On 10/02/2025 01:07, Waldek Hebisch wrote:
On Sun, Feb 09, 2025 at 09:03:57PM +0100, Waldek Hebisch wrote:
So feel free to do whatever you want with IsArrow?
Well, from the above you can see that I would write rather different
categories and domains that you did. But I do not want to break
what works, so I will probably do only minimal changes.
Attached is my proposed diff to logic.spad.
Looks good to me. I find its useful to have instances (models) of Poset,
Preorder, Lattice and similar structures in addition to their categories.
In order to implement simplicial sets I would also like a Weak Total
Order implementation.
I only mention this here as it is a potential similar example and I just
wanted to get some indication if it might be a potential addition to
FriCAS at some stage in the future. As I'm sure you can tell I have not
thought this through so don't take it too seriously.
I was thinking of using List NNI for the Rep but that does have more
information than is needed. All that is needed is how many entries and
how many duplicates of each one. I'm not sure of the most efficient way
to represent that. Then they would need to be used to index and compare
in two ways: With degeneracies included and with degeneracies excluded.
Then they would have the following functions defined on them:
)abbrev category OPF OrderPreservingFunctions
++ Order preserving functions can be decomposed into face maps d(i)
++ and degeneracy maps s(i) and they obey the following identities:
++ if i < j then d(i)*d(j) = d(j-1)*d(i)
++ if i < j then d(i)*s(j) = s(j-1)*d(i)
++ d(j)*s(j) = id = d(j+1)*s(j)
++ if i > j+1 then d(i)*s(j) = s(j)*d(i-1)
++ if i <= j then s(i)*s(j) = s(j+1)*s(i)
++ where i and j are NNI indexes and * is composition operator
OrderPreservingFunctions() : Category == Definition where
Definition ==> ???? with
d : (index : NNI, input : %) -> %
++ face maps
++ remove entry at given index.
s : (index : NNI, input : %) -> %
++ degeneracy maps
++ duplicate entry at given index
Martin
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