Subject: categories: Monoidal structure, take II

From: Francois Lamarche <[email-masked]>
Date: Thu, 18 Mar 1999 12:43:41 +0100

Given the private replies I got to yesterday's queries, it is obvious I
was not clear enough, and indeed there was an unhelpful typo.

> 
> I'm wondering, if anybody has ever described the following monoidal
> structure on the category of oriented multigraphs, what MacLane calls
> graphs, the most common kind of graph in category theory (but not

OK Saunders, from now on they're graphs. This what happens when you hang
out with combinatorists AND category theorists.

> 
> Given graphs X, Y, the set |X-oY| of vertices on  X-oY  is the set of graph
> morphisms
> X --> Y.

So right from the start this is not the usual presheaf CC structure,
where the set of vertices is the set of all functions |X| --> |Y| .

So in what follows I use categorical notation for vertices, arrows, etc.
 
> Given f,g : X --> Y the set of arrows f-->g is the set of pairs
> (p_0,p_1) of functions such that
> 
> forall  x in |X|, p_0(x) : f(x)-->g(x)
> 
> forall  k: x-->y in X, p_1(k) : f(x)-->g(y)

Now the typo has been corrected. So an arrow f --> g is like a natural
transformation, with p_0 the usual familly of arrows indexed by the
vertices/objects of X, but since things don't compose, you add the
diagonal p_1 as part of the information. There is some kinship to
homotopies, as M. Barr has remarked.

> This co-contra bifunctor has a tensor left adjoint, which is symmetric
> and monoidal.
> 

Is this more understandable?

Thanks again

Francois

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From: Michael Barr <[email-masked]>
Date: Thu, 18 Mar 1999 12:46:52 -0500 (EST)

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

-------------------------------------------------------------------
History shows that the human mind, fed by constant accessions of
knowledge, periodically grows too large for its theoretical coverings, and
bursts them asunder to appear in new habiliments, as the feeding and
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"

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From: Francois Lamarche <[email-masked]>
Date: Thu, 18 Mar 1999 19:40:57 +0100

Various replies to my query during the day (my thanks to Ronnie Brown
and Lutz Schroeder) made me realize that my SMC structure was actually
the lifting of the CC structure on reflexive graphs (those with a choice
of loop) to ordinary non-reflexive graphs. Here is a bit more detail:

We all know these two categories are categories of presheaves. So let G
and R be the categories such that Set^G is graphs and  Set^R is
reflexive graphs. There is an embedding  G --> R, which generates the
usual triple of functors between the presheaf categories. So it seems
this functorial machinery allows to transform the CC structure in Set^R
into an SMC structure in Set^G, preserving the forgetful functor. There
must be general conditions that allow this.

I'm not pursuing this any more right now, because it has to have been
done before, and in a much more general setting.

Now Michael's comment is also (among other things) about the tension
between graphs and reflexive graphs, and naturally there is Lawvere's
"Qualitative distinctions between some toposes of generalized graphs"
that says a lot about that tension.

there may be more in there than we suspect

Francois