long time and even referred to as such by Saunders Mac Lane [1], cofounder of the discipline itself! The subject is just very difficult to motivate because it’s so abstract that it’s hard to see the relevance of its results. [1] https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
- auggierose 2y ago
  Yet its origin seems to be something quite concrete and practical. For example, just recently I came across this text: "Foundations of Algebraic Topology", by Eilenberg and Steenrod. Its preamble is highly readable and engaging, see below. We have a topology and compute some algebraic structure from it. Sounds easy! ------------------------------ The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. It is the oldest and most extensively developed portion of algebraic topology, and may be regarded as the main body of the subject. The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized. It is assumed that the reader is familiar with the basic concepts of algebra and of point set topology. No attempt is made to axiomatize these subjects. This has been done extensively in the literature. Our achievement is different in kind. Homology theory is a transition (or function) from topology to algebra. It is this transition which is axiomatized. Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist's field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems. In this respect, homology theory parallels analytic geometry. How ever, unlike analytic geometry, it is not reversible. The derived algebraic system represents only an aspect of the given topological system, and is usually much simpler. This has the advantage that the geometric problem is stripped of inessential features and replaced by a familiar type of problem which one can hope to solve. It has the disadvantage that some essential feature may be lost. In spite of this, the subject has proved its value by a great variety of successful applications. Our axioms are statements of the fundamental properties of this assignment of an algebraic system to a topological system. The axioms are categorical in the sense that two such assignments give isomorphic algebraic systems.