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Stream: theory: category theory

Topic: Steiner's Augmented Directed Complexes

Tim Campion (Apr 09 2022 at 21:46):

In a beautiful paper, Steiner identifies a certain full subcategory of the category of strict $\omega$-categories with a certain full subcategory of what he calls augmented directed complexes. An augmented directed complex is a nonnegatively-graded chain complex equipped with an augmentation and a "positive cone" of elements in each degree; morphisms are the obvious things. The functor from strict $\omega$-categories to augmented directed complexes takes the free abelian group on your $\omega$-category $X$, identifies an abelian group object in $Cat_\omega$ with a nonnegatively graded chain complex via a form of the Dold-Kan correspondence, and then picks out the "obvious" augmentation and positive cones.

I've puzzled many times over the question of where does Steiner's construction come from, and I don't have an answer -- does anybody else? My biggest hint is that the adjunction $\mathbb Z[-] : Set ^\to_\leftarrow Ab : U$ -- which is well-known to be monadic -- is also comonadic. So for awhile I tried to convince myself that the induced adjunction $\mathbb Z[-] : Cat_\omega {}^\to_\leftarrow Ab_\omega : U$ is likewise comonadic, and that Steiner is describing the coalgebras for this comonad But of course this adjunction is not comonadic -- it's not conservative. For example, if $G$ is a group, considered as a 1-object category, then $\mathbb Z[G]$ remembers only the abelianization of $G$.

This leaves me wondering whether Steiner has somehow identified "a full subcategory of $Cat_\omega$ on which the restricted adjunction adjunction is comonadic", but I haven't been able to make proper sense of what this would mean. And I still haven't had much luck directly linking Steiner's description of augmented directed chain complexes to this comonad.

Just in case anybody is scared by the words "Dold-Kan correspondence" above, I'll add that that this is not bad: to compute $\mathbb Z[-] : Cat_\omega \to Ch(Ab)$, we take the free abelian group on the set of $n$-cells of our category $C$, mod out by identities and the relation $x \circ_i y = x + y$ for composition in each degree $i < n$, and write down the "obvious" differential as a difference of domain and codomain maps. To lift this to augmented directed complexes, you define the augmentation to be 1 on each object, and you take the "positive cones" to be generated by the $n$-cells for each $n$.

Morgan Rogers (he/him) (Apr 10 2022 at 08:46):

Did you try asking Steiner himself?

Tim Campion (Apr 10 2022 at 17:23):

Morgan Rogers (he/him) said:

Did you try asking Steiner himself?

What a great idea!

David Michael Roberts (Apr 10 2022 at 21:00):

He's a really nice guy, I met him once, in Sydney.

Tom Hirschowitz (Apr 12 2022 at 07:28):

Not an answer, but Simon Forest's PhD thesis might be relevant.

Tim Campion (Apr 13 2022 at 05:03):

Tom Hirschowitz said:

Not an answer, but Simon Forest's PhD thesis might be relevant.

Yeah, Forest's thesis is fantastic! I should take another look there.

Amar Hadzihasanovic (Apr 14 2022 at 17:14):

Hi Tim, I think I can answer the question. You can understand where the ADC construction comes from in the context of earlier papers by Steiner, in particular 1993's “The algebra of directed complexes”.

The original focus of Steiner's work was combinatorial presentations of freely generated $\omega$-categories aka computads or polygraphs.
In analogy with combinatorial presentations of spaces/cell complexes like combinatorial polytopes, the structures on which he originally focussed were “face posets” of $n$-categories which were additionally decorated with orientation information subdividing the boundary of a cell into a source/input half and a target/output half.
From a finitely generated $n$-category, you get one such “oriented face poset” by only remembering what generators are in the source, resp. target boundary of another generator; then you can ask in what cases the $n$-category can be uniquely reconstructed from this face poset.

Steiner found some criteria that ensure this, and these can be seen as defining only the “object” part of a faithful functor from a category of “oriented posets” to the category of $\omega$-categories. One of these criteria is that the face posets are graded: their elements have a well-defined dimension given by the length of maximal chains of faces below them.

The oriented graded poset presentation of $\omega$-categories has one advantage: the Gray product of these has a beautiful, concise presentation, so in the 2004 paper Steiner set out to use this to obtain a new definition of the Gray product of $\omega$-categories (the previous best one, from a 2001 paper by Steiner with Al-Agl and Brown, was even more indirect, going through the equivalence of “globular” with “cubical” $\omega$-categories).

However, to extend the Gray product of oriented posets to $\omega$-categories using Day's theory, he needed a dense functor, therefore a full one. Now functions of these oriented posets could not be enough, because these become functors that send generators to generators; what was missing was the ability to send a generator to a composite of generators.

So what he did was pass from “oriented graded posets” to the “free graded abelian groups” on them, so that the natural notion of morphism between them could send a generator to a sum of generators.

The oriented covering relation of a poset, which could be given as a pair of functions $\Delta^-, \Delta^+: P_n \to \mathcal{P}(P_{n-1})$ sending a rank $n$ element to the sets of its rank $(n-1)$ source, resp. target faces, can then be expressed as a pair of functions $d^-, d^+: \mathbb{Z}P_n \to \mathbb{Z}P_{n-1}$ sending a generator to the sum of its source, resp. target faces; and it turns out that setting $d := d^+ - d^-$ in each grade makes the free graded abelian groups into chain complexes.

Essentially all the additional structure & properties (the “chosen submonoid”, the unital basis, etc.) of ADCs are there to characterise those that arise as free graded abelian groups on the oriented graded posets from the earlier papers.
So in brief:

• the “oriented graded posets” are the “right objects”;
• chain complexes have the “right morphisms”;
• all the extra stuff on chain complexes is there to identify the ones that arise from the “right objects” as free graded abelian groups.

Amar Hadzihasanovic (Apr 14 2022 at 17:16):

(I should specify that “oriented graded poset” is my own terminology from my papers developing on this work and is not how Steiner called them)

Tim Campion (Apr 15 2022 at 04:35):

@Amar Hadzihasanovic Thanks, this is some fantastic context! I will need some time to digest it.

Tim Campion (Apr 15 2022 at 04:36):

I'm really amazed by how well-adapted ADC's are particularly in the context of the Gray tensor product

Tim Campion (Apr 15 2022 at 04:37):

Steiner shows that on a dense subcategory of ADC's, the equivalence to a dense subcategory of $\omega$-categories is not only fully faithful, but monoidal, and not only monoidal, but also preserves internal homs!

Tim Campion (Apr 15 2022 at 04:38):

I really don't know many nontrivial examples of functors which preserve both a monoidal product and the internal hom!

Tim Campion (Apr 15 2022 at 04:40):

Also, to be precise, chain complexes don't quite have the right morphisms -- you need the monoids of positive elements to get this, right?

Amar Hadzihasanovic (Apr 15 2022 at 06:02):

Yes, that's right. Indeed passing to free graded commutative monoids would be enough to get the "missing" morphisms but then you wouldn't get all the nice structure of chain complexes.