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September 15, 2010

Grothendieck-Maltsiniotis ∞-categories

Posted by Mike Shulman

Yesterday Georges Maltsiniotis posted a paper on the arXiv in which he presents a definition of \infty-groupoid, said to be due to Grothendieck in Pursuing Stacks, and modifies it to give a similar definition of \infty-categories. (These definitions have been available on his website for a while, but only in French.)

These “Grothendieck-Maltsiniotis” definitions are quite similar to that of Batanin, especially as modified by Leinster. The precise relationship between the two is studied in the thesis of D. Ara, a student of Maltsiniotis, but it’s not hard to get an intuitive idea of their similarities and differences.

Both definitions are alike in that they have, as underlying data, a globular set XX, equipped with extra algebraic structure. Moreover, in both cases this algebraic structure consists of operations whose

  • input is a “globular pasting diagram” of cells in XX, and whose
  • output is a single cell in XX.

Recall that a globular pasting diagram is “an arrangement of cells which would have a unique composite in a strict \infty-category.” There are lots of pictures of these in the Cheng-Lauda guidebook.

The two definitions are again similar in that they are each not really a single definition, but a family of them. In Batanin’s world you get one definition for each globular operad; in Maltsiniotis’ world you get one definition for each “globular theory.” And again in both cases, not every operad/theory gives you a good definition of “weak \infty-category;” you need two additional conditions:

  • All composites and coherences that should exist, do exist. In particular, there is a way to compose any globular pasting diagram, and any two ways to compose a given diagram are related by specified equivalences.
  • There are no unwanted equations between operations. This is what gives us a definition of weak \infty-category; in both cases, there is an operad/theory for strict \infty-categories which satisfies the first condition, so we need to exclude it somehow.

Finally, in both cases these conditions can be formulated in terms of a weak factorization system on the category of globular operads/theories, in terms of which we are interested in cofibrant replacements of the operad/theory for strict \infty-categories. The first condition says that the map T weakT strictT_{weak}\to T_{strict} is an “acyclic fibration,” while the second says that T weakT_{weak} is cofibrant.

So much for the similarities; how about the differences? It seems to me that there are two main differences. The first lies in the treatment of dimensions. In Batanin’s world, every operation takes a globular pasting diagram of some dimension nn as input and gives a cell of the same dimension nn as output. However, there are operations we expect to exist which do change dimension: for instance, the identity of an nn-cell is an (n+1)(n+1)-cell, while its source and target are (n1)(n-1)-cells. Coherence isomorphisms are also operations which raise dimension; for instance, the standard biased associator for 1-cell composition takes a 1-dimensional pasting diagram as input (three composable arrows) and gives a 2-cell as output (the associator relating two composites). Batanin’s definition deals with this in two ways:

  1. The globular-set structure of XX, including sources and targets, is given a priori before any globular operad acts on it. Every nn-dimensional operation in a globular operad also has a source and target, which are (n1)(n-1)-dimensional operations, and the action of “applying an operation to a pasting diagram” respects sources and targets.
  2. A pasting diagram comes with a specified dimension, which may be greater than the highest dimension of cell occurring in it. Thus, for instance, the associator is a 2-dimensional operation, whose source is the pasting diagram \to\to\to regarded as a 2-dimensional pasting diagram which happens to contain only 1-cells.

(Of course, phrased this way it sounds very artificial, but in fact it’s all quite natural when you write it using generalized multicategories relative to the free strict \infty-category monad. An nn-dimensional globular pasting diagram in a globular set XX is an nn-cell in the free strict \infty-category on XX, and in this strict \infty-category every nn-cell comes with an identity (n+1)(n+1)-cell, which has an identity (n+2)(n+2)-cell, and so on; these identity pasting diagrams give a very natural way to “regard a pasting diagram as having a higher dimension.”)

In the Grothendieck-Maltsiniotis definition, on the other hand, operations are not required to preserve dimension, and in fact pasting diagrams have no specified “dimension,” although we could call their dimension equal to the highest dimension of a cell occurring in them (if we so wished). Now identities, sources and targets, and coherence equivalences are all “operations” on an equal footing, and in particular the globular set structure of XX is not formally separated from the action of a globular theory on it.

The second difference lies in how “contractibility” (the existence of composites, identities, and coherence) is specified. Following Leinster, let us say that a globular operad is contractible if for any two operations uu and vv of the same nn-dimensional input shape π\pi, and any (n+1)(n+1)-dimensional input shape σ\sigma whose source and target is π\pi, there exists an operation ww of shape σ\sigma whose source and target are uu and vv. (Batanin’s notion of contractibility was more biased.) It’s easy to see how this gives us composition: given any shape σ\sigma we want to compose, if we decide on a way to compose up its boundaries, there is some compatible way to compose it as well. We get identities from “degenerate” pasting diagrams, as remarked above. Moreover, we also get coherences from degeneracies: given any two operations uu and vv which compose the same nn-dimensional shape, let σ\sigma be the corresponding degenerate (n+1)(n+1)-dimensional shape; then ww is a coherence cell mediating between uu and vv.

On the other hand, in the Grothendieck-Maltsiniotis world, we say that a globular theory is contractible if given any two operations uu and vv with the same input shape π\pi and whose outputs are parallel nn-cells, there exists an operation ww with the same input shape π\pi and whose output is an (n+1)(n+1)-cell from uu to vv. In this case, it is obvious how we get the coherences: given any two ways to compose the same shape, we get immediately a coherence cell relating them. Composition is actually also easy: if we have a pasting diagram of shape π\pi we want to compose, and we have ways uu and vv to compose up its source and target, then uu and vv are again operations with the same input shape and we get an operation ww, of output one dimension higher, from one to the other. Finally, we get an identity (n+1)(n+1)-cell for any nn-cell by taking uu and vv both to be the identity operation taking an nn-cell to itself.

Note that the fact that sources and targets are just ordinary operations in the Grothendieck-Maltsiniotis world means that a priori, they are completely symmetric. In particular, if we take uu to be the target of a single nn-cell and vv its source, then we obtain an operation assigning to any nn-cell, another nn-cell in the opposite direction, and the coherence laws make the two coherently inverse. This is why the first version of the definition is actually a definition of \infty-groupoid. To get a definition of \infty-category, you have to break the symmetry between uu and vv in an appropriate way, which Maltsiniotis describes, but which I haven’t digested yet.

To conclude, here is an attempted translation of part of Theorem 6.6.8 from Ara’s thesis: The functor MM induces an equivalence of categories between the category of homogeneous globular theories over Θ\Theta and the category of ω\omega-operads…. Moreover, if CC is a homogeneous globular theory over Θ\Theta then the categories Mod(C)Mod(C) and M CM_C-Alg are canonically isomorphic.

And here is part of Corollary 6.7.11: Let CC be a homogeneous globular theory over Θ\Theta and MM the associated ω\omega-collection. The ω\omega-collection MM is contractible if and only if the globular theory CC is a pseudo-coherator.

I haven’t yet read the definitions of “homogeneous globular theory over Θ\Theta” or the functor MM or “pseudo-coherator,” but at least “homogeneous” sounds like a condition relating to the “treatment of dimensions” discussed above. So this seems to be saying that the Grothendieck-Maltsiniotis family of definitions essentially includes Batanin’s. I haven’t reproduced the actual definitions here, but they don’t seem particularly difficult. They do feel a little more ad hoc to me than (Leinster’s rephrasing of) Batanin’s definition, but possibly that’s just what I’m familiar with; I’d be interested to hear other people’s reactions.

Posted at September 15, 2010 1:06 AM UTC

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63 Comments & 2 Trackbacks

Re: Grothendieck-Maltsiniotis ∞-categories

Thanks, Mike. While I don’t have time for this now, here is an nnLab entry:

Grothendieck-Maltsiniotis \infty-category.

By the way, the links to the ENS-server with Maltsiniotis’ previous preprint and the thesis of Ara don’t work for me. Anyone else have this problem? Maybe it’s just my machine being stubborn.

Posted by: Urs Schreiber on September 19, 2010 9:46 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

The link to Dimitri’s thesis at normalesup.org works for me. The link to Georges’s web page at math.jussieu.fr doesn’t.

Posted by: Tom Leinster on September 19, 2010 4:36 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

On page 2 Maltsiniotis recalls the complaint about non-algebraic definitions of \infty-groupoids:

The [homotopy-hypothesis] is still not proven, for any definition of \infty-groupoid giving rise to an algebraic structure species, although some progress has been done in this direction by Cisinski [4]. It becomes tautological if we define \infty-groupoids as being Kan complexes or topological spaces! But the categories of such are not locally presentable.

I am wondering if this statement is still true, in view of Nikolaus’ algebraic Kan complexes.

Posted by: Urs Schreiber on September 19, 2010 10:02 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

In particular, since we can write down retracts of simplices Δ n n+1\Delta^n \hookrightarrow \mathbb{R}^{n+1} onto horns, this is enough to make the singular set S(X)sSetS(X)\in sSet of a space XX an algebraic Kan complex. I wonder if XX is filtered (say by being a CW-complex), and one considers the simplicial set of filtered maps S filt(X):=Top filt(Δ ,X)S_{filt}(X) := Top_{filt}(\Delta^\bullet,X) (where the nn-simplex is given its canonical filtration), whether this is homotopy equivalent to the full singular set S(X)S(X)? Also, I suppose a priori one doesn’t know whether this is Kan, or if it can be made an algebraic Kan complex, but I would wager it is. This would also have nice links to how Ronnie Brown (et al) likes to approach higher groupoids, i.e. via filtered spaces.

Assuming that S filt(X)S_{filt}(X) can be made into an algebraic Kan complex, I would posit it could be half of the Quillen equivalence AlgKanCWAlgKan \leftrightarrows CW needed for the homotopy hypothesis.

But the categories of such are not locally presentable.

Is this true of algebraic Kan complexes? (It probably follows, if it does, from AlgKan being a category of algebras for a monad) If so, we should put it one the nLab page.

Posted by: David Roberts on September 19, 2010 10:50 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

In particular, since we can write down retracts of simplices Δ n n+1\Delta^n \hookrightarrow \mathbb{R}^{n+1} onto horns, this is enough to make the singular set S(X)sSetS(X) \in sSet of a space X an algebraic Kan complex.

Indeed! That’ exactly how Thomas in section 3.3. establishes the homotopy hypothesis for algebraic Kan complexes as a direct Quillen equivalence

Π :TopAlgKan:||. \Pi_\infty : Top \stackrel{\leftarrow}{\to} Alg Kan : |-| \,.

So as you write:

I would posit it could be half of the Quillen equivalence AlgKanCWAlg Kan \stackrel{\leftarrow}{\to} CW needed for the homotopy hypothesis.

Yes, that’s what Thomas proves. Have a look!

But the categories of such are not locally presentable.

Is this true of algebraic Kan complexes?

That’s exactly what i am thinking.

(It probably follows, if it does, from AlgKan being a category of algebras for a monad) If so, we should put it one the nLab page.

Yes, exactly. Thomas proves that AlgKanAlg Kan is the category of algebras over a monad on sSetsSet. Since sSetsSet is locally presentable, so AlgKanAlg Kan should be (by the standard facts collected at locally presentable category).

Posted by: Urs Schreiber on September 19, 2010 11:01 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

The relevant part is:

If TT is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category AA, then the category of algebras A TA^T is locally presentable. In particular, if AA is locally presentable and i:BAi:B\to A is a reflective subcategory, then BB is locally presentable if ii is accessible.

since sSetsSet is locally presentable, and hence accessible (I’ve just learned), then all we need is that the endofunctor UF:sSetsSetU\circ F:sSet \to sSet underlying the ‘algebraic fibrant object’ monad is κ\kappa-continuous for a regular cardinal κ\kappa. I presume this means preserving limits of diagrams of shape ‘bounded by κ\kappa’.

But since limits and colimits in AlgKanAlgKan are calculated as in sSetsSet, then UFU\circ F preserves them, and we are done: AlgKanAlgKan is locally presentable. Nice!

Posted by: David Roberts on September 19, 2010 12:15 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

But since limits and colimits in AlgKanAlg Kan are calculated as in sSetsSet

The limits and filtered colimits are. But I think that’s what we need for the proof that you sketch:

to deduce that ALgKanALg Kan is locally presentable we want that UFU \circ F preserves filtered colimits.

But FF is left adjoint and hence preserves all colimits. And UU has the special property that it preserves not only all limits but also all filtered colimits (prop 2.11).

Posted by: Urs Schreiber on September 19, 2010 12:52 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

To ask a dense question: what is κ\kappa-continuity of a functor? A priori I would assume it means preserves limits bounded by κ\kappa (in some sense), and so we don’t need to talk about colimits…

Posted by: David Roberts on September 19, 2010 1:25 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

what is κ\kappa-continuity of a functor?

It’s what you think it is. The trouble is, as I see now, that there was some typo propagating and you apparently ran into it.

The various nnLab pages give the definition serveral times, for instance at accessible functor: it has to preserve filtered/directed colimits. (Which makes sense, since their existence is the relevant properties of accessible categories.)

At “accessible category” the definition was given three times. Twice correct. Once wrong. ;-) In case that worries you, check out definition 2.16 in Adamek-Rosicky Locally presentable and accessible categories (Google books has it).

But apart from this, as Thomas points out (who can’t reply himself right now since he is busy sight-seeing in Vienna) using that UU preserves filtered colimits it is immediate to check directly that AlgKanAlg Kan is locally presentable.

Posted by: Urs on September 19, 2010 2:55 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Urs wrote:

But apart from this, as Thomas points out (who can’t reply himself right now since he is busy sight-seeing in Vienna) using that U preserves filtered colimits it is immediate to check directly that AlgKan is locally presentable.

Thank you Urs, Vienna is in fact very nice ;) And it is indeed true, that algebraic fibrant Kan complexes are locally presentable. Algebraic quasi-categories etc are also locally presentable.

Mike wrote:

I agree that algebraic Kan complexes are locally presentable (in fact, locally finitely presentable), and that this renders Maltsiniotis’ statement incorrect as literally stated. However, of course, as we’ve discussed elsewhere, there’s at least an intuitive sense in which algebraic Kan complexes are only “barely algebraic”, so that knowing the homotopy hypothesis for them is not that big of a step into the algebraic world.

I am still not sure whether I understand this point correctly. Before we had this algebraic-Kan-complex-model there was an agreement in the community (at least here) that the definition of ‘algebraic model’ is that it is monadic over a shape category. I think it would be easy to find statements as such in older cafe-posts. That is the reason I proved the statement that algebraic Kan complexes are monadic over simplicial sets. Furthermore besides this abstract fact, algebraic Kan complexes are algebraic in a very concrete sense, as Urs points out. We can get every compositor, associator or coherence cell from horn filling diagrams. We can really write it down!

Maybe the intuitive sense in which they are not algebraic is related to the shape category, i.e. simplicial sets? I also like more to think globular since that is closer to what we do in lower dimensions. But it turns out that many things are technically easier in the simplicial world…

Posted by: Thomas Nikolaus on September 20, 2010 9:18 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

I agree that algebraic Kan complexes are locally presentable (in fact, locally finitely presentable), and that this renders Maltsiniotis’ statement incorrect as literally stated. However, of course, as we’ve discussed elsewhere, there’s at least an intuitive sense in which algebraic Kan complexes are only “barely algebraic”, so that knowing the homotopy hypothesis for them is not that big of a step into the algebraic world.

Moreover, as I am fond of emphasizing, if one wants to state the homotopy hypothesis relative to “a notion of \infty-groupoid” (rather than having some specific one in mind) one has to make use of certain preconceptions about what constitutes such a notion, and furthermore (I believe) about what constitutes an “equivalence” of such groupoids. To take a trivial example, the category of all simplicial sets is certainly locally presentable, and its localization at its weak equivalences is again equivalent to that of topological spaces! But that certainly doesn’t count.

Perhaps someone should contact Maltsiniotis and suggest that he revise this sentence? We could also suggest that he modify his use of “lax” to “pseudo” / “weak” in accordance with standard categorical conventions.

Posted by: Mike Shulman on September 19, 2010 3:47 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

there’s at least an intuitive sense in which algebraic Kan complexes are only “barely algebraic”

That seems to depend on your intuition. Or can you formalize this at least roughly?

Because my intuition is very different. Think about what an algebraic Kan complex is: it is (among other things)

  • a choice of composition operation on all 1-morphisms;

  • a choice of associator 2-morphisms for all triples of 1-morphisms;

  • a choice of pentagonator 3-morphism for all quadruples of 1-morphisms;

and so on.

That’s as algebraic as it gets, it seems to me. It is in a sense more algebraic than all the operadic definitions even! Because this makes specific unique choices for all this data, where operadic definition keep around the spaces of choices without picking unqiue composites etc. The operadic definitions are really more like geometric definitions with a strong algebraic control on how the various spaces of choices interact not with an algebraic specification of the choices themselves.

Posted by: Urs Schreiber on September 19, 2010 4:17 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

It is in a sense more algebraic than all the operadic definitions even! Because this makes specific unique choices for all this data, where operadic definition keep around the spaces of choices without picking unqiue composites etc.

I don’t agree with this sentiment. I would only say there may be more operations around in the operadic definitions (where the use of the word “operation” replaces the phrase “choice of composite”).

Posted by: Todd Trimble on September 19, 2010 5:24 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

It is in a sense more algebraic than all the operadic definitions even! Because this makes specific unique choices for all this data, where operadic definition keep around the spaces of choices without picking unqiue composites etc.

I don’t agree with this sentiment. I would only say there may be more operations around in the operadic definitions (where the use of the word “operation” replaces the phrase “choice of composite”).

I certainly see what you mean. But then, in the same vein I could say this about the geometric approaches: I could say the lack of specified composite in a Kan complex really is to be read as there being many composition operations – one for every horn filler.

Methinks therefore if we want to distinguish approaches, we need a formal distinction, not one based just on intuition. So far I was aware of the formal definition that said that a definition of \infty-groupoid is algebraic, if the things that are called \infty-groupoids are precisely the algebras over a monad. That makes algebraic Kan complexes an algebraic model.

Now here is expressed the sentiment that while with this definition that’s true, somehow algebraic Kan complexes are nevertheless less algebraic than, say, Grothendieck-Maltsiniotis \infty-groupoids.

Is this just a feeling, or can one give this statement a precise meaning?

Posted by: Urs Schreiber on September 19, 2010 6:28 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Urs wrote:

But then, in the same vein I could say this about the geometric approaches: I could say the lack of specified composite in a Kan complex really is to be read as there being many composition operations – one for every horn filler.

Under the axiom of choice, yes, for the same reason that a total relation contains many functions (which is equivalent to the axiom of choice). But as far as I can see, some form of the axiom of choice is essential for that.

Normally I would consider a notion (essentially) algebraic if it can be interpreted in general categories with finite limits (or we can be generous and allow limits). I don’t see how to do that with the geometric definitions. (After algebraizing them with the help of AC, then of course you arrive at an essentially algebraic notion which can be interpreted in categories with finite limits, but that’s only after algebraizing them.)

In short, it seems to me that the geometric notions require at least geometric logic (in the sense of topos theory) in order to express them, and a choice principle to algebraize them.

Posted by: Todd Trimble on September 19, 2010 9:27 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

What exactly do you mean be “algebraizing”? Why can’t we simply make the definition of algebraic Kan complexes in categories with limits (or better fibre products).

Posted by: Thomas Nikolaus on September 20, 2010 12:10 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Sorry for the confusion. I was trying to say, or should have said (1) that the notion of geometric Kan complex itself, to be expressed internally in a category, requires geometric logic (is that right?), and (2) in order to prove that any geometric Kan complex in a category that supports this logic can be “algebraized” (can be given a compatible algebraic Kan complex structure), one is forced to make a bunch of choices of composites, which apparently requires a choice principle. That’s what I meant by algebraizing, but it came out garbled, since the notion of algebraic Kan complex is itself essentially algebraic, as you point out.

This is in response to Urs, who said, “But then, in the same vein I could say this about the geometric approaches: I could say the lack of specified composite in a Kan complex really is to be read as there being many composition operations that you could think of a geometric Kan complex as having many operations.” This seems analogous to the statement that a total relation from a set AA to a set BB is to be read as there being many functions from AA to BB dominated by that relation.

Posted by: Todd Trimble on September 20, 2010 1:24 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

I was trying to say, or should have said (1) that the notion of geometric Kan complex itself, to be expressed internally in a category, requires geometric logic (is that right?)

Let me see if I understand this:

Suppose CC a category with finite limits. X:Δ opCX :\Delta^{op} \to C a simplicial object. The objects of horns are limits of the sort X Λ 1[2]=X 1× X 0X 1X^{\Lambda^1[2]} = X_1 \times_{X_{0}} X_1 etc.

Say XX is an internal Kan complex if all morphisms X Δ[k]X Λ i[k]X^{\Delta[k]} \to X^{\Lambda^i[k]} are split epimorphims.

Say XX equipped with a bunch of splitting morphisms σ k i:X Λ i[k]X Δ[k]\sigma^i_k : X^{\Lambda^i[k]} \to X^{\Delta[k]} is an internal algebraic Kan complex.

No?

Posted by: Urs Schreiber on September 20, 2010 4:51 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

I wrote:

Let me see if I understand this

Thinking about it, I guess I am not allowed to say “split epimorphism” in an algebraic theory in the sense you are thinking of.

Posted by: Urs Schreiber on September 20, 2010 5:04 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

That puts a new spin on grammarians’ ban of “split infinitives.”

Posted by: Jonathan Vos Post on September 20, 2010 6:56 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Nothing stops you from talking about split epimorphisms - there just may not be enough of them. Or more importantly, they may not be the epimorphisms you are looking for. ;-) To compare, think about internal equivalences of groupoids compared with essential equivalences. It is the latter concept that one wants.

Posted by: David Roberts on September 21, 2010 1:10 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Nothing stops you from talking about split epimorphisms -

I think Todd’s demand not to use geometric logic (and hence no existential quantifiers) stops me.

there just may not be enough of them.

I know well that if I want to use internal \infty-groupoids I should be a bit more sophistciated. We talked about it at length before.

But here were talking about whether saying “algebraic Kan complex” requires “geometric logic”.

Posted by: Urs Schreiber on September 21, 2010 1:34 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Todd didn’t say that algebraic Kan complexes require geometric logic; he said that ordinary (geometric) ones do, and that if you want every such geometric Kan complex to be “algebraicizable” then your geometric logic needs a choice principle.

Posted by: Mike Shulman on September 21, 2010 2:07 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Todd didn’t say that algebraic Kan complexes require geometric logic

So yes or no:

do algebraic Kan complexes constitute an algebraic model for \infty-groupoids in Todd’s sense?

That’s all I am trying to understand here.

Posted by: Urs Schreiber on September 21, 2010 6:54 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

For what it’s worth (I am Todd after all!), the sense I was using for this discussion is “essentially algebraic” or “definable by a finite limit sketch”. While there might be more than one notion of algebraic Kan complex, if we fix such a notion, then it is definable by a finite limit sketch.

As for split epimorphism: I understand that as meaning there exists a section (as opposed to having a section given). This is not algebraic in the sense above. (There is also a question, as David Roberts pointed out, of what should be the right way of internalizing the notion of geometric Kan complex, but I’m not going to touch that here.)

Posted by: Todd Trimble on September 21, 2010 11:57 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Urs asked

do algebraic Kan complexes constitute an algebraic model for ∞-groupoids in Todd’s sense?

As Todd points out: if the split epimorphisms are equipped with sections as part of the data, then they would be definable ‘algebraically’ (i.e. in a finitely complete category), but not if they are just postulated to exist. But that is beside the point, if ordinary internal Kan complexes (for a given notion of epimorphism, or fibration for that matter) form a category inequivalent to the category of internal algebraic Kan complexes (equipped with sections of the split epis).

But then as Mike bemoans, I’m really more interested in the Grothendieck-Maltsiniotis definition, and how it relates to what I know. For example, what is a G-M weak 2-category, and how does it related to a bicategory? (For that matter, a G_M 1-category?) How about internal bicategories?, seeing as the G-M definition seems to ‘live in’ an ambient category (or does it?) Can we talk about higher dimensional arrows between even strict G-M n-categories/n-groupoids? A map of G-M oo-groupoids is a map of the underlying (Set-valued) presheaves. What is a 2-arrow?

Can we have a presheaf of G-M oo-groupoids? (of n-groupoids?) How does this relate to a simplicial presheaf? Is the category of G-M oo-groupoids an (oo,1)-category?

Can we replace Set (the codomain of the presheaves) with a topos? A Grothendieck topos? A monoidal category/cosmos? A finitely complete category? (This latter seems to be the weakest generically possible)

What sort of category C\mathbf{C} is part of the definition of a Gr-coherator? It needs enough colimits so that globular sums exist, but it doesn’t need all finite colimits. What if I do something crazy and replace C\mathbf{C} with an abelian category? A 2-category?

hmmm…

Posted by: David Roberts on September 21, 2010 2:02 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

what is a G-M weak 2-category, and how does it relate to a bicategory? (For that matter, a G-M 1-category?) …

Someone should be able to work that out pretty easily, similarly to how Tom worked out the 2- and 1-dimensional versions of the globular-operadic definitions in his survey.

Can we replace Set (the codomain of the presheaves) with… A finitely complete category?

Indeed, I believe the paper actually gives the definition in that context.

Your other questions are also good ones! I expect that they are of about the same difficulty as the same questions for Batanin/Leinster ∞-categories, and in that case I believe no definitive answers are yet known.

Posted by: Mike Shulman on September 22, 2010 3:09 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

in the same vein I could say this about the geometric approaches: I could say the lack of specified composite in a Kan complex really is to be read as there being many composition operations — one for every horn filler.

I think the essence of an “algebraic” notion is that the collection of operations is given as an aspect of the theory, rather than separately for each object. In a Batanin ω-category, there may be many ways to compose a diagram of a given shape, but all Batanin ω-categories (relative to the same operad) have the same collection of ways to compose any diagram of that same shape. By contrast, a given pair of composable arrows in one Kan complex may be composable in only one way, whereas another pair of composable arrows in another Kan complex (or even in the same one) may be composable in ten, or 10 10010^100, or 0\aleph_0 ways.

(By the way, this is why Definition L’, although very similar in some ways to the globular-operadic approaches, should actually be regarded as a non-algebraic definition of ω-category.)

Of course, algebraic Kan complexes are certainly algebraic in this sense, and probably in any other precise sense one could write down, so perhaps “algebraic” was the wrong word to use. I used that because Tom labeled his continuum with “Algebra/Logic” on one side and “Topology” on the other. (Maybe he can explain what he had in mind by saying that algebraic Kan complexes are towards the non-algebraic side of the continuum of algebraic notions of ∞-groupoid.)

Note that Grothendieck is quoted as saying your suggestion that Kan complexes are “the ultimate in lax [sic] ∞-groupoids” does not in any way meet with what I am really looking for, and this for a variety of reasons…. I presume that the “variety” of reasons goes beyond algebraicity in any precise sense, and are probably closer to what I had in mind.

Posted by: Mike Shulman on September 20, 2010 7:31 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Note that Grothendieck is quoted as saying…

I did note that with interest, yes. Made me think that maybe an opportunity in history was missed.

your suggestion that Kan complexes are “the ultimate in lax ∞-groupoids” does not in any way meet with what I am really looking for, and this for a variety of reasons

Does anyone happen to know the list of reasons?

Posted by: Urs Schreiber on September 20, 2010 7:40 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Hi Mike,

Maybe he can explain what he had in mind by saying that algebraic Kan complexes are towards the non-algebraic side of the continuum of algebraic notions of \infty-groupoid.

Explanation coming up—though I’m meant to be doing something else for the next few days, so it might be a while before I read any replies.

The word ‘algebraic’ was never supposed to be precise, but I tend to use it to mean something like ‘the category of nn-categories and strict nn-functors is monadic over nn-globular sets’. Here n{}n \in \mathbb{N} \cup \{\infty\}, and ‘categories’ could be changed to ‘groupoids’.

(This presupposes that there is a notion of strict nn-functor, which isn’t the case for all the proposed definitions. The existence of such a notion is also a necessary condition for being algebraic.)

I’d also expect the monad to be finitary and, in the case of nn-categories rather than nn-groupoids, cartesian.

As far as I know, the category of algebraic Kan complexes isn’t algebraic in that sense. That’s a reason for not putting them on the algebraic side. This is related to the last paragraph of Thomas’s comment.

On the other hand, there’s an obvious sense in which algebraic Kan complexes are indeed an ‘algebraic’ version of Kan complexes. That’s a reason for putting them somewhere around the middle.

I don’t know how much significance to attach to the fact that algebraic Kan complexes are monadic over simplicial sets. (I mean significance in terms of this discussion over whether to call them ‘algebraic’, not in any wider sense.) Simplicial sets do have something in common with globular sets. But: the category of simplicial sets is, trivially, also monadic over simplicial sets, and presumably no one wants to call simplicial sets an algebraic notion of \infty-groupoid. (Cf. Mike’s point.)

(It’s maybe worth noticing that being monadic over a presheaf category does not necessarily make you look like a variety of algebras. Indeed, the reflection between sheaves and presheaves shows that any Grothendieck topos is monadic over a presheaf category.)

Having said all that, I’m not convinced that it’s important to debate where on the spectrum a particular notion of nn-category or nn-groupoid should be placed. What’s much more important — and of course, way harder — is to show that everything on the spectrum is ‘equivalent’: that the unambiguously algebraic notions such as Penon’s, Batanin’s and Trimble’s are essentially the same as the unambiguously non-algebraic notions. Modulo the difference between \infty-categories and \infty-groupoids, that’s essentially the Homotopy Hypothesis. Once that’s established, it won’t really matter where on the spectrum things are placed, because the spectrum will have been collapsed to a point.

Posted by: Tom Leinster on September 20, 2010 10:04 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

The word ‘algebraic’ was never supposed to be precise, but I tend to use it to mean something like ‘the category of n-categories and strict n-functors is monadic over globular sets’.

How does globular shape get singled out by algebraicity? What about cubical sets? Which shape categories do you consider algebraic. And why?

I’d think pairing Thomas’ result with one of Jardine’s, one gets a monaic model of \infty-groupoids over cubical sets together with a proof of the homotopy hypothesis.

And a morphism between algebraically fibrant simplicial/cubical sets takes composites to composites, associators to associators, pentagonators to pentagonators, and so on. Isn’t that a “strict functor”?

Posted by: Urs Schreiber on September 21, 2010 1:43 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Hi Urs,

How does globular shape get singled out by algebraicity?

It doesn’t. It gets singled out by the fact that we’re discussing theories of nn-category; and whatever an nn-category is, it should certainly have an underlying nn-globular set.

What about cubical sets?

If we were discussing nn-tuple categories (a.k.a. cubical nn-categories) then I’d probably use the word algebraic in much the same way as for nn-categories, changing ‘globular set’ to ‘cubical set’ throughout. That is, I’d probably call a theory of nn-tuple categories ‘algebraic’ if the category of nn-tuple categories and strict functors between them were (nicely) monadic over nn-cubical sets.

Which shape categories do you consider algebraic?

I don’t consider any shape category to be either algebraic or non-algebraic, as I hope the comments above explain.

Posted by: Tom Leinster on September 21, 2010 3:34 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

How does globular shape get singled out by algebraicity?

It doesn’t. It gets singled out by the fact that we’re discussing theories of nn-category; and whatever an nn-category is, it should certainly have an underlying nn-globular set.

Using the orientals, every simplicial model for nn-category has an underlying globular set.

I don’t quite see how the specific shape used to organize the data is an invariant. For instance the Duskin nerve identifies globular bigroupoids with precisely simplicial 2-hypergroupoids.

Posted by: Urs Schreiber on September 21, 2010 7:03 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

I’d probably call a theory of nn-tuple categories ‘algebraic’ if the category of nn-tuple categories and strict functors between them were (nicely) monadic over nn-cubical sets.

If for a theory of \infty-fold groupoids satsifying this criterion the equivalence to topological spacesis proven, woult this count for you as a genuine proof of the homotopy hypothesis?

Posted by: Urs Schreiber on September 21, 2010 7:18 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Perhaps someone should contact Maltsiniotis and suggest that he revise this sentence?

I can’t find his email address. His website seems to have been down all day at least. (?)

Posted by: Urs Schreiber on September 19, 2010 4:26 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

It’s maltsin#math,jussieu,fr, making the obvious substitutions.

Posted by: Tom Leinster on September 19, 2010 10:17 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

It’s maltsin#math,jussieu,fr, making the obvious substitutions.

Thanks, I’ve emailed him.

Posted by: Urs Schreiber on September 19, 2010 10:35 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

If I’m reading what you are saying correctly, Mike, then the main difference seems to be that globular operads correspond to monads on globular sets whose arities come from Θ 0\Theta_0 and which are parametric right adjoint, whereas Grothendieck-Maltsiniotis omega-categories correspond to monads on globular sets whose arities still come from Θ 0\Theta_0, but are not necessarily parametric right adjoint. I have encountered the word homogeneous in things that Clemens wrote where I think it amounts to the existence of a generic/free factorisation on the theory, i.e., the corresponding monad being parametric right adjoint. So the theorem you cite sounds pretty reasonable.

Posted by: Richard Garner on September 19, 2010 11:49 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

That sounds plausible, although it’s not immediately obvious to me because of how the source/target structure maps of a globular set are rolled into the rest of the Grothendieck-Maltsiniotis definition, rather than being given first with the rest of the data as added structure. I haven’t thought a lot about it, though, and my understanding of p.r.a. functors is still in the early stages.

Posted by: Mike Shulman on September 20, 2010 7:17 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Well having had a bit more of a look at the paper I think you could argue either way as to whether the source and target maps are rolled in or whether they are there “beforehand” as it were. A Grothendieck-style omega-groupoid in Set\mathbf{Set} is defined as a model of a “Gr-coherator”, which is a category C\mathbf{C} equipped with a functor 𝔾C\mathbb{G} \to \mathbf{C} satisfying some properties. Given such a coherator, a model of it is a functor C opSet\mathbf{C}^op \to \mathbf{Set} which preserves certain kinds of limits (“globular products”). But any such model will certainly give rise to a functor 𝔾 opSet\mathbb{G}^op \to \mathbf{Set} by precomposition with 𝔾C\mathbb{G} \to \mathbf{C}. So any model is a globular set. I guess it is fair to argue that it is only a globular set after the fact: but I think that if you want to, you can switch things around as follows.

One of the properties of a “Gr-coherator” F:𝔾CF : \mathbb{G} \to \mathbf{C} is that we can take the left Kan extension of FF along the inclusion 𝔾Θ 0\mathbb{G} \to \Theta_0 to get a functor F¯:Θ 0C\bar F \colon \Theta_0 \to \mathbf{C}. In terms of this functor we can define a model to be the following: it’s a presheaf in [C op,Set][\mathbf{C}^op, \mathrm{Set}] which when restricted along F¯\bar F to a presheaf in [Θ 0 op,Set][\Theta_0^op, \mathrm{Set}], lies in the image of the singular functor 𝔾^[Θ 0 op,Set]\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}] induced by the inclusion Θ 0𝔾^\Theta_0 \to \hat{\mathbb{G}}. That is, the category of models \mathcal{M} for the coherator is given by the pullback of 𝔾^[Θ 0 op,Set]\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}] along (F¯) *:[C op,Set][Θ 0 op,Set](\bar{F})^\ast \colon [\mathbf{C}^op, \mathrm{Set}] \to [\Theta_0^op, \mathrm{Set}]. But this latter functor (F¯) *(\bar{F})^\ast is monadic so long as we assume that F¯\bar{F} is bijective on objects (something which seems entirely sensible to do to me, I didn’t see it in the paper but perhaps I didn’t look closely enough); in any case, assuming (F¯) *(\bar{F})^\ast is monadic then when we pull it back along 𝔾^[Θ 0 op,Set]\hat{\mathbb{G}} \to [\Theta_0^op, \mathrm{Set}], we get a functor 𝔾^\mathcal{M} \to \hat{\mathbb{G}} which is monadic as long as it has a left adjoint, which it clearly does (mumble mumble locally presentable mumble mumble filtered colimits). And once we know that \mathcal{M} is monadic over globular sets, we can equally well view a model of a coherator as a globular set with extra structure, and can recast all the stuff about coherators as stuff about the corresponding monads on globular sets.

(This sort of argumentation is pretty much how Clemens recasts all the globular operad stuff in the things he’s written on this topic).

Posted by: Richard on September 21, 2010 2:01 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

so long as we assume that F¯\bar{F} is bijective on objects (something which seems entirely sensible to do to me, I didn’t see it in the paper but perhaps I didn’t look closely enough)

I think it’s there in the middle of page 4. A Gr-coherator is defined to be essentially a countable cell complex in the category of globular theories, which starts with Θ 0\Theta_0 and successively attaches liftings for pairs of parallel arrows: 𝔾Θ 0C 0C 1C=colimC n\mathbb{G} \to \Theta_0 \simeq C_0 \to C_1 \to \dots \to C = \colim C_n and he then says

Condition (b 0)(b_0) implies that the category C 0C_0 is equivalent to Θ 0\Theta_0. We will usually assume that C 0C_0 is equal to Θ 0\Theta_0… It can be easily seen that the functors C nC n+1C_n \to C_{n+1} induce bijections on the sets of objects, so that we can suppose that all categories C nC_n and CC have same objects…

In particular, your functor F¯\bar{F} is the inclusion Θ 0=C 0C\Theta_0 = C_0 \to C, so these remarks imply it can be taken to be bijective on objects.


I didn’t actually have much doubt that the definition could be recast to be monadic over globular sets via some argument of this sort, but thanks for spelling it out! I was more saying that given how formal the construction is (cf. mumble mumble) I don’t see immediately how properties of the Gr-coherator translate into properties of the resulting monad on globular sets, such as “having arities from Θ 0\Theta_0.” What you say certainly seems plausible, though.

Posted by: Mike Shulman on September 21, 2010 5:44 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Well I guess the arities bit is probably the easier part. From any bijective on objects functor Θ 0C\Theta_0 \to \mathbf{C} you get a monad on 𝔾^\hat{\mathbb{G}} by the construction I outlined above. Conversely from any monad TT on 𝔾^\hat{\mathbb{G}} you get a bijective on objects functor Θ 0C\Theta_0 \to \mathbf{C} by taking C\mathbf{C} to be the full subcategory of the Kleisli category of TT spanned by the objects in Θ 0\Theta_0. I think this gives you an idempotent adjunction, and the fix points on each side are the monads with arities from Θ 0\Theta_0, respectively the functors Θ 0C\Theta_0 \to \mathbf{C} which preserve globular sums. Actually this might be quite a nice way to think about the general nerve theorem.

Posted by: Richard on September 21, 2010 10:15 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

So this has a certain ring of plausibility to it. If i:𝒞𝒟i \colon \mathcal{C} \to \mathcal{D} exhibits 𝒞\mathcal{C} as a full small dense subcategory of 𝒟\mathcal{D}, then to ask for a monad TT on 𝒟\mathcal{D} to have arities from 𝒞\mathcal{C} is equally well to ask that the induced monad [𝒞 op,Set]()i𝒟T𝒟𝒟(i,1)[𝒞 op,Set][\mathcal{C}^op, \mathbf{Set}] \xrightarrow{(-) \star i} \mathcal{D} \xrightarrow{T} \mathcal{D} \xrightarrow{\mathcal{D}(i,1)} [\mathcal{C}^op, \mathbf{Set}] on [𝒞 op,Set][\mathcal{C}^op, \mathbf{Set}] be cocontinuous (assuming here that 𝒟\mathcal{D} is cocomplete; we don’t strictly need to but it makes things easier). But such a cocontinuous monad is equally well a monad on 𝒞\mathcal{C} in Prof\mathbf{Prof}, which is equally well a category 𝒯\mathcal{T} with a bijective-on-objects functor 𝒞𝒯\mathcal{C} \to \mathcal{T}. In the displayed situation, that category 𝒯\mathcal{T} will be precisely the full subcategory of Kl(T)\mathbf{Kl}(T) spanned by the objects from 𝒞\mathcal{C}, as in my earlier message. Hmmmm…

Posted by: Richard on September 21, 2010 11:04 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

So the class of monads on 𝒟\mathcal{D} with arities from 𝒞\mathcal{C} is actually independent of 𝒟\mathcal{D}?

Posted by: Mike Shulman on September 25, 2010 8:03 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

No, as not every bijective on objects functor F:𝒞𝒞F \colon \mathcal{C} \to \mathcal{C}' will arise from a monad on 𝒟\mathcal{D}. Those which do are precisely those FF for which the corresponding cocontinuous monad on [𝒞 op,Set][\mathcal{C}^\op, \mathbf{Set}] maps the reflective subcategory 𝒟\mathcal{D} into itself.

I proved the conjecture I made above, by the way. Functors Θ 0C\Theta_0 \to \mathbf{C} which preserve globular sums correspond precisely to monads with arities from Θ 0\Theta_0, and the algebras for the monad are the same as models for the globular theory. And this has a correlate for the general nerve theorem which is quite cool.

Posted by: Richard Garner on September 29, 2010 1:06 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Another way to say this is that F:𝒞𝒞F \colon \mathcal{C} \to \mathcal{C}' bijective on objects arises from a monad on 𝒟\mathcal{D} iff the singular functor 𝒞(F,1):𝒞[𝒞 op,Set]\mathcal{C}'(F, 1) \colon \mathcal{C}' \to [\mathcal{C}^\op, \mathbf{Set}] factors through the inclusion of the full subcategory 𝒟\mathcal{D}.

Posted by: Richard Garner on September 29, 2010 1:37 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Great! So to be precise, is it correct that you’ve proven that what Maltsiniotis calls a globular theory is the same as a monad on globular sets with arities from Θ 0\Theta_0? Whereas Batanin’s globular operads can be identified with such monads that are additionally p.r.a.? Then I guess the next question is how the contractibility conditions compare.

Posted by: Mike Shulman on September 30, 2010 8:46 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Yes, that’s right on both counts. (Actually now that I look this is essentially known, at least the second part, from the first section of Clemens’ paper “A cellular nerve for higher categories”.) Comparing the contractability conditions shouldn’t be that hard I would hope!

Posted by: Richard Garner on October 1, 2010 6:56 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Actually, what do you mean by “arities”? I can guess what the arities of a p.r.a. functor on a presheaf category are — namely, the objects E T(x)E_T(x) occurring in the equivalence of such functors with diagrams indexed by a category of elements — and I can see how requiring those to come from Θ 0\Theta_0 (meaning the category of globular pasting diagrams, right?) would make a monad equivalent to a globular operad (perhaps via both being equivalent to a cartesian monad with a cartesian map to the “free strict ω-category” monad). But what are the arities of a non-p.r.a. monad?

Posted by: Mike Shulman on September 20, 2010 8:16 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Ah right well this is part of the nerve theorem which Mark Weber proved. Suppose you have a category 𝒟\mathcal{D} and a small dense subcategory 𝒞\mathcal{C} of it. Then there is a technical notion of a monad TT on 𝒟\mathcal{D} having arities from 𝒞\mathcal{C} which allows you to prove all sorts of wonderful things about it. The condition is very roughly that if when you restrict TT to a functor 𝒞𝒟\mathcal{C} \to \mathcal{D} and then left Kan extend back to an endofunctor of 𝒟\mathcal{D}, you get back the TT you started with. Actually, this isn’t quite strong enough, it turns out; what you actually need is that when you take the singular functor 𝒟(T,1):𝒟[𝒟 op,Set]\mathcal{D}(T,1) \colon \mathcal{D} \to [\mathcal{D}^op, \mathrm{Set}], restrict it along 𝒞𝒟\mathcal{C} \hookrightarrow \mathcal{D} and then left Kan extend, you get back to where you started. The whole thing is in in Mark’s paper called “Familial 2-functors” etc.

Posted by: Richard on September 21, 2010 2:14 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Sorry, I mangled the last bit really badly there. The correct condition is that the composite functor 𝒟T𝒟𝒟(i,1)[𝒞 op,Set]\mathcal{D} \xrightarrow{T} \mathcal{D} \xrightarrow{\mathcal{D}(i,1)} [\mathcal{C}^\op, \mathbf{Set}] should be equal to the left Kan extension of its restriction along i:𝒞𝒟i \colon \mathcal{C} \to \mathcal{D}.

Posted by: Richard on September 21, 2010 3:01 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Ah, okay, thanks. Does it agree with what I guessed in the case of a p.r.a. functor?

Posted by: Mike Shulman on September 21, 2010 5:47 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Yes, that’s one of the leading examples in Mark’s paper

Posted by: Richard on September 21, 2010 9:56 PM | Permalink | Reply to this
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Re: Grothendieck-Maltsiniotis ∞-categories

I’m kind of sad that the discussion so far has consisted mostly of rehashing old disagreements about what’s “algebraic” and whether Kan complexes are a good notion of ∞-groupoid. I’m not blaming anyone in particular – I’ve been contributing too – but I feel like we’re turning into philosophers (with apologies to the actual philosophers present) who endlessly debate the meanings of words without actually getting anything done.

Is anyone besides Richard interested in the actual definition? It’s true that we already have no lack of definitions of ∞-category, but a new one with Grothendieck’s name attached to it seems worthy of some attention. In particular, I’m interested that this definition seems to require less technical machinery than Batanin’s, while apparently describing essentially the same notion (or a slightly more general one). It does seem to me a little more ad hoc, as I said, but that might just be an artifact of the presentation; Richard’s comment seems like a promising direction.

One thing that just occurred to me is that it seems that Maltsiniotis’ notion of a “Cat-coherator” (the analogue of a globular operad) is strictly more general than a “Gr-coherator”. If that’s true, it means that among the “notions of ∞-category” described by Cat-coherators are included all the “notions of ∞-groupoid” described by Gr-coherators. So we would really need some additional condition on a Cat-coherator in order to really want to call it a notion of ∞-category. But maybe I’m reading the definitions wrong?

Posted by: Mike Shulman on September 21, 2010 8:50 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

I am actually quite interested in the definition itself, but have not had time to read it in any detail. I would love to see someone give it Tom’s treatment and write up a 2-ish page bare bones approach to the definition.

Posted by: Nick Gurski on September 21, 2010 11:41 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

That’s pretty much what it is in the paper anyway: you just need to read pp. 3–4 for the omega-groupoids, and pp. 18–19 for the omega-categories.

Posted by: Richard on September 21, 2010 11:56 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

the discussion so far has consisted mostly of rehashing old disagreements

To me it seems instead there is the claim that a certain important problem is open and an attempt to learn what precisely the statement of the problem is supposed to be.

I quite gather from reactions here and private conversation that a widespread impression is that there are solutions to the problem that are way more simple and tractable than others, and the feeling is that therefore they are too simple to be good.

There are two possibilities: the simple solutions are really too simple to be good. Or the complicated approaches are actually too complicated to be good.

I think for the field it would be good to clarify this, by making the question more precise. But I’ll not persue this here anymore, not to highjack your entry.

Posted by: Urs Schreiber on September 21, 2010 4:56 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

To me it seems instead there is the claim that a certain important problem is open and an attempt to learn what precisely the statement of the problem is supposed to be.

Maybe we can just state the problem as “prove the homotopy hypothesis for all definitions of ∞-groupoid,” or more generally “prove the equivalence of all definitions of ∞-category”?

There are two possibilities: the simple solutions are really too simple to be good. Or the complicated approaches are actually too complicated to be good.

Or a third possibility, which I think is likely to be the case: the simple solutions are more tractable and thus easier for many purposes, but there are some purposes for which they are too simple, and for which more complicated solutions are necessary.

This is the case in some other situations where there are many different definitions of a concept, e.g. spectra. The definition of an Ω\Omega-spectrum is the simplest and easiest to write down. But they don’t give you a monoidal model category. Probably the simplest monoidal model category of spectra is symmetric spectra. But the homotopy groups of a symmetric spectrum are hard to calculate, and they don’t “equivariantize” very well. Orthogonal spectra are a little more complicated and solve these problems, but not all orthogonal spectra are fibrant and so the 0th space functor can be tricky to deal with. EKMM S-modules solve that problem, but are yet more complicated.

Posted by: Mike Shulman on September 21, 2010 6:02 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Or a third possibility, which I think is likely to be the case: the simple solutions are more tractable and thus easier for many purposes, but there are some purposes for which they are too simple, and for which more complicated solutions are necessary.

Okay, great, this is what I am trying to find out here, while coming across as somebody who is rehashing old disagreements.

I want to know: what is a good motivation for these complicated definitions of \infty-groupoids?

I’ll play advocatus diaboli , to highlight what I am after, so what I say now is intentionally made to sound a bit strong, to trigger reactions:

we have complicated structures here, whose definition alone fills articles. There are no examples for these objects given. There are no applications of these objects given. There is no theory about these objects.

All we have is one single motivation: the desire to find an “algebraic” left hand in the homotopy hypotesis GrpdTop\infty Grpd \simeq Top. But then, the definitions given are not shown to solve this!

So what is it we have achieved here? How do we know this is not a dead end, with so little in our hands?

But now I look at it and find: there is already something solving this motivational problem. And it is elegant, useful, has examples and applications and comes with a good theory.

So of course I am wondering: where are we headed: why do we need these complicated definitions?

There is probably a good answer to this. Probably if one formulates sharply what one is really looking for, what it really is that existing algebraic models do not satisfy, why one needs this and what it is good for, then probably one will have good motivation.

Cause, there must be better motivation than just “Grothendieck said so, too”. If in physics you say this with “Einstein” instead, you score on the crackpot index.

All right, so that was the harsh version. I don’t actually strictly feel this way. But maybe putting it this way can serve a purpose.

Posted by: Urs Schreiber on September 22, 2010 11:01 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

What exactly are you referring to as the thing which exists and solves the motivational problem with all the good properties? We’ve been talking about Kan complexes as models of ∞-groupoids so far, but if that’s what you mean, then one serious thing missing is an understanding of a notion of ∞-category which specializes to Kan complexes as the ∞-groupoids. Quasicategories get us to (∞,1)-categories, but it seems to get much more difficult after that. Weak complicial sets are at least a first step.

Another thing which comes to mind that I think a good theory of ∞-categories should have, which I don’t see working very smoothly in the world of Kan complexes, is an enriched version.

Posted by: Mike Shulman on September 25, 2010 7:55 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

This is a very delayed comment on the above discussion. But today I was looking back through this discussion and the article, and noticed that I am not sure if the term algebraic structure species on page 2 is being used in a formal sense, and if so what the precise definition is. I’d be grateful if somebody could briefly tell me.

Posted by: Urs Schreiber on December 6, 2010 10:38 AM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

Google doesn’t suggest to me that “algebraic structure species” is an established term in English. It does sound as though he had a specific meaning in mind, though. Perhaps it is a translation from French of a phrase more frequently translated as something else.

Posted by: Mike Shulman on December 6, 2010 3:18 PM | Permalink | Reply to this

Re: Grothendieck-Maltsiniotis ∞-categories

@Urs: This is a Bourbaki term. It should be read as “algebraic species of structure”. This is *sigh* defined in Bourbaki’s book on set theory, in the section on the theory of structures.

Posted by: Harry on March 5, 2011 1:52 PM | Permalink | Reply to this

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