Homotopy Theory

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3:26 PM

I imagine the answer is no. But is there a connection between the following two definitions of a fat point (at least morally, since not mathematically):

1) (Say for a $k$-scheme $X$) a map $\text{Spec}(k[x]/(x^n))\to X$ for $n>1$ (or whatever is appropriate more generally, but we're mapping from a point scheme with a non-reduced structure)

2) (Let $S$ be a noetherian scheme) A fat point laying over a $k$-point $x:\text{Spec}(k)\to S$, is a triple $(x_0,x_1,R)$ where $R$ is a DVR, $x_0:\text{Spec}(k)\to \text{Spec}(R)$, and $x_1:\text{Spec}(R)\to S$ are such that 1) $x=x_1\circ x_0$, 2) Th…

2 hours later…

Denis Nardin

5:35 PM

@NeverEnoughTime They are not really related. I don't know of any standard name for the second one, but I'd call it a "curve germ" through the point since that's what it essentially is..

2 hours later…

Saal Hardali

7:35 PM

Dumb Question: Is there a characterization of paracompactness of (sober) topological spaces in terms of properties of their corresponding $\infty$-topoi?

Saal Hardali

7:50 PM

Literal translation leads me to something along the lines of "sheafification almost preserves filtered colimits" (where "almost" here is in the sense of truncation). Hoping there's something more elegant and/or precise/true to say here...

3 hours later…

Jonathan Beardsley

11:08 PM

This is a kind of silly technical question which I could probably work out if I put some tedious time into it, but does anyone know if this definition of "loops" given by Moore in his notes "Algebraic Homotopy Theory" is equal/isomorphic to one of the standard definitions?

And, moreover, what construction would be its left adjoint (if it has a left adjoint)?

(note that I'm genuinely interested in, like, sSet isomorphism here, not homotopy equivalence)

Like, it looks like the definition given in Goerss and Jardine by pulling back the path space:

(but I actually still don't know what the strict left adjoint to that construction is)

Piotr Pstrągowski

I think it's not quite isomorphic to the path-space construction from Goerss-Jardine, rather it's more like the over/undercategories of Joyal which can also be used to compute mapping spaces.

Jonathan Beardsley

@PiotrPstrągowski do you happen to have a link to where this appears in Joyal's work?

Piotr Pstrągowski

If $(X, x)$ is a pointed simplicial set, it seems to me that Moore defines $n$-simplices of $\Omega X$ to be those (n+1)-simplices of $X$ such that: 1) they take the zero-th vertex to $x$, 2) they take the face opposite the zero-th vertex to the constant/degenerate n-simplex on $x$.

If we instead look at those (n+1)-simplices which only satisfy 1), we'd get exactly the overcategory $X_{/x}$ as in HTT (this certainly goes to Joyal, if not further).

Jonathan Beardsley

11:23 PM

Oh interesting.

Piotr Pstrągowski

Also asking for 2) is looking at the preimage of $\{ x \}$ under the fibration $X_{/x} \rightarrow X$

So if $X$ was a Kan complex (or even an $\infty$-category), you would indeed compute $map_{X}(x, x)$, ie. the loop space. It is possible this gives you the right homotopy type even when $X$ is not Kan, I don't know.

This is one of the left/right mapping spaces in sense of Lurie. It's somewhere in HTT that this is the same, up to homotopy equivalence, as the symmetric "path"-mapping space a la Goerss-Jardine.

Jonathan Beardsley

Well, yeah I'm working in a weird context where I'm not even sure what "homotopy type" is supposed to mean actually, and I'm definitely not working with Kan complexes, but your description is really helpful.

Piotr Pstrągowski

You must be reading a real classic here, which is really cool! I am aware that people used to call simplicial sets different things back in the day, but I never went that far back myself.

Jonathan Beardsley

Haha, well... I'm reading something much more recent, but it references the old stuff, basically because it needs to get away from the modern references which are really focused on Kan complexes and quasicategories, I think.

Oh yeah, okay Goerss and Jardine's loop space is what Lurie calls "Hom_S(x,x)" in HTT

And in the case that S is a quasicategory this is equivalent to the left and right mapping spaces.

That's 4.2.1.8

@PiotrPstrągowski I think this is an even sillier question, but how do you interpret that statement about iterated face maps to be saying that we take the zeroth vertex to the base point?

I'm a little confused by it... I feel like it's saying every possible vertex of our n+1-simplex is the base point?

Piotr Pstrągowski

11:48 PM

Yes, I was being a little bit brief - I think Moore is saying that all vertices are taken to the basepoint.

Jonathan Beardsley

Oh good, okay.

Piotr Pstrągowski

But you already know all except the 0-th one will anyway, by the other condition (the face opposite of the 0-th vertex is degenerate).

Jonathan Beardsley

Ohhhhh... nice, okay.

Piotr Pstrągowski

And if you split conditions like this it's easier to see it's the fibre of $X_{/x} \rightarrow X$.