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I’m (slowly) working on writing an introduction for chapter 1 of my thesis so it can go on the arXiv as a standalone paper, then for submission to TAC. (The working title is ’Internal anafunctors I’, but if this seems rather drab or a turn-off, I’ll work through some more options - I’m open to suggestions). I know why I care about internal anafunctors, but I know there are other points of view, and I want to at least touch on them to catch peoples’ interest. Here are some examples, sometimes only talking about special cases.
anafunctors correspond to:
Is this a decent enough list to catch the eye of the casual category theorist? Can anyone think of any more?
There is more material on internal anafunctors floating around my notes and in my head, at least for an IA2, and I hope to eventually get it out in the public eye, but this needs doing “now” so it can be reference by someone’s upcoming paper on butterflies.
I think I’d also like (and this is directed at Cafe hosts) to talk about anafunctors in a post on the cafe, perhaps covering some of the above examples - if I can garner a few more examples/applications that way I’d be happy.
I think I’d also like (and this is directed at Cafe hosts) to talk about anafunctors in a post on the cafe,
Sure, just send the code by email.
From my notes:
A (say principal) $G$-bundle over $B$ (for $G$ a group and $B$ a space) is the same as an anafunctor from $B$ (thought of as a discrete groupoid) to $G$ (thought as a one-object groupoid), given essentially by transition functions on a cover of $B$.
John and Urs (HGT2) know how to generalise the above to a bundle with connection.
$2$-term chain complexes are internal categories, and anafunctors are classified by $Ext$.
A (say principal) G-bundle over B (for G a group and B a space) is the same as an anafunctor from B (thought of as a discrete groupoid) to G (thought as a one-object groupoid), given essentially by transition functions on a cover of B.
I can’t believe I forgot this one! I usually trot out the ’anafunctors are just a generalisation of Cech cocycles’, but I forgot.
2-term chain complexes are internal categories, and anafunctors are classified by Ext.
I know the first bit, but what about the second? I’m guessing it relates to the link between butterflies (weak maps between crossed modules) and saturated anafunctors, and that butterflies contain an extension as part of their data. Am I right?
Am I right?
Eh … I forget. I’ll see if I can work it out again tomorrow, but I need to go to bed.
one more, which subsumes a bunch or maybe all the previous ones: ana-morphisms are representatives of morphisms in the homotopy category of a category of fibrant objects.
Thanks - that puts my fourth point in context. But I don’t know if internal categories form a category of fibrant objects - I believe internal groupoids do, but I should sit down and figure out if the axioms hold.
Don’t worry, I just convinced myself - by inspection - that the axioms do hold. :S silly me for doubting
Actually, I don’t know if the internal arrow category is a path object, or at least, it isn’t a path object with the fibrations I was proposing.
Just to clear things up, is there a category of fibrant objects structure on the 1-category Cat? I thought perhaps there was with fibrations the surjective on objects functors, but I can’t see how to verify the path object axiom. Perhaps my brain has gone home early…
Yes, so I think an important point in this business is this one:
in an arbitrary category with weak equivalences, we know from Dwyer-Kan that the generalized morphisms are zig-zags of arbitrary length.
$\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \cdots \to \,.$The point is that with a little bit of extra assumptions, it is sufficient to consider zig-zags of much shorter length.
For instance if in addition we have the structure of a model category, then zig-zags
$\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow}$are sufficient. If we are in a category of fibrant objects, then even zig-zags
$\stackrel{\simeq}{\leftarrow} \to$are sufficient.
Just to clear things up, is there a category of fibrant objects structure on the 1-category Cat?
Yes, because there is even a model category structure on Cat (the canonical model structure) in which all objects are fibrant. Every full subcategory on fibrant objects in a model category is a category-of-fibrant-objects.
Same for the standard model structures on 2-categories and strict omega-categories.
It’s the fact that all these are “algebraic” definitions of higher categories that makes all objects fibrant.
But also the full subcategory of quasi-categories inside all simplicial sets is a category of fibrant objects, being the non-trivial full subcategory on the fibrant objects of the Joyal model structure on simplicial sets.
And on very general grounds this means that categories of presheaves with values in these models for higher categories form categories of fibrant objects. And your categories of internal categories all sit inside such more general categories of category-valued presheaves.
Ah, in case you are not aware of it: check out the link folk model structure
I knew about the folk model structure, I was just being a bit dense as to what the fibrations in it are. I should point out that the article by Everaert-Kieboom-van der Linden linked to at canonical model structure only proves that the Quillen model structure exists for certain sites, not including obvious sites like Top or Diff.
It is not clear to me that the model structures obtained by embedding internal categories in a presheaf category lift (descend?) to the original category. For example, one has to prove representability of the various constructions, which I think is quite non-trivial (perhaps this has been done and I am unaware of it - a not-unlikely possibility).
Based on some quick scribblings on the train this morning, I think that in certain circumstances anafunctors compute the homotopy category of a category with fibrant objects, but the link is not direct, in that the original acyclic fibrations don’t necessarily line up with the original anafunctors. Additionally, in the case that the ambient category is not finitely complete there is a little bit of to-and-fro before the result settles down, and it may be that not quite all of the original weak equivalences are inverted by anafunctors. I haven’t checked all the details though.
A quick and basic question on the homotopy category associated to a category of fibrant objects: how, precisely, are arrows equivalence classes of spans with left leg an acyclic fibration?
the article by Everaert-Kieboom-van der Linden linked to at canonical model structure
I am being blind. I can’t see this article. Could you post the link here?
It is not clear to me that the model structures obtained by embedding internal categories in a presheaf category lift (descend?) to the original category.
The model structures are very unlikely to restrict to the internal objects, but structures of categories of fibrant objects have a chance to.
how, precisely, are arrows equivalence classes of spans with left leg an acyclic fibration?
Here is how: let $C$ be a category of fibrant objects. Write $\pi C$ for the category obtained from this by identifying parallel morphisms that become right homotopic after pullback to a joint refinemnet of their domain.
This is described at Homotopies in the entry on cats of fib objects.
This $\pi C$ inherits its weak equivalences from $C$ and in $\pi C$ they form a left multiplicative system. It follows that the homotopy category $Ho_C$ has hom-sets
$Ho_C(A,B) \simeq colim_{\hat A \stackrel{w \in \pi W}{\to} A} \pi C(\hat A, B) \,.$This is described at The homotopy category.
One comment about that intermediate step to $\pi C$: the point of this is to identify more left legs. It’s not important for the right legs of the anafunctors span, because there the homotopies are taken care of be the colimit anyway.
The point is to identify homotopies between covers: if $p_1 : \hat A \to A$ is an cover (acyclic fibration) over $A$, then there may be another one $p_2 : \hat A \to A$ that is homotopic to the first. So we will want to identify them as being the same resolution $\hat A$ of $A$. But just forming $colim_{\hat A \stackrel{w \in W}{\to}} C(\hat A , B)$ cannot divide out these homotopies between the left legs. So this set is a bit too large. It may distibuish a cocycle $\omega : \hat A \to B$ from that same cocycle itself, if it regards one as coming from a projection map $p_1$ and the other from a homotopic projection map $p_2$. So that’s what this intermediate step of $\pi C$ takes care of.
I can’t see this article. Could you post the link here?
from the page:
I thought this article was the best thing since sliced bread, until I figured out they were pretty much thinking of semiabelian ambient categories, and just working in a little more generality, which didn’t include Top. I’ll still have a think about the category with fibrant objects approach, though, as I think it really points the way to working with the localised bicategory of anafunctors, rather than the homotopy category construction given at The homotopy category.
I think I get it: identify arrows which are ’locally homotopic’ using the path space, and then form the category of spans of this category. Then identify those spans which are joined by a span (imagine the relevant diamond-filled-with-a-cross shaped diagram, if you will).
I think I get it: identify arrows which are ’locally homotopic’ using the path space, and then form the category of spans of this category. Then identify those spans which are joined by a span (imagine the relevant diamond-filled-with-a-cross shaped diagram, if you will).
Yes.
I think it really points the way to working with the localised bicategory of anafunctors, rather than the homotopy category construction given at The homotopy category.
Oh, yes, that’s of course true: the homotopy category is just the full decategorification, we really have an $\infty$-category in general, which may be a 2-category if we start with the model structure on categories. Yes.
Probably your “localized bicateory of anafunctors” is this (2,1)-category. And that would be very good.
Maybe just for empahsis again: we know from Dwyer-Kan that given any category with weak equivalences, we get a localized $(\infty,1)$-category whose morphisms are zig-zags
$\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \cdots \to$and whose higher cells are that in the Kan fibrant replacement of the category of “hammocks” of zigzags between such zig-zags. (square lattices of zig-zags vertically and horizonatlly, that at the left and right end are fixed on the source and target object, respectvely).
What the structure of a category of fibrant objects buys is that it tells us that doing this construction with just single zig-zags $\stackrel{\simeq}{\leftarrow} \to$ is sufficient. But the higher morphisms space of this is given by the Dwyer-Kan construction only in a not so usable way. So one should look for more efficient ways to encode this, and quite likely your construciton of a 2-localization is just that.
from the page:
A general internal version relative to a Grothendieck coverage can be found here.
Ah. As a public service, I have now spelled out the reference in the entry.
FWIW, I’m not really fond of paper titles that end in “I”. It seems like the following “II” usually takes longer to appear than the author expects (which is probably true of most mathematical papers), and occasionally never appears or gets renamed before it appears, thereby sending someone on a wild goose chase if they expect (reasonably) that if there is a “Widget theory I” there must also be a “Widget theory II”. Also, even if both end up existing, I think it’s harder to remember the distinction between the two – was the fundamental theorem of widget theory proven in “Widget theory I” or in “Widget theory II”? I’d suggest just calling the first one “Internal anafunctors” and finding a more descriptive name for the second one.
Re: #18, it seems that any purely homotopical construction would give only a (2,1)-category, whereas one wants the localized bicategory of anafunctors to be a (2,2)-category with noninvertible 2-cells. I think Dorette Pronk has a paper about localizing bicategories that admit a calculus of fractions, which is probably related.
I would be very interested to see a careful general development of something like “homotopy bicategories of model 2-categories” (or 2-categories of fibrant objects, etc.). There are other things that should be examples of some such construction. For instance, the 2-category T Alg of algebras and pseudomorphisms over a 2-monad should be equivalent to the homotopy bicategory of the Lack model structure on the 2-category $T Alg_s$ of algebras and strict morphisms. And going up a categorical level, the tricategory Bicat should be the homotopy tricategory of the Lack model structure on the model Gray-category Gray.
@Mike,
thanks for the advice on papers called Blah I and Blah II - given my situation I think it is too risky to claim that I can get out a second paper in reasonable time (if at all). Even Kervaire and Milnor put out a paper (reading up for the exotic differentiable structure references) called ’Groups of homotopy spheres I’ with no sequel.
Regarding (2,1)- versus (2,2)-categories, the results from my thesis do give a 2-category proper. The homotopy construction that Urs was describing only gives the Poincare category of this 2-category, so doesn’t even see the 2-arrows. As for Pronk - I have certainly read it! One point of my paper is to give a simpler construction of the localisation of $Cat(S)$ than in her paper (and in more situations than she proves it exists).
Does Poincare category = homotopy category? In #18 Urs did describe a construction that would (probably) give a (2,1)-category, namely the (∞,1)-categorical localization of Cat(S) at the weak equivalences (which would presumably actually be a (2,1)-category and coincide with the local core of the (2,2)-category you want — it certainly does for Cat = Cat(Set)).
How specific to Cat(S) are your proofs? Could they be applied in some axiomatic framework for homotopy 2-categories?
The terminology ’Poincare category’ goes back to Benabou’s monograph on bicategories, and is the 1-category with hom-sets the isomorphism classes of the hom-categories of the 2-category. I might put in a nLab page if there isn’t one.
Unfortunately my proofs involve anafunctors, which are pretty specific to internal categories. But there are some nice properties of Cat(S) that don’t apply to more general 2-categories with fractions, which could be abstracted in combination with the properties of Cat(S) as a category of fibrant objects. Some of the techniques might be useful for homotopy 2-cats. One long-term project in my folders is to construct the localisation of an arbitrary (up to size issues) 2-category at a class of 1-arrows. Are there any particular ideas about homotopy 2-categories that you have in mind?
Well, one thing I’d like to know is that Cat-enriched weighted homotopy limits in a Cat-enriched model/homotopical category represent bilimits in its homotopy bicategory.
Hi Mike,
just found this old discussion. Now that I know how to work with more general 2-categories, perhaps it is time to think about it again. I can’t promise that the business with Cat-enriched things works, but it’s worth a try.
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