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Atrium > Mathematics, Physics & Philosophy: 'Model' of an n-category versus model category
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- CommentRowNumber1.
- CommentAuthorHarry Gindi
- CommentTimeDec 14th 2009
I recently ran into a problem on MathOverflow, where I was talking about both model categories, and "models" of an n-category (specifically (\infty,1) in this case, but it doesn't matter). Is there a better word for a "model", used just as an english word here? I mean, it's somewhat like "construction," but stronger, while it's not as strong as a "characterization". Any ideas? Are there any conventions on this? -
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeDec 14th 2009
For emphasis one could say that among all the models for higher categories, there is particularly Quillen's model for (oo,1)-categories.
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- CommentRowNumber3.
- CommentAuthorHarry Gindi
- CommentTimeDec 14th 2009
Is that the best answer you can give? -
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeDec 14th 2009
<div> <blockquote> Is that the best answer you can give? </blockquote> <p>I am sensing a certain dissatisfaction with that answer. :-)</p> <p>I must say I do like the answer I gave. It is rarely put explicitly this way, but it deserves to be said that "model" in "Quillen model category" means precisely (with a bit of historical hindsight) "Quillens model for (oo,1)-categories".</p> <p>So instead of inventing new terminology, why not embrace the rare coincidence that historically evolved terminology does happen to be logical and fitting?</p> </div> -
- CommentRowNumber5.
- CommentAuthorTobyBartels
- CommentTimeDec 14th 2009
But etymologically, the ‘model’ in ‘model category’ means that the model category is a 1-categorial model for a specific (oo,1)-category, while Harry wants to say that model categories comprise a model for the entire concept of (oo,1)-categories.
Besides ‘construction’, Harry, how about ‘definition’? That is, one could write ‘An (oo,1)-category is [...].’, substituting the definition of model category, quasicategory, simplicially enriched category, or so on. Of course, some of these might not make very good defintions, which is why ‘model’ is a better word, but one might naïvely try them.
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- CommentRowNumber6.
- CommentAuthorHarry Gindi
- CommentTimeDec 14th 2009
- (edited Dec 14th 2009)
No, I don't want to change the terminology! I just want a term for "model" as in the sense of "a construction" that is completely explicit. 'Definition' won't work either, because the notion of an (infty,1) category is (should be?) independent of the 'model' we choose for it. That is, in 'the book', there's the perfect notion of an (infty,1) category, and all of these very different constructions are just trying to 'model' this perfect definition of (infty,1)-categoryness. -
- CommentRowNumber7.
- CommentAuthorTobyBartels
- CommentTimeDec 14th 2009
I would contrast a ‘notion’ (which should be perfect) with a ‘definition’ (which may never be perfect). The latter term is pretty well established when talking about definitions of n-category.
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- CommentRowNumber8.
- CommentAuthorHarry Gindi
- CommentTimeDec 14th 2009
- (edited Dec 14th 2009)
Oh, by the way, is defining infinity categories axiomatically pretty much accepted to be an intractable problem? Due to combinatorial explosions and all that. -
- CommentRowNumber9.
- CommentAuthorTobyBartels
- CommentTimeDec 14th 2009
You just need some clever idea to handle those. Coming up with such an idea is the hard part, but not necessarily intractable. I don't know what the cutting edge is on the question, but maybe somebody else will speak up.
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- CommentRowNumber10.
- CommentAuthorMike Shulman
- CommentTimeDec 14th 2009
I think that actually, the "model" in "model category" refers to the objects of the model category as "models" for homotopy types (or objects of some other homotopy theory). So "model category" really means "category of models."
I agree with Toby that "definition" can be used for the precise ones, with "notion" or "idea" or "concept" for the thing that they're all trying to model.
Harry, I'm not sure what you mean by "defining
-categories axiomatically" -- is there something "non-axiomatic" about all the existing definitions listed at n-category? -
- CommentRowNumber11.
- CommentAuthorHarry Gindi
- CommentTimeDec 15th 2009
I mean sure, those are axiomatic, but they require you to have lots of background to even understand the definition. I mean something simple. I mean, it's the difference of defintitions between varieties and manifolds. Sure, theyr'e very similar, but it was much harder to generalize a manifold because it couldn't be defined purely algebraically (hush about the zariski topology, you know the difference!). That's the distinction I''m trying to make. -
- CommentRowNumber12.
- CommentAuthorTobyBartels
- CommentTimeDec 15th 2009
I took what Harry said to mean axiomatising the notion of
-category in a way analagous to the way in which ETCS axiomatises the notion of 
-category (set); my answer #9 should be interpreted in that light. -
- CommentRowNumber13.
- CommentAuthorHarry Gindi
- CommentTimeDec 15th 2009
That would work too. -
- CommentRowNumber14.
- CommentAuthorMike Shulman
- CommentTimeDec 15th 2009
So Harry, it sounds like your real question is, is there a simple definition of
-category? I think the answer to that question is that no one has yet come up with one. -
- CommentRowNumber15.
- CommentAuthorHarry Gindi
- CommentTimeDec 15th 2009
What I meant was, "Has it basically been shown that it's not likely that one exists because of the whole combinatorial explosion problem?" -
- CommentRowNumber16.
- CommentAuthorTobyBartels
- CommentTimeDec 15th 2009
No, I don't think that anybody would accept that reason. After all, people have come up with definitions of
-category, and while none of these is exactly simple, still there are precise finite deifnitions that don't involve infinitely many clauses due to combinatorial explosion. That is, all of them take steps to tie up the necessary combinatorial data into a finite package. -
- CommentRowNumber17.
- CommentAuthorzskoda
- CommentTimeDec 15th 2009
Joyal says at one place that quasi-categories ar an example of an infinity category. Maybe there will be useful versions/examples at some point which are not equivalent to others. Saying model presumes the conjecture that all the infinity categories if they have the same numbers a la Baez (infty, r) then they are "the same" in approrpiate sense (equivalence of model categories of such is not sufficient as this is just (infty,1)-equivalence, one should have equivalence of (infty,r+1)-categories of (infty,r)-categories, so r jumps by one in true statement hopefully).
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- CommentRowNumber18.
- CommentAuthorUrs
- CommentTimeDec 15th 2009
<div> <blockquote> the whole combinatorial explosion problem?" </blockquote> <p>Is the explosion necessarily so explosive?</p> <p>The definition of a <a href="https://ncatlab.org/nlab/show/weak+compicial+set">weak compicial set</a>, for instance, is only slightly more intricate than that of a Kan complex, and still supposedly models arbitrary oo-categories.</p> <p>Or for instance the recursive definition of <a href="https://ncatlab.org/nlab/show/Trimble+n-category">Trimble n-category</a> is very succinct, and also supposed to give a definition of general oo-categories.</p> </div> -
- CommentRowNumber19.
- CommentAuthorUrs
- CommentTimeDec 15th 2009
- (edited Dec 15th 2009)
<div> <blockquote> I think that actually, the "model" in "model category" refers to the objects of the model category as "models" for homotopy types </blockquote> <p>Can we find out from a historical source? I always felt "model" was meant as "model for homotopy theory". Otherwise, the term "category of models" would have been more fitting.</p> </div> -
- CommentRowNumber20.
- CommentAuthorUrs
- CommentTimeDec 15th 2009
<div> <blockquote> Has it basically been shown that it's not likely that one exists because of the whole combinatorial explosion problem? </blockquote> <p>Just to amplify: no, on the contrary: it has been shown that reasonably tractable definitions of oo-categories do exist. Another point in case is Rezk's <a href="https://ncatlab.org/nlab/show/Theta+space">Theta space</a>s, which are claimed to model (n,r) categories for all n and r, including infinity. This is a pretty cool model as it lends itself seamlessly to a definition of a (oo,n)-stack in terms of a <a href="https://ncatlab.org/nlab/show/homotopical+presheaf">homotopical presheaf</a>.</p> </div> -
- CommentRowNumber21.
- CommentAuthorMike Shulman
- CommentTimeDec 15th 2009
@Zoran: I would argue that if two purported definitions of n-category turn out not to be equivalent, then at least one of them should be called something else, though without necessarily implying that it is any less interesting. As everyone probably knows by now, I definitely believe that there are very interesting and useful "higher categorical structures" that are not n-categories. But I do believe that there is a unique particular notion out there which should be given the name "n-category."
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