Re: Monoidal structure, take II

[Michael Barr <barr@triples.math.mcgill.ca>]

I did not quite understand Francois' construction.  However, my first
reaction to a question like that is that it ought to be a homotopy.  So
I will say what a homotopy reduces to in this case and leave it to
Francois to decide if this is what he has.

I suspect that rather few people know what a simplicial homotopy is and,
of those, rather few have ever actually verified one.  I am in that
minority^2, so perhaps I tend to see them where they are not the most
natural, but I think it quite remarkable that they can arise where no
real topology (but some combinatorics) is present.

I have to begin by saying how a graph becomes a simplicial set.
Actually, that is a lie, since unless you are dealing with reflexive
graphs--that are equipped with a selected loop at each vertex--you will
only get a face complex.  But homotopies are still definable.  A
category is a simplicial set by taking for n-simplexes composable
n-tuples of arrows.  This doesn't work for graphs, since the "interior
faces" (all except the lowest and highest numbered) all depend on
composition.  But there is a face complex in which an n-simplex is
simply an n-simplex in the graph.  So a 2-simplex is a
triangle--obviously non-commutative and a 3-simplex is a tetrahedron and
so on.  You can describe a composable n-tuple in a category as
commutative n-simplex, so this isn't so different.  Now given this, if
f,g:  X --> Y are graph morphisms, what is a homotopy?  Well, write X as
d^0,d^1:  X_1 --> X_0 and similarly for Y. Then f consists of f_0:  X_0
--> Y_0 and f_1:  X_1 --> Y_1 giving a serially commutative square.
Just a homomtopy between functors turns out to be simply a natural
transformation, a homotopy in this case turns out to consist of a
function p_0:  X_0 --> Y_1 and a function p_1:  X_1 --> Y_1 such that
there
is a diagram (not, of course commutative; what a diagram does is specify
source and target) as follows.  In this diagram I assume x:  x^0 --> x^1
in X, and f(x):  y^0 --> y^1 and g(x):  z^0 --> z^1 in Y.
                            f(x)
                    y^0 -----------> y^1
                     | \              |
                     |  \             |
                     |   \            |
                     |    \           |
                     |     \          |
                     |      \         |
                     |       \        |
             p_0(x^0)|  p_1(x)\       |p_0(x^1)
                     |         \      |
                     |          \     |
                     |           \    |
                     |            \   |
                     |             \  |
                     |              \ |
                     v      g(x)     vv
                    z^0 -----------> z^1

So if this is what Francois was saying, then the answer is it a homotopy
of face complexes.  Of course, if you replaced X_1 by X_1 + X_0, you
would have a reflexive graph and I assume (I have not checked this) you
would then get a simplicial homotopy.

BTW, homotopies do not generally compose--and the ones described here do
not appear to either.  Categories are special because of their internal
composition.  It makes me wonder if the well-known failure of dinats to
compose could be related to this in some way.

Having seen Francois' clarification, I think this is exactly what he
had.

Michael



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History shows that the human mind, fed by constant accessions of
knowledge, periodically grows too large for its theoretical coverings, and
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growing grub, at intervals, casts its too narrow skin and assumes
another... Truly the imago state of Man seems to be terribly distant, but
every moult is a step gained. 

- Charles Darwin, from "The Origin of Species"