expliciting the hom-set condition for precategories about unimath HOT 10 CLOSED

If you're interested in the attempt in the HoTT/HoTT library, I have a branch, too, that makes this assumption explicit, up to the Yoneda lemma. The first place that I needed truncatedness was to prove the composition laws for natural transformations, and thus for building functor categories.

from unimath.

On 11/25/2014 06:54 PM, Jason Gross wrote:

If you're interested in the attempt in the HoTT library, I have a
branch,
too, that makes this assumption explicit, up to the Yoneda lemma.
The first place that I needed truncatedness was to prove the
composition laws for natural transformations, and thus for building
functor categories.

Thanks for the link! The situation in RC is the same.

from unimath.

Is this an issue? If so, what action should we consider?

from unimath.

On 11/25/2014 08:18 PM, Daniel R. Grayson wrote:

Is this an issue? If so, what action should we consider?

Yes, this is an issue, rather than a PR, since I do not request to pull
the code yet, but would like to have an opinion or two first.
There is a lot of code concerned, in particular yours.

from unimath.

@benediktahrens, did you find any (significant?) slowdown, in particular around functor categories? (I have some timing scripts that work if you have a make TIMED=1 variable that works like the TIMED=1 variable of Coq trunk's coq_makefile-made makefile, and will auto-display a nice diff like this.)

from unimath.

On 11/25/2014 08:44 PM, Jason Gross wrote:

@benediktahrens, did you find any (significant?) slowdown, in
particular around functor categories? (I have some timing
scripts
that work if you have a make TIMED=1 variable that works like the
TIMED=1 variable of Coq trunk's coq_makefile-made makefile, and
will auto-display a nice diff like
this.)

No, I have not noticed any slowdown.
I had already admired the sophisticated setup you have for the HoTT
library - some of the features will hopefully be available in UniMath at
some point. In case your stuff is free software, I would like to take a
look at it (not right now, though).

from unimath.

Great!

I think the definition of an iso needs to be corrected and this should allow to remove the has_homsets from all the
places related to isos.

The first place (other than the statements that themselves concern the h-level of various objects) where the condition
has_homsets should be used in a substantial way is, as far as I could see, this one:

is_precategory_functor_precategory_data

and this is very suggestive.

Vladimir.

On Nov 25, 2014, at 12:57 PM, Benedikt Ahrens [email protected] wrote:

I have a branch where the condition of hom-types being sets is factored out, i.e. a precategory now has mor: ob -> ob -> UU. Hence an assumption (hs: has_homsets C) has to be added wherever necessary.
https://github.com/benediktahrens/UniMath/tree/hom-hset-explicit https://github.com/benediktahrens/UniMath/tree/hom-hset-explicit
Most constructions actually need that assumption of hom-types being sets; or at least I have not made an effort to do without. That makes the code a bit cumbersome.
Also, the definition of isomorphism is not the right one without that assumption, but I have not given another one yet.
I am not sure whether this change should go into UniMath.
On the other hand, the hom-set condition is part of a more general "saturation" condition, see https://github.com/benediktahrens/folds https://github.com/benediktahrens/folds .
In that spirit, it would make sense to have it separated from the definition of a precategory.

—
Reply to this email directly or view it on GitHub #54.

from unimath.

On Nov 25, 2014, at 5:54 PM, Jason Gross [email protected] wrote:

The first place that I needed truncatedness was to prove the composition laws for natural transformations,

It should be possible to do the composition itself without, only the unit and associativity structures in [C, C’] should require
has_homsets for C’ or the next level of the coherence conditions on the composition in C’.

from unimath.

It should be possible to do the composition itself without, only the unit and associativity structures

Yes. Unit and associativity structures is what I meant by "composition laws".

from unimath.

Done.

from unimath.