The n-Category Café

Let’s see. Parts of my questions have an obvious answer. Let me try:

Let $V$ be a closed monoidal category such that we can enrich over it. Assume in addition that $V$ has duals on objects.

Then for $C$ any $V$-category with composition $${\circ }_{x,y,z}:\mathrm{Hom}(x,y)\otimes \mathrm{Hom}(y,z)\to \mathrm{Hom}(x,z)$$

we obtain a dual co-catgeory enriched over $V$ simply by hitting everything in sight with the duality operation. So we get a co-composition $${\circ }_{x,y,z}^{*}:\mathrm{Hom}(x,z{)}^{*}\to \mathrm{Hom}(x,y{)}^{*}\otimes \mathrm{Hom}(y,z{)}^{*}$$ obviously.

Hm. And I guess if I do everything internal to $V$, the same general statement is true. And if I then imagine I have an $\omega$-category internal to $V$, the same kind of statement is still true.

That was easy enough.

So next suppose that we have one $\omega$-category acting on another. Say $G$ is an $\omega$-category with a single object and we have a morphism

$$G\to \mathrm{End}(C).$$

Then, somehow, we want to dualize $G$, probably obtaining a co-$\omega$-category this way , and then merge the result with $C$ to some single new entity.

What might that be?

Urs wrote:

Great cosmic coincidence! Let’s talk about this stuff.

Okay, great! You seem to have sensed that “week258” was secretly about $\omega$-categories — or maybe you just read the little hint in this blog entry. I didn’t want to get into that aspect explicitly in “week258”, since it would further clutter what was already an overly jargonesque explanation of what ultimately is a very simple idea.

But, whenever I think about chain complexes, I’m secretly thinking of them as specially nice strict $\omega$-categories.

Okay…

So let $C$ be our abelian category.

The Dold-Kan correspondence says that we have an equivalence of categories $$\mathrm{Simp}(C)\simeq {\mathrm{Ch}}_{•}(C)$$ between simplicial objects in $C$ and non-negatively graded chain complexes in $C$. The direction $$\mathrm{Simp}(C)\to {\mathrm{Ch}}_{•}(C)$$ is forming the “normalized chain complex” which corresponds to throwing away lots of face maps except for one at each level, restricting that to the kernel of some of the others, and regarding it as the differential in our complex.

Right.

Now, first question: how precisely do I connect this to higher categories now?

I want to combine this with the nerve functor $$\omega \mathrm{Cat}(C)\stackrel{\mid \cdot \mid }{\to }\mathrm{Simp}(C)$$ which sends an $\omega$-category internal to $C$ to its simplicial nerve. I want to get an equivalence $$\omega \mathrm{Cat}(C)\simeq {\mathrm{Ch}}_{•}(C)$$ this way. Do I?

Yes you do! Here I’m assuming you mean strict $\omega$-categories.

You can use a nerve construction as you outline. But, it’s equally easy to see directly that the category

• $\omega \mathrm{Cat}(C)$: strict $\omega$-categories internal to the abelian category $C$, with strict $\omega$-functors as morphisms.

is equivalent to the category

• ${\mathrm{Ch}}_{•}(C)$: nonnegatively graded chain complexes in $C$, with chain maps as morphisms.

It’s fun and instructive to build this equivalence ‘by hand’, reinterpreting globes in terms of chains. But if you’re too busy, there’s a paper by Brown and Higgins that does it:

• Ronald Brown and Philip J. Higgins, Cubical abelian groups with connections are equivalent to chain complexes.
  
  Abstract: The theorem of the title is deduced from the equivalence between crossed complexes and cubical $\omega$-groupoids with connections proved by the authors in 1981. In fact we prove the equivalence of five categories defined internally to an additive category with kernels.

I believe that three of these five are $\omega \mathrm{Cat}(C)$, ${\mathrm{Ch}}_{•}(C)$ and $\mathrm{Simp}(C)$, while the others have a cubical flavor. But if you prefer octahedra and their higher-dimensional generalizations, you can probably use them too — this idea is incredibly robust!

You might enjoy this passage in HDA6 more now that you’re thinking about this stuff:

Since 2-vector spaces are equivalent to 2-term chain complexes, as described in Section 3, it should not be surprising that $L_{\mathrm{\infty }}$-algebras are related to the categorified Lie algebras we are discussing here. An elegant but rather highbrow way to approach this is to use the theory of operads [MSS]. An $L_{\mathrm{\infty }}$-algebra is actually an algebra of a certain operad in the symmetric monoidal category of chain complexes, called the ‘$L_{\mathrm{\infty }}$ operad’. Just as categories in $Vect$ are equivalent to 2-term chain complexes, strict $\omega$-categories in $Vect$ can be shown equivalent to general chain complexes, by a similar argument [BH]. Using this equivalence, we can transfer the $L_{\mathrm{\infty }}$ operad from the world of chain complexes to the world of strict $\omega$-category objects in $Vect$, and define a semistrict Lie $\omega$-algebra to be an algebra of the resulting operad.

I also sketched how to see a chain complex in $C$ as a strict $\omega$-category in $C$ starting at the bottom of page 18 here:

• John Baez, Higher Gauge Theory, Homotopy and $n$-Categories.

This is getting long, so I’ll quit this comment here and talk about other things in another comment. Your question:

So: in the context that $\omega$-categories correspond to non-negatively graded chain complexes, what corresponds to positively graded chain complexes?

was very much on my mind while writing “week258”. Or at least it would have been if you meant to write “what corresponds to non-positively graded chain complexes?” Is that what you meant?

Urs meant to write:

So: in the context that $\omega$-categories correspond to non-negatively graded chain complexes, what corresponds to non-positively graded chain complexes?

Maybe $\omega$-co-categories??

One way to tackle this uses the fact that whenever $C$ is an abelian category, so is $C^{\mathrm{op}}$. The people who invented this concept cleverly made the definition self-dual!

So, here goes:

First, a non-positively graded chain complex in some abelian category $C$ is the same as a non-negatively graded cochain complex in $C$.

Second, a non-negatively graded cochain complex in $C$ is the same as a non-negatively graded chain complex in $C^{\mathrm{op}}$.

Third, a non-negatively graded chain complex in $C^{\mathrm{op}}$ is the same as a strict $\omega$-category in $C^{\mathrm{op}}$.

So, a non-positively graded chain complex in $C$ is the same as a strict $\omega$-category in $C^{\mathrm{op}}$.

I suppose you could call this an ‘$\omega$-co-category’ in $C$, but fewer people would understand you — like, 3 instead of 10.

However, we don’t always want to separate the positive and negative parts of a chain complex and treat them separately! What if we’ve got a full-fledged, $\mathbb{Z}$-graded chain complex in $C$?

This gives a strict $\mathbb{Z}$-category in $C$!

A $\mathbb{Z}$-category has $n$-morphisms for $n$ going down all the way to $-\mathrm{\infty }$, as well as all the up to $+\mathrm{\infty }$.

Alas, the only people I know who have discussed $\mathbb{Z}$-categories are James Dolan and myself. And, we haven’t written too much about them — just a few remarks in various papers.

It’s not hard to define strict $\mathbb{Z}$-categories, internalize them, and prove that a $\mathbb{Z}$-graded chain complex in $C$ is the same as a strict $\mathbb{Z}$-category in $C$.

The trickier part is to make sure that a strict $\mathbb{Z}$-groupoid works out to be precisely a $\mathbb{Z}$-graded chain complex of abelian groups. You need to make sure the Eckmann–Hilton theorem kicks in and makes everything abelian.

The really tricky thing would be to define weak $\mathbb{Z}$-groupoids in a purely algebraic way, and then make sure they’re the same as spectra. In the field of ‘stable homotopy theory’, algebraic topologists use spectra as a generalization of chain complexes in precisely this way. The word ‘stable’ means ‘with homotopy groups going infinitely far down’. But, these guys don’t define spectra starting from a theory of weak $\mathbb{Z}$-categories. Not yet, anyway.

There’s a lot more to say about this… Most things about weak $\mathbb{Z}$-categories would be hard to make precise at present. But, many things involving strict $\mathbb{Z}$-categories should be quite easy to make precise.

Thanks, Mikael — that’s extremely helpful!

I think this means that the $A_{\mathrm{\infty }}$-algebra structure that everyone uses on ${\mathrm{Ext}}_R^{*}(M,M)$ agrees with the ‘conceptually simple’ one that I proposed in “week258”.

My way — for which the details haven’t been worked out — was to associate an $A_{\mathrm{\infty }}$-category to any sufficiently nice abelian category. For simplicity, let’s start with the category of $R$-modules for some ring $R$. Then my idea was to build an $A_{\mathrm{\infty }}$-category with $R$-modules as objects, but a new hom, the ‘derived function complex’

$$\mathrm{HOM}(x,y)=\mathrm{hom}(Px,y)$$

Here $Px$ is a nice functorial projective resolution of $x$, say the free resolution. We think of $Px$ and $y$ as chain complexes of $R$-modules, and let $\mathrm{hom}(Px,y)$ be the cochain complex of abelian groups where the $n$-chains are degree-$(-n)$ maps of graded $R$-modules from $Px$ to $y$… not necessarily chain maps!

I believe there’s a reasonable way to define a composition

$$\circ :\mathrm{HOM}(x,y)\otimes \mathrm{HOM}(y,z)\to \mathrm{HOM}(x,z)$$

and I think this gives an $A_{\mathrm{\infty }}$-category. It could in fact be a mere differential graded category, which would make what I’m saying sound a bit pompous. But either way, this would make

$$\mathrm{HOM}(x,x)$$

into an $A_{\mathrm{\infty }}$-algebra (for example a dg algebra). Then, taking cohomology and using the Kadeishvili theorem, we see that

$${\mathrm{Ext}}^{*}(x,x)=H^{*}(\mathrm{HOM}(x,x))$$

is an $A_{\mathrm{\infty }}$-algebra.

I guess the only difference between this stuff and what you’re saying is that I’m using

$$\mathrm{hom}(Px,x)$$

where you are using

$$\mathrm{hom}(Px,Px)$$

Hmm! These should be equivalent as $A_{\mathrm{\infty }}$-algebras, but the multiplication is obviously associative ‘on the nose’ in yours, so yours is simpler.

So, instead of defining

$$\mathrm{HOM}(x,y)=\mathrm{hom}(Px,y)$$

maybe I should define

$$\mathrm{HOM}(x,y)=\mathrm{hom}(Px,Py)$$

to make composition easier to define, and get a dg category instead of an $A_{\mathrm{\infty }}$-category. It’s probably ‘equivalent’ (in some relaxed sense of equivalence suited to $A_{\mathrm{\infty }}$-categories), but more manageable.

Indeed, everything I’m doing has probably been done better already by the experts.