terminology for simplicial sets

terminology for simplicial sets

[Prof. Peter Johnstone]

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


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Re: terminology for simplicial sets

[Peter May]

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
>


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Re: terminology for simplicial sets

[Richard Garner]

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
>

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Re: terminology for simplicial sets

[Peter May]

I agree with Peter J. that a good name is a good thing.

However, he has suggested that ``regular'' is surely better than
``Property A'', and I have to say that it most assuredly is not.
Regular CW complex has a standard meaning, and there is a
related standard meaning for regular simplicial set, which is
recalled on page 4 of the link I sent originally
   (http://www.math.uchicago.edu/~may
Categories, posets, Alexandrov spaces, simplicial complexes,
with emphasis on finite spaces. Buenos Aires, November 10, 2008)

A nondegenerate  n-simplex x in a simplicial set K is regular if the
subcomplex [x] that it generates is the pushout of  the last face
inclusion  \Delta_{n-1}  \to  \Delta_{n}  along  the last face
d_n x : \Delta[n-1] \to [d_nx].   K itself is regular if all of its
nondegenerate simplices are regular.

The subdivision of any simplicial set is regular.

This definition is standard because  the realization of a regular
simplicial
set is a regular CW complex, and regular CW complexes are triangulable,
that is homeomorphic to the realization of a simplicial set coming from a
classical simplicial complex.  This is all classical, and I could cite a
number
of sources. A modern one (1990) with a good treatment is Fritsch and
Piccinini,  Cellular structures in topology.  See p. 208.

I hold no particular brief for Property A (and B and C), but they
will do until something definitely better comes along. Richard has
suggested semisimplicial, but that to my mind is certainly not better
(but then I'm old enough to remember when semisimplicial meant
what we now call simplicial, to differentiate from classical simplicial
complexes).

Peter


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Re: terminology for simplicial sets

[Eduardo J. Dubuc]

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
  >

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