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Inspired by the discussion at directed n-graph and finite category, added some examples and further explanation to computad.
Thanks, Mike. I have added a section on computads as cofibrant resolutions -- at least for strict omega-categories.
I am still wondering about that question I had back then when we had the blog discussion on this: how does free omega-category on a "globular infinity-graph" = computad relate to free category on a "simplicial oo-graph" = simplicial set.
It would be strange if both concepts were not equivalent. Here Michael Barr said that he "strongly suspects" that the free omega-category on a simplicial set is not cofibrant. I haven't made an effort to attack this question formally, but purely intuitively I would on the contrary strongly suspect that it is. What am I missing?
"Globular -graph" seems to me like a poor name for "computad"—it makes me think instead of just a globular set. Why not just say "computad"?
The only thing I can think of right now that could spoil it for simplicial sets might be the presence of degeneracy maps, which computads don't have.
Daniel Schaeppi gives a detailed reply to Mike's question at computad
I have queried the discussion of terminology in computad. It seems to suggest that in general one should use an earlier term, (and I have not checked the original definition of computad in Street's paper). When is the earliest mention of polygraph in the literature?
A related question is when should one term be considered `better' than another? Some of the terms used in our various areas are, quite frankly, dreadful! The intuition behind 'computad' was to generalise monad in some way, but I was not that convinced when I heard it. ...?
You are likely quite right here, Tim. I think I added that comment on terminology originally, because when I once used the term "polygraph" in a blog entry, people jumped at me and told me I should better be saying "computad".
You should feel free to edit the entry accordingly, I think. You don't need to enclose your comment in a query box if chances are high that you are the person with thourough understanding of the subject.
Computad seems like the name for a robot villain in a campy science fiction movie. It seems to associate higher category theory with fads in programming (probably the people who jumped down your throat at the café). I much prefer to call it a polygraph.
I would suggest using computad in the context of describing categories and n-categories, whilst polygraph for the (same) general notion when applied to rewriting. I am working with Philippe Malbos and Yves Guiraud who work in Rewriting and tend to use polygraph, hence my prejudice! I have put a `synonym' page at polygraph, with that in mind.
"Polygraph" makes me think of the version of graphs which underlie polycategories.
Is there a good reason to use a "synonym" page rather than a redirect? I mean, if the notions are really the same, shouldn't we just have one page which mentions both terminologies?
bump. no one answered my question in #9.
I personally feel a redirect suffices (unless “polygraph” needs disambiguation – it doesn’t that I know of).
To quote from computad:
The goal there [Street’s article on 2-computads] was to describe which 2-categories are “finitely presented” (the presentation being given by a 2-computad) in order to describe the correct notion of “finite 2-limit”.
How weak is the notion of finiteness here? Is ’posessing a finite number of isomorphism classes and each hom-set finite’ ok? What about other cardinality bounds? (countable, regular,…) It took me far too long, but I just realised I need 2-computads for a project on 2-category localisations, and as with 1-categories, the boundary where size issues matter needs to be mapped out. For example, the 2-category of internal groupoids, anafunctors and transformations is locally essentially small (and locally locally small) when (external) Choice and internal WISC holds, and this is a localisation (as you well know) of the 2-category of internal categories. In this instance there is a 2-category of fractions, but in general this hasn’t been done - let alone size issues.
A 2-category with finitely many objects, arrows, and 2-cells is finitely presented in Street’s sense; the point is that the converse need not hold (as a trivial example, $B\mathbb{N}$ is finitely presented by a single object and a single endomorphism, but not finite). I think the converse is more or less true for $\kappa$-presented categories for $\kappa\gt\omega$, though.
What I meant was that what if one has a non-finite 2-computad such that the free 2-category on it is equivalent to a finite 2-category? Is this enough? (ditto for higher cardinals) I imagine there’s a notion of equivalence of 2-computads, is this defined intrinsically or depending on equivalence of free 2-categories on them? (I’ll think about the 1-computad=directed graph case in the meantime).
Really I think I only care about the distinction between small and non-small, so (a pair of) inaccesible cardinals should be enough to ponder.
The limits in question are indexed by diagrams of 2-categories, not by diagrams of 2-computads. So what matters is the existence of a computad of a certain cardinality generating the 2-category, and any finite 2-category can certainly be presented in terms of finite computads. It makes no difference whether the finite 2-category could also be presented using infinite computads.
If what you care about is limits over categories that are not finite but are equivalent to finite ones, then of course the distinction doesn’t matter if you’re looking at fully weak “bi” limits, whereas if you care about strict 2-limits then I wouldn’t expect knowing something about an equivalent 2-category to tell you much of anything.
Ah, we have a bit of communication problem. I’m not interested particularly in limits, but in bounded (finite or small) presentation. If I construct a 2-computad of a certain size, hopefully small, how big is the free 2-category on it? (did I say weak 2-cat? I meant it - I must be turning into an Australian category theorist of sorts) In other words, does the adjunction between 2Comp and 2Cat (hmm, which 2Cat?) restrict to one between small 2-computads and locally essentially small 2-categories?
Ah. The free 2-category on a 2-computad of size $\lt\kappa$ will always also be of size $\lt \kappa$ for any uncountably infinite cardinal $\kappa$, since there are only ever countably many free composites to add in. The only mild subtlety is that a finite 2-computad can generate a countably infinite 2-category. Does that answer your question?
Yes, thanks.
Thanks.
I added to computad a sketch of a generalization from globular sets to arbitrary inverse diagrams.
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