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David Egolf (Jan 27 2021 at 05:07):For context, I'm interested in thinking about imaging using concepts from category theory.
We can describe an image (say, a grid of intensity values) by specifying its intensity values. The more intensity values we know, the better we can describe the image. Similarly, we can partially characterize an unknown imaging target by bouncing ultrasound echoes off it from a few angles. A function can be described by how it acts on each element in its source set. A linear map can be described by how it acts on each basis vector in its source vector space. A vector can be described by how it maps under different linear mappings.
The examples above illustrate the concept of partially characterizing something by looking at parts of it, or by observing the results of different measurements of it. Is there a categorical concept corresponding to this "partial characterization"? For example, one could imagine saying that an object in a category is better characterized as we are told more about the morphisms from or to .
Matteo Capucci (he/him) (Jan 27 2021 at 08:55):Sheaves could be what you're looking for. This paper by Robinson explicitly talks about images, if I recall correctly.
Matteo Capucci (he/him) (Jan 27 2021 at 08:57):There's also the concept of generator (or separator, in nLabian) which captures what you described for set maps and vector spaces maps.
fosco (Jan 27 2021 at 09:07):Well, an object of a category is better characterised as we are told more about the morphisms from any other object to it. It's called "Yoneda lemma"
Fawzi Hreiki (Jan 27 2021 at 09:34):Maybe the concept of a dense subcategory could help. The idea is that all the objects in the category are constructed by glueing together some basic ones (in a canonical way).
Fawzi Hreiki (Jan 27 2021 at 09:38):For example, the one point set is dense in the category of sets (since a set is just a bunch of points next to each other).
David Egolf (Jan 27 2021 at 16:23):Thanks everyone. Those all look interesting, in particular the paper by Robinson.
fosco said:
Well, an object of a category is better characterised as we are told more about the morphisms from any other object to it. It's called "Yoneda lemma"
I have (sadly) yet to fully follow any presentation of the Yoneda lemma. My understanding, though, was that it talks about totally characterizing an object once we know all the morphisms to it. Does it also describe how this characterization develops when we are given some partial information about the morphisms to it?
Joe Moeller (Jan 27 2021 at 16:33):Fawzi Hreiki said:
Maybe the concept of a dense subcategory could help. The idea is that all the objects in the category are constructed by glueing together some basic ones (in a canonical way).
I noticed there are no examples on that nLab page. You should add a few :smile:
Fawzi Hreiki (Jan 27 2021 at 17:15):There is also this page which has the following fact: if is an algebraic theory which has a presentation with operations of at most arity and if is its category of models, then the full subcategory of on the free -algebra on generators is dense. So is dense in the category of -vector spaces, etc..
Fawzi Hreiki (Jan 27 2021 at 17:17):So that basically covers most examples from algebra.
Fawzi Hreiki (Jan 27 2021 at 17:18):The density theorem states that the representables in a presheaf category are dense.
Fawzi Hreiki (Jan 27 2021 at 17:22):I'm pretty sure that the simplices are dense in the category of simplicial complexes. The same kind of thing should be true for other categories of combinatorial spaces (delta complexes, CW complexes, etc..) but I haven't checked.