## July 21, 2014

### Pullbacks That Preserve Weak Equivalences

#### Posted by Mike Shulman

The following concept seems to have been reinvented a bunch of times by a bunch of people, and every time they give it a different name.

Definition: Let $C$ be a category with pullbacks and a class of weak equivalences. A morphism $f:A\to B$ is a [insert name here] if the pullback functor $f^\ast:C/B \to C/A$ preserves weak equivalences.

In a right proper model category, every fibration is one of these. But even in that case, there are usually more of these than just the fibrations. There is of course also a dual notion in which pullbacks are replaced by pushouts, and every cofibration in a left proper model category is one of those.

What should we call them?

The names that I’m aware of that have so far been given to these things are:

1. sharp map, by Charles Rezk. This is a dualization of the terminology flat map used for the dual notion by Mike Hopkins (I don’t know a reference, does anyone?). I presume that Hopkins’ motivation was that a ring homomorphism is flat if tensoring with it (which is the pushout in the category of commutative rings) is exact, hence preserves weak equivalences of chain complexes.

However, “flat” has the problem of being a rather overused word. For instance, we may want to talk about these objects in the canonical model structure on $Cat$ (where in fact it turns out that every such functor is a cofibration), but flat functor has a very different meaning. David White has pointed out that “flat” would also make sense to use for the monoid axiom in monoidal model categories.

2. right proper, by Andrei Radulescu-Banu. This is presumably motivated by the above-mentioned fact that fibrations in right proper model categories are such. Unfortunately, proper map also has another meaning.

3. $h$-fibration, by Berger and Batanin. This is presumably motivated by the fact that “$h$-cofibration” has been used by May and Sigurdsson for an intrinsic notion of cofibration in topologically enriched categories, that specializes in compactly generated spaces to closed Hurewicz cofibrations, and pushouts along the latter preserve weak homotopy equivalences. However, it makes more sense to me to keep “$h$-cofibration” with May and Sigurdsson’s original meaning.

4. Grothendieck $W$-fibration (where $W$ is the class of weak equivalences on $C$), by Ara and Maltsiniotis. Apparently this comes from unpublished work of Grothendieck. Here I guess the motivation is that these maps are “like fibrations” and are determined by the class $W$ of weak equivalences.

Does anyone know of other references for this notion, perhaps with other names? And any opinions on what the best name is? I’m currently inclined towards “$W$-fibration” mainly because it doesn’t clash with anything else, but I could be convinced otherwise.

Posted at July 21, 2014 11:03 PM UTC

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### Re: Pullbacks That Preserve Weak Equivalences

It’s tempting to call these “quasifibrations”. Unfortunately, actual quasifibrations (in the sense of Dold and Thom) need not have this property (though all maps of this kind are quasifibrations).

Posted by: Charles Rezk on July 22, 2014 3:03 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

For that matter, what’s wrong with “sharp”? You could use “cosharp” for the dual notion, if you don’t like “flat”.

Posted by: Charles Rezk on July 22, 2014 3:08 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

I agree that “quasifibration” is tempting but probably unacceptable for the same reason you gave. And you’re right that “sharp” is at least non-clashing on its own, and “cosharp” would be fine. The only problem I see is that with “flat” out of the picture, there’s not much in the way of motivation for “sharp”.

Posted by: Mike Shulman on July 22, 2014 3:52 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Sharp sounds good in connection with slices.

Posted by: Joachim Kock on July 22, 2014 5:58 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Slices are also flat.

Posted by: Tom Leinster on July 22, 2014 6:04 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Barwick and Kan call these maps fibrillations, see
Definition 9.2 in arXiv:1208.1777.

Cisinski calls these maps weak fibrations (fibrations faible),
see 3.11 in his paper “Invariance de la K-théorie par équivalences dérivées”.

Posted by: Dmitri Pavlov on July 22, 2014 11:47 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

According to Wikipedia,

fibrillation is the rapid, irregular and unsynchronized contraction of muscle fibers.

It’s easy to imagine mathematicians wanting to talk about contracting fibres, but is that actually what’s going on in this definition? And is there anything irregular or unsynchronized about it?

When I first saw Dmitri’s post, I assumed that Barwick and Kan were just coining a new word that resembled “fibration” (maybe with defibrillators vaguely in mind). But perhaps there’s more to it…?

Posted by: Tom Leinster on July 22, 2014 5:23 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Barwick and Kan mention fibrations as a motivation
for the name “fibrillation”.

By the way, the term “cofibrillation” (which is what
Barwick and Kan use for the dual notion)
is also used in the medical literature:
see http://www.pnas.org/content/101/1/87.full
and http://www.ncbi.nlm.nih.gov/pubmed/19427320.

For any fibrillation the map from its ordinary
fibers to its homotopy fibers is a weak equivalence.
For acyclic fibrillations these fibers are contractible,
so the term does seem relevant!

Posted by: Dmitri Pavlov on July 22, 2014 5:38 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Relevant… but, it seems to me, highly misleading, since the maps in question are in general not acyclic!

Posted by: Mike Shulman on July 22, 2014 8:26 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Preserving weak equivalences is something like preserving isomorphism (which all pullbacks do, as do all functors), which means it’s something like ensuring that pullback descends to a functor on the underhomotopy categories $ho C/[A] \to ho C /[B]$ in a nice way — that is, if the homotopy category has pullbacks along $[f]$, then they preserve equivalences of course, but to have it reflected in the model, before localizing…

I’m not going anywhere with this, I’m just riffing.

Posted by: Jesse C. McKeown on July 23, 2014 3:12 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Arguably, the real problem is how to get everyone to agree on one name for these things, whatever that name might be. I think almost any of the above names would be better, if used consistently by everyone, than six or more different names used by different people!

Posted by: Mike Shulman on July 23, 2014 4:42 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

I agree…

Posted by: Tom Leinster on July 23, 2014 11:45 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

I actually like the idea of coflat, since flat ring homomorphism is the first thing I think of when I think “flat”. I don’t think there’s a problem with overusing vocabulary if the concepts are closely related. Moreover, model categories are close to homological algebra.

Posted by: Jason Polak on July 23, 2014 8:39 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Hi, Mike. I just want to explain our motivations with Clemens about calling those maps h-cofibration more clearly. You are right, one motivation was to keep connection with May and Sigurdsson. But more serious was a need to have a short and easy to pronounce name for a new notion of monoidal model category which behaves nicely with respect to homotopy. This nice behavior means that tensor of a (trivial) cofibration on another object is a (trivial) h-cofibration. So, somehow we wanted that the name of such a category remembers it. After numerous experiments with various names for h-cofibrations and these monoidal categories we came to conclusion that the names h-cofibration for these maps and h-monoidal model category for our concept of “good” monoidal model category would be very convenient. We are still convinced that this is a nice combination.

Michael.

Posted by: Michael Batanin on August 13, 2014 1:08 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Thanks for stopping by to explain! However, as I said, I would rather not try to appropriate an existing terminology for something different. But your point does seem to rule out some of the possibilities, e.g. “left proper monoidal model category” doesn’t mean what you want at all.

What would you think about “flat-monoidal model category” or “$w$-monoidal model category”? I kind of like the latter, coming from a slight modification of Grothendieck’s terminology with “$w$-cofibration” and “$w$-fibration” following the same general pattern as May-Sigurdsson’s $h$-cofibration, $q$-cofibration, $m$-cofibration, etc. The prefix “$w$-” also suggests Cisinski’s “weak fibration”.

Posted by: Mike Shulman on August 13, 2014 5:06 AM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Yes, I agree that it is better to avoid changing the existing terminology. But is this case we had a curious trilemma. First, as you already pointed out there was no generally accepted terminology. Second, we wanted to keep connections with May and Sigurdsson for historical and other reasons. And third we wanted something easy to pronounce and remember even for non English speakers. We considered different possibilities including those you mentioned in your reply. Finally we decided in favour of “h-cofibrations” and “h-monoidal categories” just because it was the most convenient to use during our (almost four years long) work on the paper. I agree that it is not ideal but it is very practical.

Michael.

Posted by: Michael Batanin on August 13, 2014 3:14 PM | Permalink | Reply to this

### Re: Pullbacks That Preserve Weak Equivalences

Here’s an idea, it has a disadvantage that it introduces a completely new name that people might not like, but I think it was never used for anything in that part of mathematics.

I like the name “sharp morphisms” very much and got used to it already and will probably keep using it. It arose by dualizing “flat morphisms”, but as we agreed “flat” is way overused. So how about we dualize back with a little shift to obtain “blunt morphisms”?

Posted by: Karol Szumiło on August 23, 2014 9:14 AM | Permalink | Reply to this

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