On 27/08/16 17:05, Waldek Hebisch wrote:
Martin, what gave you idea that cochain complex is obtained
by inverting matrices? Chain complex requires that product
of two consequtive matrices is 0, so normally matrices
are noninvertible (some may be invertible, but this is an
exception).
I eventually worked that out (especially since the matrices are not even
square). I guess my thinking was that, to turn something into
cosomething, I need to reverse the arrows and, in this case the arrows
are linear maps, so I need to reverse the linear maps and since the
linear maps are represented by matrices I need to invert the matrix.
BTW I do remember writing something about inverting the matrix but I
can't find it now. It does not matter if its in this forum but if its in
the code then I should correct it.
You could obtain cochain complex by transposing matrices,
or more generally applying HOM functor with fixed second
argument, but that is not the only way to produce
cochain complexes.
How can I work with hom functors in FriCAS?
Let me take a very simple example of a cochain complex. Say we have a
simple electrical network consisting of resistors and batteries (a
discrete version Maxwells equations in Kurt s excellent tutorial).
In the chain the nodes are represented by indexes:NNI in the Rep.
So in the cochain, everywhere the node indexes appear in Rep it is now
replaced by NNI->Float which maps the node to its voltage value.
Similarly edges are replaced by NNI->Float which represents the voltage
drop across the resistor.
A hollow triangle would give zero because the voltage drop around the
loop is zero.
So to model this in FriCAS I need to have function types like NNI->Float
in the Rep and, most importantly, I need to be able to compose these
functions. So to get the edge:(NNI->Float) I need to be able to compose
the node(NNI->Float) with the boundary function.
Do you think this is practical to implement in FriCAS?
Martin B
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