Ah, I just thought the whole system was having a clock malfunction and re-sending a discussion from 2016… :-P On the serious side: one motivation for “identification” not yet mentioned is that it’s more in line with traditional mathematical usage. Saying that the circle defined as R/Z or as a subset of R^2 are “equal” clashes badly with the traditional mathematical usage of words, and (for at least some mathematicians) their mental picture of mathematics — but saying that we can identify them, and that all nice properties respect such identifications, is completely consistent with traditional writing and worldviews. Also, the fact that we may identify things in multiple ways, and so have to be a little careful about how we make use of such identifications, fits with traditional intuition and practice. So saying “identifications” may require less rewiring in our brains than saying “equalities” or “paths”, at least in types of algebraic structures and similar. (For synthetic homotopy theory, of course, “paths” often fits closer to established usage.) –p. On Mon, May 4, 2020 at 6:07 PM Steve Awodey wrote: > I’m afraid that someone may have hacked Thorsten’s email account. The real > Thorsten went through all this with us many years ago. > : - ) > > > > On May 4, 2020, at 12:00, Michael Shulman wrote: > > > > The word "path" is closely tied to the homotopy interpretation, and to > > the classical perspective of oo-groupoids presented via topological > > spaces, which has various problems. This is particularly an issue in > > cohesive type theory, where there is a separate "point-set level" > > notion of path that it is important to distinguish from > > identifications. > > > >> On Mon, May 4, 2020 at 7:48 AM Stefan Monnier > wrote: > >> > >>> I don't think using "identification" necessarily implies any > >>> difference between "identification" and "equality". I don't think of > >>> it that way. For me the point is just to have a word that refers to > >>> an *element* of an identity type. Calling it "an equality" can have > >>> the wrong connotation because classically, an equality is just a > >>> proposition (or a true proposition), whereas an element of an identity > >>> type carries information. Calling it "an identification" suggests > >>> exactly the information that it carries: a way of identifying two > >>> things. > >> > >> I thought that's what "path" was for? > >> > >> > >> Stefan "who really doesn't know what he's talking about" > >> > > > > -- > > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyT...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQx_2TinRHBrmOAZFnmFp8VVQ-yMcPvtKFtX-d9wGsD%2B2Q%40mail.gmail.com > . > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyT...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/E31B538B-E0C9-4D4F-A4F1-4335E59CAE0D%40gmail.com > . >