terminology for simplicial sets
In something I've been thinking about recently, the condition on a simplicial set that all faces of non-degenerate simplices are non-degenerate seems to play a significant role. Does anyone know whether this condition has been considered previously, and if so whether it has a standard name? The condition is of course satisfied by those simplicial sets which are derived from simplicial complexes in the standard way, but it's more general: it allows the possibility that two (formally) different faces of a non-degenerate simplex might coincide, as long as they're not degenerate. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Peter, Ah, that is a condition I know well, thanks to work of Rina Foygel, a one-time student of mine now in our Statistics department. I had asked her to study the combinatorics of subdivision of categories. You can find a link to a talk that discusses the condition (starting on page 5) on my web page: http://www.math.uchicago.edu/~may Categories, posets, Alexandrov spaces, simplicial complexes, with emphasis on finite spaces. Buenos Aires, November 10, 2008 (dvi)(pdf) The property you ask about is called property A there, and using certain related properties B and C one can prove Theorem. A simplicial set K has property A if and only if its second barycentric subdivision Sd^2(K) is the simplicial set associated to a classical (ordered) simplicial complex. Another result is that if K does not have A, then Sd(K) cannot be a quasi-category. Still another is that if K has A, then Sd(K) is the nerve of a category. One transfers properties A, B, and C to categories via the nerve functor N. Using them, one proves Theorem. The second subdivision sd^2(C) of any category C is a poset. Theorem. For any category C, sd(C) is isomorphic to the `fundamental category' \tau_1(Sd(NC)). Theorem. A category C has property A if and only if Sd(NC) is isomorphic to N(sdC). Moreover, posets are equivalent to Alexandrov $T_0$-spaces, which have weakly homotopy equivalent classical simplicial complexes. These results, and others related to them, shed light on the Thomason model structure on Cat. Peter May On 10/19/10 5:54 AM, Prof. Peter Johnstone wrote:
In something I've been thinking about recently, the condition on a simplicial set that all faces of non-degenerate simplices are non-degenerate seems to play a significant role. Does anyone know whether this condition has been considered previously, and if so whether it has a standard name?
The condition is of course satisfied by those simplicial sets which are derived from simplicial complexes in the standard way, but it's more general: it allows the possibility that two (formally) different faces of a non-degenerate simplex might coincide, as long as they're not degenerate.
Peter Johnstone
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Peter, Though I do not have a direct answer to your question the following seems at least relevant. By a semi-simplicial set, we mean a presheaf on Delta_f, the lluf subcategory of Delta spanned by the face operators (I think the terminology here is nowadays standard). The inclusion i: Delta_f -> Delta induces by left Kan extension a functor Lan_i from the category of semi-simplicial sets to the category of simplicial sets, which is faithful and, though not full, at least full on isomorphisms. Now a simplicial set satisfies the condition you name just when it lies in the replete image of this functor. This suggests that one might reasonably call a simplicial set satisfying your condition "semi-simplicial", and a map between two such "semi-simplicial" when it maps non-degenerate simplices to non-degenerate simplices. Another way of looking at it is that the semi-simplicial objects are those admitting coalgebra structure for the comonad (i^* o Lan_i) on simplicial sets; since Lan_i is full on isomorphisms, such structure will be unique up to unique isomorphism when it exists. The semi-simplicial maps between such objects are those which are coalgebra homomorphisms for some (and hence every) choice of coalgebra structure on their domain and codomain. Richard On 19 October 2010 21:54, Prof. Peter Johnstone <P.T.Johnstone@dpmms.cam.ac.uk> wrote:
In something I've been thinking about recently, the condition on a simplicial set that all faces of non-degenerate simplices are non-degenerate seems to play a significant role. Does anyone know whether this condition has been considered previously, and if so whether it has a standard name?
The condition is of course satisfied by those simplicial sets which are derived from simplicial complexes in the standard way, but it's more general: it allows the possibility that two (formally) different faces of a non-degenerate simplex might coincide, as long as they're not degenerate.
Peter Johnstone
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Richard, Yes, that is relevant. But I don't think one can really use "semi-simplicial" as an adjective to qualify "simplicial set"; and I don't want to call them "semi-simplicial sets" since I want to think of them as simplicial sets with an additional property. I hesitate before adding one more mathematical usage of the word "regular"; but Myles Tierney informs me that it does at least have a respectable pedigree in this context (it was used by Steenrod), and it's surely better than Peter May's "Property A". Peter ---------------- On Wed, 20 Oct 2010, Richard Garner wrote:
Dear Peter,
Though I do not have a direct answer to your question the following seems at least relevant. By a semi-simplicial set, we mean a presheaf on Delta_f, the lluf subcategory of Delta spanned by the face operators (I think the terminology here is nowadays standard). The inclusion i: Delta_f -> Delta induces by left Kan extension a functor Lan_i from the category of semi-simplicial sets to the category of simplicial sets, which is faithful and, though not full, at least full on isomorphisms. Now a simplicial set satisfies the condition you name just when it lies in the replete image of this functor. This suggests that one might reasonably call a simplicial set satisfying your condition "semi-simplicial", and a map between two such "semi-simplicial" when it maps non-degenerate simplices to non-degenerate simplices. Another way of looking at it is that the semi-simplicial objects are those admitting coalgebra structure for the comonad (i^* o Lan_i) on simplicial sets; since Lan_i is full on isomorphisms, such structure will be unique up to unique isomorphism when it exists. The semi-simplicial maps between such objects are those which are coalgebra homomorphisms for some (and hence every) choice of coalgebra structure on their domain and codomain.
Richard
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Peter, "Regular" occurred to me too; for example, a CW complex structure is said to be regular if each of its attaching maps is a homeomorphism. Best, Todd ----- Original Message ----- From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk> To: "Richard Garner" <richard.garner@mq.edu.au> Sent: Wednesday, October 20, 2010 6:50 AM Subject: categories: Re: terminology for simplicial sets
Dear Richard,
Yes, that is relevant. But I don't think one can really use "semi-simplicial" as an adjective to qualify "simplicial set"; and I don't want to call them "semi-simplicial sets" since I want to think of them as simplicial sets with an additional property.
I hesitate before adding one more mathematical usage of the word "regular"; but Myles Tierney informs me that it does at least have a respectable pedigree in this context (it was used by Steenrod), and it's surely better than Peter May's "Property A".
Peter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Regarding the terminology "regular" suggested by Myles, I had written "a CW complex structure is said to be regular if each of its attaching maps is a homeomorphism" but of course I meant "homeomorphism onto its image". Apologies for the confusion. Todd ----- Original Message ----- From: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk> To: "Richard Garner" <richard.garner@mq.edu.au> Sent: Wednesday, October 20, 2010 6:50 AM Subject: categories: Re: terminology for simplicial sets
Dear Richard,
Yes, that is relevant. But I don't think one can really use "semi-simplicial" as an adjective to qualify "simplicial set"; and I don't want to call them "semi-simplicial sets" since I want to think of them as simplicial sets with an additional property.
I hesitate before adding one more mathematical usage of the word "regular"; but Myles Tierney informs me that it does at least have a respectable pedigree in this context (it was used by Steenrod), and it's surely better than Peter May's "Property A".
Peter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Peter (which one ?) After the recent discussions on "evil", I hope we all realized that to have a "descriptive name" is very often conflictive (and evil !!). I would say also that it is always misleading. Welcome Peter May for such a simple and brief name, "Property A" is perfect !!, why hoping for something else ?. Well, seriously, very often in mathematical practice "descriptive" names are not convenient because there is no name that adapts really to the property or concept. In that case, it is much better to use a "neutral" non descriptive name, even just a letter or a meaningles combination of 2 or 3 letters and numbers (may be related to the property, like AB5 for example). I propose the following: Call the property of being invariant under equivalence of categories property IEC, and instead of "evil" use "not IEC". Greetings to all e.j. Prof. Peter Johnstone wrote:
Dear Peter,
Many thanks. Naturally, I'd been hoping that it might have a more descriptive name than "Property A", but if that is what it's called ...
The reason I got interested in it: if you consider the total category of the discrete fibration (over the simplicial category Delta) corresponding to a given simplicial set, the full subcategory whose objects are the non-degenerate simplices is reflective iff Property A holds.
Peter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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Eduardo J. Dubuc -
Peter May -
Prof. Peter Johnstone -
Richard Garner -
Todd Trimble