Homomorphisms as structure-preserving maps

[quasar987]

Homotopy types: The homotopy category is not concrete -- roughly speaking its objects cannot be represented as "sets with structure", no matter clever a notion of structure you might come up with.

That's interesting.. i wonder how one goes about proving such a bizarre statement.

(I imagine that the homotopy category has as its objects the topological spaces and as its morphisms the homotopy classes of continuous maps)

By the way, I have studied some mathematical logic since the other discussion, so I'm now familiar with how structures/algebras are defined in model theory/universal algebra. [...]

What reference would you recommend ?

 

[Landau]

(I imagine that the homotopy category has as its objects the topological spaces and as its morphisms the homotopy classes of continuous maps)

Yes.

That's interesting.. i wonder how one goes about proving such a bizarre statement.

The precise statement is that \textbf{Toph} is not concretizable, meaning there does not exist a faithful functor F:\textbf{Toph}\to\textbf{Set}. A proof of this statement can be found here.

 

[Fredrik]

What reference would you recommend ?

Sorry, I didn't see the question until now. I first read about first-order formal languages, "structures" and what it means for a set of sentences to "logically imply" another sentence, in Enderton, and in Rautenberg. I don't think either of them is really easy to follow, so it was really helpful to have access to both. Then I bought Kunen's book "Foundations of mathematics". That's a very good book, so I can recommend it without reservations. But I got the feeling that a lot of the stuff that I found easy to understand would have been quite hard to understand if I hadn't read the about the stuff I mentioned above in Enderton/Rautenberg first.