My question is rather imprecise and open to modification. I am not entirely sure what I am looking for but the question seemed interesting enough to ask:
The opposite category of rings is the category of affine schemes. This is usually thought of as the category of spaces. Can we run the construction backwards for categories usually thought of as containing spaces?
For instance, does $\operatorname{Top}^{\operatorname{op}}$ have a nice description as some "algebraic" category?
Note that it does not seem easy to describe the opposite category of all schemes. Therefore, the above question might be asking too much. Perhaps the following is a more tractable (or not) question:
Can we find an "algebraic" category $C$ such that we can embed $C^{\operatorname{op}}$ in $\operatorname{Top}$ such that every topological space can be covered by objects in $C^{\operatorname{op}}$? Perhaps one would like to replace this criterion of being covered by objects by a more robust notion in general.
One can repeat the question for other categories of spaces like:
- Category of manifolds (perhaps closer to schemes than general topological spaces)
- Compactly generated spaces
- Simplicial Sets
and so on. A perhaps interesting example is the category of finite sets, it's opposite category is the category of finite Boolean algebras.