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Given an $\infty$-topos, an object $X$, and a $f$ 1-monomorphism (i.e. (-1)-truncated) is the internal hom $[X,f]$ again a 1-mono?
And dually for $f$ 1-epimorphism (i.e. effective epimorphism) do we have some extra conditions such that $[f,X]$ is 1-mono?
For the first part, this follows because the fiber of post-composition with $f : A \to B$ at some $h : X \to B$, $fib_{f\circ}(h)$, is equivalent to the Pi-type $\prod_{x : X} fib_f (h x)$.
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: $Hom(Y,[X,f]) = Hom(Y\times X, f)$.
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.
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 $X$ is one for which the map $X \to \Im X$ 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 $X$ is that the 0-section of its cotangent stack is a 1-mono. This turns out to be implied if $X \to \Im X$ is the op-ed version of a 1-mono.
Now i know for some situations of intrest that $X \to \Im X$ 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.
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