well, at the moment i'm actually studying what i guess one would call deformation theory
there are these things from number theory called formal groups, and i'm more or less studying deformations of them
and formal groups end up being useful in algebraic topology
but i'm not really studying formal groups, i'm studying approximations to formal groups, so like, this thing called a formal lie variety can take an isomorphism from the so-called formal affine line, and this kind of defines a curve on this formal lie variety dude, and then i'm kind of studying like, approximations to that curve
@Eric I would guess it would be in some version of an ando, blumberg, and gepner paper... but I think you wouldn't be asking if it were. What do you mean by V3? the new articles from a couple years ago?
but i don't really know how they arise in number theory. apparently the things i'm talking about are used to prove the main theorem in local class field theory
@JonBeardsley: there are 3 theories running around (more, but whatev) and 2 of them are bousfield equivalent. I thought one of them was a $K$, but it isn't
@JonBeardsley I don't quite understand... We can use it to approximate elliptic curves/integrals and then, if the bounds are sufficiently sharp, get lattice points with such-and-such properties? Perhaps this is a possible application in number theory?
@JonBeardsley Wikipedia page in homotopy shows a continuous transformation from one path to another. "Clearly" (intuitively) there are many ways to do this and many "intermediate steps". Are you taking a well-behaved curve and trying to deform it into a pathological one (with interesting properties?) And how do you construct these transformations explicitly?
@eric: the multiplication-by-eta map is not so bad to compute from $\pi_0$ to $\pi_1$. many people refer to this under the term of Picard groupoids.
essentially, if you have a symmetric monoidal category with $\pi_0$ a group, then there's a map from $\pi_0$ to $\pi_1$
given an element $c$ in the category, there's a canonical identification $Aut(I) \to Aut(c)$ and so you get an element $\tau \in Aut(c \otimes c) \cong Aut(I)$, where $I$ is the unit
for the spectrum associated to this category, $Aut(I)$ is $\pi_1$, and this map $\pi_0 \to \pi_1$ sending $c$ to this twist is multiplication by eta.
ah! right. so Pic(R) is this space parametrizing invertible R-modules; if you pass to the homotopy category you toss away everything except $\pi_0$ and $\pi_1$ of Pic(R)
boy. I don't know a coherent answer to that which doesn't involve calculations with Charles' logarithm
@SeanTilson there is higher homotopy - you just toss it away when you pass to the homotopy category (sorry, I forgot to say that you're taking the category of invertible objects and weak equivalences, not just maps)
even for KU this issue is a little sticky. using log calculations (which are not so bad) I think you can write down the fiber of the map $KU \to L_{E(1)} KU$, which has only finitely many homotopy groups; this is also an equivariant description
the fiber of the log (2-locally) has homotopy groups Z/2, Z/2, Z, Z in degrees -1..3, and so it almost gets you the splitting you need from the bottom of the postnikov tower back to the spectrum
you just need to take the connective cover, and then show that the Z in degree ... argh, in degree ONE splits off as a wedge summand. (sorry, it's -1..2)
@SeanTilson It identifies $\pi_1 Pic(R)$ with $\pi_0(R)^\times$ and $\pi_{n+1} Pic(R)$ with $\pi_n(R)$ for $n > 0$
this is basically some assertion like: the space of $R$-module maps $R \to R$ is $\Omega^\infty R$, and the space of self-equivalence just consists of those path components corresponding to units of $R$
the $\infty$ categorical way is just some way of saying the same thing - this mostly goes back to work on orientation theory (e.g. may-quinn-ray, I think)
it happens to be a very convenient way that avoids a lot of technical details
@Eric so far as KU, there is an analogous result for $GL_1(KU)$ - this is what gets leveraged by the "twisted K-theory" people - essentially one splits off a $Z/2$ semidirect product with a $K(Z,2)$ from $GL_1(KU)$
so far as for the E_n, that's an interesting question
Westerland-Sati split off this "top" homotopy group of $gl_1 E_n$
@Eric mostly trying to help myself by getting someone to figure out these mysteries anyway
@DylanWilson so far as excisiveness, you should be able to get pretty explicit estimates because the homotopy groups of $GL_n$ are explicit in terms of those of $R$
unfortunately, these are functors on rings, and homotopy groups aren't linear in the ring; the correction term is quadratic, though, so calculus should be absolutely applicable
@Drew it turns out not to be a super exciting statement, but it is in section 7 or 8 or whatever the smallness-and-dualizability-phenomena section of hovey & strickland's morava K-theories and localization
it's not phrased quite like that; it's something like "when K(n)_* X is even then E_n^\vee X is too, and also vice versa, and also also E_n^\vee X / I_n = K(n)_* X"
@SeanTilson Waldhausen was the first to define GL_1, although I think that some ABG** variant talks about BGL_n as a space of self-equivalences of modules equivalent to R^n
I thought that was important... but it certainly seems like you should be able to construct GL_n the way you suggest, or BGL_n rather.
right.
I mean, they fit together to form something, like an operad.
that is irrelevant probably.
OK!!! excellent, and that is part of the reason why there is no spectrum bgl_n, because we can't take advantage of two different products that distribute over eachother!
ok, I feel better now.
ie the eckman-hilton dance that andrew stacey has an animation of.
@JonBeardsley in general there's only a space of maps between ring spectra, not a spectrum of them. If S[x] means the free ring on S, then this functor is $\Omega^\infty$, but if it means the monoid algebra $\Sigma^\infty_+ \mathbb{N}$, then it's something much more mysterious.
@TylerLawson I would think it would mean the second. I think what jon is after is some sort of underlying "additive group" of a spectrum. I guess we would need a diagonal on S[x], which we do have since it is a suspension spectrum.
But the monoid algebra structure is inherently multiplicative -- $E_\infty$-maps from $\Sigma^\infty_+ \mathbb{Z}_{\geq 0}$ into an $E_\infty$-ring $R$ are the same as maps of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to \Omega^\infty R$, where $\Omega^\infty R$ is given the multiplicative infinite loop structure, so algebraically this (with the usual diagonal) corresponds to the multiplicative monoid rather than the additive monoid. Is there a way around this?
i'm after something that should be the spectral additive group. however, i've been thinking more about this, and i'm thinking this might be stupid to do
sort of for the reason that spectra are already linear.
i'm not sure though. I just don't really know enough about all of this. and yeah, @TylerLawson I guess I mean the group ring of $\mathbb{N}$, i'm just.... can I think of this as $\mathbb{S}[x]$ by identifying $n$ with $x^n$ or something?
it's also fascinating to me, though perhaps not to others, that $\mathbb{S}[x]$, as $\mathbb{P}(\mathbb{S})$ (this is another notation for what you mean by free ring?), somehow gives you back $\Omega^\infty$. Is that obvious?
@seantilson i think this harper hess paper is a really big effing deal. like. i really really like this framework. it's like quillen's cohomology of commutative rings on steroids
they said in that paper that they're going to write some more about that framework and goodwillie derivatives
so, i'm anticipating that
it ALSO explains Schwede's DB spectra, and sort of why $\pi_\ast(DB)=TAQ(HB)$
but Schwede's picture is really nice, in the sense that does things in terms of gamma rings, which is sort of... more algebro-geometric
or like... sehafy
*sheafy
moreover, i suspect that in the context of $\infty$-categories, one could talk about the completion tower of a category, combining harper and hess' stuff as well as lurie's stuff on deformation theory
it's funny, I think Lazard's old stuff is super applicable here, but just needs to be generalized, he basically talks about tangent vectors of curves on formal varieties, which is not super interesting because over smooth things this is just (somehow) the additive group, basically (abelianization-ish stuff??), and goodwillie calculus as deformation theory is basically the derived version of this
