/sci/ - Science & Math

>>8508411
Thanks! I can give you a brief explanation, a bit vague because the details are technical and not that interesting.

I borrowed some ideas from topology to define nice subcategories for my categories with already a nice structure. I then used an equivalence relation to make them small enough to be used properly. I then used this https://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem to connect my categories to rings, which I then reduced to their isomorphism classes. Using these isomorphism classes, I was able to define a semigroup, and then the Grothendieck group of this semigroup I used to get a topological space for these categories. The details were a bit hazy, but i refined them so that it was close enough to work, but then I found out the same thing was already done with a slightly different way, this thing https://en.wikipedia.org/wiki/Q-construction by Quillen.

I then returned to the very beginning, and started following another path I could follow. I have so far been able to construct these topological spaces out of the categories with a lot easier method, and my current project is to show I can induce a topology in my small abelian subcategories using this space. Since I defined my subcategory so that its properties are independent of its objects, I could then make a local topological/topology-like structure over this category using these categorical deformation retracts I have, and, with luck, this may even give me something analogous to a manifold. I just need to find a way to formalize my ideas properly.

Sorry if my text is hard to follow, I've slept about 11 hours this week.

>>8450196
I like Rotman's book. It is good for self studying.