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Spurred on by the vigourous discussion on what directed homotopy should be, I have started a page looking at some of the motivation that people have used for directed homotopy theory. The present emphasis is on Goubault’ s work about 10 years ago on modelling concurrency, ( see here Models for concurrency), and I will build new entries on various of those models, such as higher transition systems which are based on cubical sets.
I have left a query on the Models for concurrency entry since I do not really know what is meant by ’Rendezvous model’ and that description needs some work done on it.
I have started a page
Here is the link to the page that Tim started: Motivation for directed homotopy
A question, on the application in concurrency:
what is the application of directed homotopies (as opposed to just directed spaces)? What do they model?
is it the directed homotopy type of a space that matters, or the directed space itself?
P.S. By the way, we have a kind of convention not to capitalize page titles unless there is a particular reason. Might it be good to rename your entries and de-capitalize them?
Capitalisation… will do. i was forgetting that.
The concurrent system is often used to define a directed homotopy model (Goubault uses homotopies through directed paths but the homotopies themselves are through directed paths but are not themselves directed, so he uses ’, not directed , at least in the paper I am consulting.)
The directed homotopies at this initial level model equivalence between directed paths, say, when using interleaving between the various directions. The directed homotopy type is modelled using a ’compressed’ version of the fundamental category. (I hope to get onto that later on, but need to revise the papers that discuss it.) The idea is to look for ’deadlock’ and ’inaccessible’ states by using directed homotopy invariants.
By the way, we have a kind of convention not to capitalize page titles unless there is a particular reason. Might it be good to rename your entries and de-capitalize them?
Actually, in this case I think there is a good reason to keep that page capitalized. Over the weekend I even went back and changed the capitalization of
and the “understanding” entries at
There is a difference between a “dictionary like” page and an “article” or “chapter” like page. I think if the title of a page is longish like that and can be thought of as a chapter title or something, it should be capitalized. It makes sense for simple nouns (as in dictionary-like) entries to be lower case, but some page titles look awkward lower case. This and the earlier “Motivation” are example titles that look awkward lower case.
I was even going to start a page called “Motivations” or something and describe a series of articles that are distinct from but complementary to the corresponding entries. This page would link to a (hopefully growing) list of motivational pages.
For a page that is better thought of as an article or chapter, maybe we can create a new category or something. Pages in this category would be capitalized. I was using “category: reference”, but maybe something like “category: article” or “category: chapter” or something might be better. Oh! Maybe “category: motivation” :)
What do you think?
I have put a query in models for concurrency as I am not sure about a description there. Help please!
I think that new categories for motivation and even understanding .. (sort of ’topos theory for dummies’ a la n-lab would become topos theory:understanding) or some similar title may be a good idea.
Have created a stub for Marzurkiewicz trace theory, and a new entry on trace alphabets
This is a page in need of some care. Perhaps we should take what’s valuable and integrate it into directed homotopy theory.
have cleared the page and am merging relevant content into directed homotopy theory
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