Quivers: A year of linear algebra by drawing arrows(lisyarus.github.io)
lisyarus.github.io
Quivers: A year of linear algebra by drawing arrows
https://lisyarus.github.io/blog/posts/quivers-a-year-of-linear-algebra-by-drawing-arrows.html
7 comments
Looks like an introductory tutorial on the application of category theory to linear algebra. These quivers are lovely small categorical constructions. Actually, q.uiver.app is a visual tool to use the quiver package in latex, which is used to draw commutative diagrams in category theory.
Where are the theorems and the proofs? Can the usual theorems of the "year of linear algebra" be proved using these arrows?
To understand what's happening here, I'd recommend you to first understand Representation theory: https://en.wikipedia.org/wiki/Representation_theory
In representation theory we reduce problems of algebra to problems of linear algebra. E.g. the standard example is to find representations of groups, this way we can represent group operations as matrix operations. We do this because (1) linear algebra is mathematically very well-understood, (2) in terms of applications, linear algebra is computationally fast, faster than implementing the group with code manually (at least, in general).
In the OP post, author reduces quiver (which is a particular kind of algebra) to linear algebra. Once this is done, the intention is to solve problems of quivers in the language of linear algebra.
In representation theory we reduce problems of algebra to problems of linear algebra. E.g. the standard example is to find representations of groups, this way we can represent group operations as matrix operations. We do this because (1) linear algebra is mathematically very well-understood, (2) in terms of applications, linear algebra is computationally fast, faster than implementing the group with code manually (at least, in general).
In the OP post, author reduces quiver (which is a particular kind of algebra) to linear algebra. Once this is done, the intention is to solve problems of quivers in the language of linear algebra.
If you're interested in actual linear algebra with diagrams, I think https://graphicallinearalgebra.net/ is the thing to look at.
I like directed acyclic graphs and/or DAG because it's a succinct description and contract. Trying to change the name of it makes me quiver with uncertainty.
Quiver (a directed graph with multiple edges) is a standard mathematical term:
https://en.wikipedia.org/wiki/Quiver_(mathematics)
https://ncatlab.org/nlab/show/quiver
A quiver is simply just the data of a category, i.e. a "category" without any of the laws, namely identity and composition.
They're not isomorphic to DAGs since Quivers can have multiple edges between the same set of vertices, directed multigraphs, if you will. There is also no requirement of acyclicity (DAGs are acyclic).
For example, in the category of Sets, vertices are sets and edges are functions between sets, so between e.g. N and N there will be infinitely many edges (all functions between natural numbers) with a particular distinguished identity edge that maps f(n) = n due to category laws. So if you turn the category of Sets to a quiver, you'll have infinitely many edges N -> N and one of them will happen to be the identity function `f(n) = n` but you "forgot" its "identity" relationship/law when you reduced the category to a quiver. This is not a graph, since within your data you need to express that there are other edges between N -> N for example `f(n) = 2*n` is another edge (we can call these multigraphs).
https://en.wikipedia.org/wiki/Quiver_(mathematics)
https://ncatlab.org/nlab/show/quiver
A quiver is simply just the data of a category, i.e. a "category" without any of the laws, namely identity and composition.
They're not isomorphic to DAGs since Quivers can have multiple edges between the same set of vertices, directed multigraphs, if you will. There is also no requirement of acyclicity (DAGs are acyclic).
For example, in the category of Sets, vertices are sets and edges are functions between sets, so between e.g. N and N there will be infinitely many edges (all functions between natural numbers) with a particular distinguished identity edge that maps f(n) = n due to category laws. So if you turn the category of Sets to a quiver, you'll have infinitely many edges N -> N and one of them will happen to be the identity function `f(n) = n` but you "forgot" its "identity" relationship/law when you reduced the category to a quiver. This is not a graph, since within your data you need to express that there are other edges between N -> N for example `f(n) = 2*n` is another edge (we can call these multigraphs).
But they aren't DAGs. They are multidigraphs.