Naming of LEM and AC about book

[awodey]

On Mar 8, 2013, at 10:08 AM, Mike Shulman [email protected] wrote:

At tea yesterday it was suggested to name LEM_n the version of LEM restricted to n-types, and AC_n the version of AC that involves the n-truncation. Then LEM_infty and AC_infty would be the PAT versions (so AC_infty is provable and LEM_infty is inconsistent with univalence) and LEM_{-1} and AC_{-1} would be the mere-propositional versions (which hold in the sSet model and thus may consistently be assumed).

One technical problem I see with this is that LEM_infty and AC_infty might also be interpreted to refer to the infty-truncation (hypercompletion) rather than no truncation at all.

wouldn't these be LEM_\omega, etc.?
However, that seems unlikely to be a serious problem, since the untruncated meanings of LEM_infty and AC_infty aren't intended for wide use.

A more serious problem is that there are also another parameters in AC. The version of AC in section 2.8.5 assumes that X is a set, each A(x) is a set, and each P(x,a) is a mere proposition; thus this statement lives entirely in the world of sets. If we remove the requirement that X be a set, the statement becomes inconsistent, since we could take A to be the universal cover of the circle. However, I believe the statement is true in the simplicial model when X is a set and each P(x,a) is a mere proposition, with no truncation hypotheses on A (and also still with propositional truncation in the statement), and at least a priori this seems to be a stronger statement. So there is at least another whole axis of truncation indices that we could vary along. There might even be a third one: I don't know what happens if we remove the truncation hypothesis on P.

interesting.
I guess these newly visible "generalied choice principles" will just have to be specified more carefully individually, when they are studied or used?

[mikeshulman]

On Mar 8, 2013 10:17 AM, "Steve Awodey" [email protected] wrote:

wouldn't these be LEM_\omega, etc.?

That's one possible choice, but one might also argue for the reverse.
Infinity is already used in higher topos theory when referring to
hypercomplete objects.

I guess these newly visible "generalied choice principles" will just have
to be specified more carefully individually, when they are studied or used?

Yeah. I think now that because AC-infinity is an equivalence, the relevant
parameter is not the truncation of A or P, but of Sigma(a:A),P. So there
are actually three parameters.