Author: Ali Caglayan Format: TextSo the representation theory of quivers is a bunch of useful tools for understanding certain algebras and categories of modules over them. Not to mention in Algebraic Geometry where they can be used to study the derived categories of certain varieties. (Essentially allowing for a concise presentation of information (and more?)).
Now I have had the idea of "higher" quivers rattling around in my head for the past year or two, but I have never really given it much serious thought. However I am now reading about globular sets and these are exactly the "higher" quivers I was thinking about. Just like normal quivers they aren't quite categories, (in this case the higher ones are strict). But there is a faithful embedding from Quiv into Cat, and I assume the same thing for Glob into (oo,1)Cat (This isn't full). These are both adjoint to the obvious forgetful functors.
Now in the quiver case we can think about representations in two equivalent ways: as functors from the free category generated by the quiver to a categry of k-modules, or we can take the free k-module genereated by the set of morphisms of the free category of the quiver and define an algebra with product composition and zero when it doesn't compose, representations are modules over this "path algebra". This is the idea of quiver representations being path algebra modules.
Now generalising to globular sets. The free category of a globular set should just be the strict oo-category it produces. Representations are therefore oo-functors from the free oo-category to some Aoo-module category (not sure how to define this). We can also try to produce some sort of higher path algebra similarly to how it was done in the quiver case. And hopefully these two notions coincide to give a unified higher representation of globular sets.
I have no idea what the category of these things might look like, but I hope for example it is a stable category because then we can define a "derived category" via stable Dold-Kan. Which would have some sort of cohomology of globular sets. (This could also be completely vacous).
These are just some eveloped ideas I have had floating around, hopefully someone here can formalise or point to a developed reference.
So the representation theory of quivers is a bunch of useful tools for understanding certain algebras and categories of modules over them. Not to mention in Algebraic Geometry where they can be used to study the derived categories of certain varieties. (Essentially allowing for a concise presentation of information (and more?)).
Now I have had the idea of "higher" quivers rattling around in my head for the past year or two, but I have never really given it much serious thought. However I am now reading about globular sets and these are exactly the "higher" quivers I was thinking about. Just like normal quivers they aren't quite categories, (in this case the higher ones are strict). But there is a faithful embedding from Quiv into Cat, and I assume the same thing for Glob into (oo,1)Cat (This isn't full). These are both adjoint to the obvious forgetful functors.
Now in the quiver case we can think about representations in two equivalent ways: as functors from the free category generated by the quiver to a categry of k-modules, or we can take the free k-module genereated by the set of morphisms of the free category of the quiver and define an algebra with product composition and zero when it doesn't compose, representations are modules over this "path algebra". This is the idea of quiver representations being path algebra modules.
Now generalising to globular sets. The free category of a globular set should just be the strict oo-category it produces. Representations are therefore oo-functors from the free oo-category to some Aoo-module category (not sure how to define this). We can also try to produce some sort of higher path algebra similarly to how it was done in the quiver case. And hopefully these two notions coincide to give a unified higher representation of globular sets.
I have no idea what the category of these things might look like, but I hope for example it is a stable category because then we can define a "derived category" via stable Dold-Kan. Which would have some sort of cohomology of globular sets. (This could also be completely vacous).
These are just some eveloped ideas I have had floating around, hopefully someone here can formalise or point to a developed reference.