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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 26th 2009
• (edited Nov 26th 2009)

I see Mike's 1-category equipment

May I vote for the following: we should "play Bourbaki" and correct the naming mistake made here. The obvious name one should use is "pro-morphism structure".

We equip a category with pro-morphisms.

We equip a category with a pro-morphism structure.

Or, if you insist,

We equip a category with pro-arrows.

We equip a category with a pro-arrow structure.

But the day will come when you want a pro-2-morphism structure. And then one will regret having used "arrow" instead of "morphism".

I mean, compared to issues like "presentable" versus "locally presentable", this idea of saying just "equipment" is a bit drastic, to my mind.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeNov 26th 2009
What is your proposed word for "a category equipped with pro-morphism structure"?

I really don't see much difference between "morphism" and "arrow." What would be wrong with a pro-2-arrow equipment? There are nowadays people who talk about "2-arrows" where other people say "2-morphisms."

One difference between this and the case of locally presentable categories is that "locally presentable category" is standard everywhere in the literature except Lurie, whereas "(proarrow) equipment" is also standard everywhere in the literature.
• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeNov 26th 2009

I have to say that I agree with Mike here on both points here. I don't see why "arrow" is inferior to "morphism". And as far as I know, he is reporting accurately on what is standard.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 26th 2009
• (edited Nov 26th 2009)
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> And as far as I know, he is reporting accurately on what is standard. </blockquote> <p>That's for sure. I am suggesting that the standard use is odd.</p> <blockquote> What is your proposed word for "a category equipped with pro-morphism structure"? </blockquote> <p>I am suggesting that the entry should be titled somehting like <a href="http://ncatlab.org/nlab/show/pro-arrow+structure">pro-arrow structure</a> and then go like:</p> <p>"A (2-)category equipped with a pro-arrow structure is ...suchandsuch... It is common practice to call a pro-arrow structure on a (2-)category an equipment ".</p> <p>That seems to be the very obvious thing to me. But if you don't see it the same way, let's not fight over it. There are more important things to take care of.</p> </div>
• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeNov 26th 2009
• (edited Nov 26th 2009)

Even if John The User learns today what si "equipment" the word will create no association after one year of not using that term. Like much of other terminology which is forgettable. I vote for Urs's suggestion in the first comment in this entry. Morphism vs. arrow I do not care, but word pro should be in the name to suggest what is it about. Equipment is ambiguous. In fact even structure is superfluous. "Category with pro-arrows" or alike is enough. Like "if we add a pro-arrow structure to a category" we get a "category with pro-arrows"...

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeNov 26th 2009

How about ‘pro-arrow equipment’ or ‘pro-morphism equipment’? We don't have to say ‘structure’, and ‘equipment’ will keep it familiar for those who already know the standard term.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeNov 27th 2009

I am very happy to discuss possible better terminology. Many people that I've talked to seem to dislike "equipment," and although I don't personally understand the vehemence with which some people abhor it, I agree that it would be nice to have something better. But if we're going to do that, let's slow down, do some brainstorming, and consider each other's suggestions carefully. I am always turned off to a productive discussion when someone says that their personal viewpoint is "obvious."

"Pro-arrow equipment" is already the name in use; "equipment" is just an abbreviation for that (although perhaps an ill-chosen one). I really don't like "(2-)category with pro-arrows" or "(2-)category with pro-morphism structure," though it's hard to say exactly why. For one thing, I think the notion is so important that it deserves a shorter and easier-to-say name, one which makes it sound like a single notion rather than a category with something else stuck on. One precise expression of that is that a proarrow equipment is not actually "structure" on a 2-category in the sense of stuff, structure, property but rather it is (not even "stuff" but) eka-stuff. That is, the forgetful functor from the 3-category of proarrow equipments to the 3-category 2Cat is faithful on 3-cells, but no more.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 27th 2009

Some thoughts:

The importance of a concept may be clear to those working on it, and they may use abbreviated terminology for it. But this may not be the best official terminology.

For instance most people working on that topic say "presentable oo-category". Here on the Lab we agreed to use the more properly descriptive terms "locally presentable (oo,1)-category". Nobody else uses that. When I talk to others working on this, they ask me to stop using this pedantic terminology (like Ieke did the other day). Still, I think it is good that our nLab entries adhere to this more precise terminology (or some do, I still need to fill in the "locally" at a few places). Not the least because you insisted on it.

While abbreviated terminology is necessary and unavoidable in daily use by the inside circle, undescriptive terminology is a bad advertisement to the outside. When you started telling us about "equipments" first, my first gut reaction was: "uh, who needs another un-natural bit of structure". The term doesn't resonate with anything, and there are so many other things that want to be understood. Why bother about some random equipment?

But, then, ah, this is about pro-morphisms. The word alone gives me an idea of what is going on. I know pro-functors and how they are of utmost importance. So immedietly I'll be interested in pro-morphisms.

Given this experience I admit that I was surprised by the terminology issue under discussion here. You might just take it as a maybe noteworthy empirical data point that somebody like me, not actively working on the topic, feels the terminology is "obvious"ly, yes, not adhering to usual and reasonable terminology patterns.

But apart from all this, it seems that we can probably easily agree on using the full "proarrow equipment" in the entry title and have a remark on terminology usage in the body?

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeNov 28th 2009

I'm not saying that the terminology shouldn't be changed! I'm just saying it's not obvious to me what the best thing is to change it to. For now, I'll certainly go along with "proarrow equipment" as better than "equipment".

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeNov 29th 2009

I am still for simply "(2-)category with pro-arrows".

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeDec 2nd 2009

Thanks everyone for your feedback! I have changed the page name to 2-category equipped with proarrows, with "proarrow equipment" used occasionally to describe the entire structure. Does that seem acceptable?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeDec 2nd 2009

Yes, thanks!

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeDec 4th 2009

Looks good to me.