this is a bit of a vague question so let me describe a bit what motivates it: Yesterday I was reading the Wikipedia article about perfect numbers, where I find the section https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers quite interesting. It appears that we know a good amount of properties that odd perfect numbers must satisfy, but what we don't know is if there actualy are odd perfect numbers.
What I then asked myself is if this phenomenon occurs on the level of category theory, i.e. is there a (say concrete) category, which is quite well understood (where, I don't know, limits, products and some structure theorems might exist), but where we don't know a single example of its objects.
I mean, usually all introductory courses in any mathematical topic start in rather the same pattern: first we define the category of objects we're interested in, say topological spaces or groups or manifolds, then we give concrete examples (say R with the topology induced by the euclidian metric for a topological space, symmetric groups for groups or the n-sphere for manifolds) and then we show some stability properties such as "the cartesian product of two topological spaces with the product topology is again a topological space" et cetera.
But do you know about a topic, where step 1 and 3 can be done, but not step 2 because we don't know any concrete example?
Sorry for the vagueness of the question, but I hope you'll understand what I mean.
Thank you!
Best, jgrk
It is a soft question, so here is my personal opinion:
No. When you think to know a lot about a category, but don't know if it has any objects, you don't know a lot (in fact, very little) about that category.