Fixed typos in transcript:

Just generally speaking, what can I do with this sheaf-theoretic data structure that I've got? Okay, [I'll] remind you what is a sheaf. A sheaf is something that allows me to translate between physical sources or physical realms of data [in the left diagram] and the data that are associated with those physical regions [in the right diagram]

So these [on the left] are various open sets [an example being] simplices in a [simplicial complex which is an example of a] topological space.

And these [on the right] are the data spaces and I'm able to make some translation between [the left and the right diagrams] by taking a look at restrictions of overlaps [a on the left] and inferring back to the union.

So that's what a sheaf is [regarding data structures]. It's something that allows me to make an inference, an inferential machine.