Not signed in (Sign In)

# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Sign in using OpenID

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeDec 15th 2018

Add missing axiom that weak equivalences are stable under filtered colimits

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeDec 15th 2018

Added an explanation of how the DK result implies that sSet is excellent.

• CommentRowNumber3.
• CommentAuthorDylan Wilson
• CommentTimeDec 16th 2018
Since this came up somewhere else, it's probably good to note that Tyler Lawson proved that the difficult-to-check "invertibility hypothesis" actually follows from the other axioms: http://www-users.math.umn.edu/~tlawson/papers/invertibility.pdf
• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeDec 16th 2018

Nice, thanks! Do you feel like updating the page?

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeDec 17th 2018

I’m not sure what I think about this proof. It seems to go like this:

1. Simplicial sets satisfy the invertibility hypothesis (by work of Dwyer and Kan).
2. Cubical sets are monoidally Quillen equivalent to simplicial sets, hence also satisfy the invertibility hypothesis.
3. Any otherwise-excellent model category admits a monoidal Quillen functor from cubical sets, hence also satisfies the invertibility hypothesis.

Part of me feels as though the use of simplicial and cubical sets ought to be $\beta$-reducible out of this proof to obtain a proof that actually works directly with an otherwise-excellent model category. But another part of me finds it a pleasing application of the “classifier principle” that if you can find a universal object with some structure (e.g. classifying space, classifying topos, syntactic category of a type theory), then working directly with that universal object can be an easier way to prove that something holds in all such structures.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJan 17th 2019

Added more examples and a properties section, and also some discussion of Lawson’s proof that the invertibility hypothesis is redundant.

1. Added clarification about Lumsdaine–Shulman usage, to forestall confusion

Peter LeFanu Lumsdaine

2. fixed bad wiki link from prev edit

Peter LeFanu Lumsdaine

Add your comments
• Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
• To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

• (Help)