Why presheaves? (Episode #1) Diagrams, actions, and representability

[@luminator2145]

Great series! Your example of graphs made me wonder: In what sense is it natural to see graphs as presheaves, rather than just a special case of an algebraic theory, i.e. a product-preserving functor on a syntactic category? Can we always see the syntactic category of a Lawvere theory as a site, such that the resulting sheaves are exactly the product-preserving functors? (Of course this does not exactly fit the situation you have, as the syntatic category for the Lawvere theory of graphs is product-generated by V and E)

[@carloangiuli]

That's an interesting thought, I'm not sure! But I think another way to look at the graph example is to think of categories as multi-sorted Lawvere theories with only unary operations, in which case diagrams are the models.

[@ValentinRobert]

Very nice! I don't know whether this is a "bug" in the video, but the display seems to be way behind the commentary? For instance, you mention the context "x : A, y : B" at 6:07, but it only gets displayed at timestamp 6:53. This is quite jarring as what is being said and what is being displayed start diverging a lot from about 5 minutes in.

[@carloangiuli]

Thanks! I recorded the audio and video separately and then sped up the video in places to keep them roughly in sync. I got much better at editing as I went along, so hopefully that's the only spot where it's an issue. (A few people have mentioned this spot out to me but nowhere else, so fingers crossed.)

[@CyberneticOrganism01]

Does G already contain enough information to define a directed graph? So the presheaf F defines a directed multi-graph ?

[@yegorwienski1236]

1:03 Isn't "topoi" the plural for "topos"? What's "toposes" then? Do they differ in singular as well? How do I learn all about all of these wonderful things? ... or is it just a mistake in the slide?

[@carloangiuli]

Ah, this was just a bit of a joke :) They are both used as plurals of "topos," and many people have strong feelings about which one is better.

[@ranaldclouston9325]

Clobbers?

[@carloangiuli]

In the sense that the freely-added colimits never coincide with whatever colimits existed in the base category: the old sums are replaced by a new sum, etc.

[@tizusa]

@@carloangiuli You mean like, say, よ(a+b) ≠ よ(a) + よ(b), and we replace the LHS with the RHS?

[@carloangiuli]

​@@tizusa I'm thinking of it as a kind of "inheritance" with Ĉ.product := C.product (if it exists) but Ĉ.coproduct = (not C.coproduct, even if it exists).