Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 21 of 21
For emphasis one could say that among all the models for higher categories, there is particularly Quillen's model for (oo,1)-categories.
<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>
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.
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.
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.
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?
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.
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.
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.
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).
<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>
<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>
<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>
@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."
1 to 21 of 21