hi all, are there etale suspension/loop functors which modulate the dimension of the etale fundamental group as $\Sigma,\Omega$ do for $\pi_n(X)$?

@MattCuffaro Functors on what? The étale fundamental group is the fundamental group of a profinite space (the so-called profinite shape of your scheme), and profinite spaces definitely have a suspension-loop adjunction, but maybe you were thinking of something different

do you mind referring me to a source about that adjunction on profinite spaces? I was reading Schmidt and Isaksen where each claim that etale homotopy factors through A1 homotopy, so if A1 homotopy has their own suspension-loop adjunction I was wondering how to extend that to the etale case

to be clear i have a faint idea of what im talking about

Uhm... every pointed model category (or even pointed ∞-category with all finite limits & colimits) has a loop-suspension adjunction. So both profinite spaces and motivic spaces have one. I'm not sure the étale fundamental group factors through motivic spaces though: it is not A¹-invariant (unless in char 0)

See for example Hovey, Model categories 6.1 or Lurie, Higher Algebra 1.1.2.6 (and the paragraph immediately before)

certainly. this is one of the sources I was referring to: mathi.uni-heidelberg.de/~schmidt/papers/63117.pdf

To be clear, ΣX is defined as the (homotopy) pushout of 0←X→0, and similarly ΩX is the (homotopy) pullback of 0→X←0

@MattCuffaro Ah, they complete away from the characteristic, then I agree :)

what does it mean to complete away from a characteristic?

I think they just mean completing at a prime different from the characteristic

i love the spatial flavor of mathematical thinking

to complete "away" from

Unless I'm reading it wrong they are using a very unrefined notion of profinite space in that paper (they're just taking a pro-object in the homotopy category...). I doubt it is important for the purposes of that paper, but we have more refined notions today

what would be a good source for this? as an outsider to the field i have a sort of undifferentiated understanding of the important sources

Hmm.. It depends how aggressively modern you want to be

interesting parameter

let's say very

If you want to work with model categories, Isaksen has a few papers where he sets up the whole thing (e.g. arxiv.org/abs/math/0106152, arxiv.org/abs/math/0404303). But my favourite treatment uses shape theory for ∞-topoi, so that would be section 7.1.6 in Higher Topos Theory and appendix E of Spectral Algebraic Geometry

nontechnical question: how does Lurie and Isaksen get that wonderful curly C for category in LaTeX?

I don't know, sorry :)

It's probably an euscript font. maths.usyd.edu.au/u/SMS/texdoc/euscript.pdf

You might also find the introduction of the recent paper by Barwick, Glasman & Haine helpful: arxiv.org/abs/1807.03281 (this is also in the "aggressively modern" bin)

\usepackage{eucal}

modernity is not how you define things but which latex font you choose

thank you @DenisNardin