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Stream: learning: questions

Topic: Spectra and Linearity

Keith Elliott Peterson (Mar 26 2025 at 19:19):

In 'Higher Topos Theory in Physics', Urs points out that linearity means abelian group structure and in higher gauge theory.

Ignoring all the physics for the moment, and sticking to linear algebra, how exactly does linear structure come about by delooping, and what is the exact relation between abelian/symmetric structure and linearity?

I mean, I doubt many people think of linear algebra as being about abelian structure, at least not it the immediate. Can someone make this a bit more explicit as to what is going on?

Kevin Carlson (Mar 26 2025 at 20:24):

There's an important classical theorem that infinite loop spaces are the same as $E_\infty$-spaces, which are spaces with a multiplication operation that's infinitely coherently homotopy-associative and -commutative. Does that answer the first question?

Kevin Carlson (Mar 26 2025 at 20:26):

I don't really know a precise notion of "linearity" that's relevant to compare to "abelian/symmetric" structure, other than that words like "linear" are often used as rough synonyms for stability, which is the property of $\infty$-categories in which delooping is an auto-equivalence, which in particular gives every object an internal $E_\infty$-structure.

John Baez (Mar 27 2025 at 23:59):

Keith wrote:

I mean, I doubt many people think of linear algebra as being about abelian structure, at least not it the immediate.

Most ordinary folks think about linear algebra as being about vector spaces, but many important features of linear algebra generalize to modules of an arbitrary commutative ring $R$, and when we take $R = \mathbb{Z}$ these are none other than abelian groups. That's the simplest reason why "abelian" and "linear algebra" sound very similar to Urs (and me).

Next, the category of $R$-modules for any ring $R$ is an [[abelian category]].

Then comes the relation to delooping, but I'm out of time for now!

So when Urs is talking about "linearity" he's talking about

Kevin Carlson (Mar 28 2025 at 04:09):

What a cliffhanger.

David Corfield (Mar 28 2025 at 07:42):

One way to see the proximity between the spectral side and linearity is through a construction known as the [[heart]], which picks out the "0-sector" of stable $\infty$-categories. E.g., the heart of the $\infty$-category of $H \mathbb{K}$ module spectra is the category of $\mathbb{K}$-vector spaces. You can see these module spectra as a kind of categorified vector space.

I was having an idle wonder about "truncations" to larger sections via other t-structures. Couldn't we see @John Baez's Baez-Crans 2-vector spaces appear?

John Baez (Mar 30 2025 at 14:41):

That's an interesting thought. I don't know much about t-structures and their hearts, but it looks like there's a problem: A [[t-structure]] on a triangulated category $C$ is a pair $\mathfrak{t}=(C_{\ge 0}, C_{\le 0})$ of [[strictly full subcategories]]

$C_{\geq 0}, C_{\leq 0} \hookrightarrow C$

such that various conditions hold, and then their heart is

$C_{\geq 0} \cap C_{\leq 0}$

If we take $C$ to be the category of $\mathbb{Z}$-graded chain complexes of vector spaces we can take $C_{\geq 0}$ to be the complexes supported on grades $\ge 0$ and $C_{\leq 0}$ to be the complexes supported on grades $\le 0$, and then their intersection is $\mathrm{Vect}$.

We would get 2-term chain complexes (aka 'Baez-Crans 2-vector spaces' it we let $$C_{\geg 0}$$ be the complexes supported on grades $\ge -1$ and keep $C_{\leq 0}$ the same.

However, the definition of a t-structure requires some conditions, including:

• for all $X \in C_{\geq 0}$ and $Y \in C_{\leq 0}$ the hom object is the zero object: $Hom_{C}(X, Y[-1]) = 0$

and that fails with this choice of $C_{\ge 0}, C_{\le 0})$: the overlap is too big to make the overlap go away by shifting $C_{\le 0}$ down one notch!

John Baez (Mar 30 2025 at 15:05):

John Baez said:

Most ordinary folks think about linear algebra as being about vector spaces, but many important features of linear algebra generalize to modules of an arbitrary commutative ring $R$, and when we take $R = \mathbb{Z}$ these are none other than abelian groups. That's the simplest reason why "abelian" and "linear algebra" sound very similar to Urs (and me).

Next, the category of $R$-modules for any ring $R$ is an [[abelian category]].

Then comes the relation to delooping, but I'm out of time for now!

So when Urs is talking about "linearity" he's talking about

I just forgot to delete that last sentence in my rush to finish up!

I've tried to indicate how we can do a generalized version of 'linear algebra' with abelian groups replacing vector spaces. Mathematicians do this all the time; they just don't tend to call it linear algebra.

Then, we can $\infty$-categorify! The simplest way is to replace abelian groups by $\mathbb{N}$-graded chain complexes of abelian groups, since these are the same as

• strict $\infty$-groupoids in the category of abelian groups

or

• abelian group objects in the category of strict $\infty$-groupoids

Here the abelianness is 'tacked on' to the concept of strict $\infty$-groupoid. But - though this may or may not be quite worked out yet - we can look at $\mathbb{Z}$-groupoids, which are a generalization of $\infty$-groupoids that have $n$-morphisms for all $n \in \mathbb{Z}$. A strict $\mathbb{Z}$-groupoid with only one morphism for each $n \lt 0$ deserves to be called a strict stable $\infty$-groupoid, and these should be the same as $\mathbb{N}$-graded chain complexes of abelian groups!

So here abelianness is arising from 'stabilization': the fact that we have a single morphism for each $n \lt 0$. This happens thanks to a version of the [[Eckmann-Hilton argument]].

However, we can get more interesting examples from [[infinite loop spaces]]: these should be the same as non-strict stable $\infty$-groupoids!

The 'infinite loop' stuff is the topological way of thinking about putting in only one morphism for each $n \lt 0$.

So, infinite loop spaces are like a glorified version of chain complexes of abelian groups, and thus yet another setting for glorified 'linear algebra'.

All this generalizes further to spectra. And just as abelian groups generalize to $R$-modules for a commutative ring $R$, which are a nice context for 'linear algebra', spectra generalize to modules of an $E_\infty$ ring spectrum $R$.

I'm pretty sure Urs would consider modules of any $E_\infty$ ring spectrum an example of a context for doing 'linear algebra'.

Keith Elliott Peterson (Mar 31 2025 at 00:33):

May I request this be a "Week"? Sounds interesting.

John Baez (Mar 31 2025 at 05:53):

When did the last issue of "This Week's Finds" come out?

David Corfield (Mar 31 2025 at 06:56):

John Baez said:

However, the definition of a t-structure requires some conditions, including:

• for all $X \in C_{\geq 0}$ and $Y \in C_{\leq 0}$ the hom object is the zero object: $Hom_{C}(X, Y[-1]) = 0$

and that fails with this choice of $C_{\ge 0}, C_{\le 0})$: the overlap is too big to make the overlap go away by shifting $C_{\le 0}$ down one notch!

Yes, it's not t-structures directly we want, but poking around I think I saw people looking to truncate less severely. There should be a truncation to $[0, 1]$, and in some simple case, this should be 2-term chain complexes.