Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2017
    • (edited Mar 22nd 2017)

    Given an \infty-topos, an object XX, and a ff 1-monomorphism (i.e. (-1)-truncated) is the internal hom [X,f][X,f] again a 1-mono?

    And dually for ff 1-epimorphism (i.e. effective epimorphism) do we have some extra conditions such that [f,X][f,X] is 1-mono?

    • CommentRowNumber2.
    • CommentAuthorUlrik
    • CommentTimeMar 22nd 2017

    For the first part, this follows because the fiber of post-composition with f:ABf : A \to B at some h:XBh : X \to B, fib f(h)fib_{f\circ}(h), is equivalent to the Pi-type x:Xfib f(hx)\prod_{x : X} fib_f (h x).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeMar 22nd 2017

    Speaking categorically, the corresponding statement for the external-hom is essentially the definition of 1-monomorphism, and the internal part internalizes in the same way this sort of stuff usually does: Hom(Y,[X,f])=Hom(Y×X,f)Hom(Y,[X,f]) = Hom(Y\times X, f).

    For the second part, I don’t think an effective epi should be called a 1-epi; a 1-epi should be a map that is 1-mono in the opposite category. As you know, unlike in the 1-categorical case, effective epis are not in general 1-epi in this sense, even for the external-hom, and I don’t know any natural conditions ensuring that they are.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 22nd 2017
    • (edited Mar 22nd 2017)

    Right, thanks, I am concerned with the op-ed version of 1-mono.

    This came from thinking about the definition of formally smooth stacks. How do people usually define formally smooth stacks?

    A formally smooth scheme XX is one for which the map XXX \to \Im X to its de Rham space “infinitesimal shape modality” is an epimorphism in the 1-topos.

    The question is what is the right way to generalize this from schemes to stacks.

    I came to think that a key property of a would-be formally smooth stack XX is that the 0-section of its cotangent stack is a 1-mono. This turns out to be implied if XXX \to \Im X is the op-ed version of a 1-mono.

    Now i know for some situations of intrest that XXX \to \Im X is an effective epi. I was hoping that I could add a little bit of something to see that it’s an op-ed 1-mono. But I gather I should instead re-think the whole situation in terms of op-ed 1-monos in the first place.