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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeApr 28th 2016

At path category there is disambiguation between several senses, none of which seems to be what Benno van den Berg is using in Path categories and propositional identity types.

The notion of a path category is a slight strengthening of Brown’s classic notion of a category of fibrant objects [6] and was introduced in [5], where also many of its basic properties were established.

[5] is B. van den Berg and I. Moerdijk. Exact completion of path categories and algebraic set theory, arXiv:1603.02456, while [6] is BrownAHT. I see in [5] they compare their approach to Mike’s in ’Univalence for inverse diagrams and homotopy canonicity’.

I’ll leave it to the better qualified to decide what there is to be added to nLab.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeApr 28th 2016

I haven’t had the chance to read that paper carefully yet, but as they say, their “path categories” are similar to what other people call “categories of fibrant objects” or “fibration categories”. I guess time to disambiguate yet again.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 28th 2016

By the way, the entry fibration category is waiting for somebody to fill in more variants.

• CommentRowNumber4.
• CommentAuthorspitters
• CommentTimeJun 1st 2016

path category refers to Gabriel and Zisman for the free category on a directed graph. However, I could not find that terminology in the index, neither do I have a searchable PDF. Does someone have a quick reference?

• CommentRowNumber5.
• CommentAuthorThomas Holder
• CommentTimeJun 1st 2016
• (edited Jun 1st 2016)

They call it the category of paths and it occurs at various places in their dictionary with the definition right on page 1. More elaborate discussions of the free category can be found in e.g. the Mac Lane book, Borceux vol.1, or the Barr-Wells book ’CT for computer science’.

Related things are done with free cats with extra-structure & their path cats in two papers by Walters in no. 62 (1989) of JPAA or by Latch here. All three concern context free grammars.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 1st 2016

Just as a kind reminder: one of you should now go and add this information to the nLab entry. Thanks!

• CommentRowNumber7.
• CommentAuthorspitters
• CommentTimeJun 1st 2016

Thanks! I’ve added it to the page.

• CommentRowNumber8.
• CommentAuthorspitters
• CommentTimeJun 1st 2016

The page consistently has lists: $a_2,f_1,a_0$. I guess this should be $a_1,f_1,a_0$. Or am I overlooking something?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 1st 2016

Clearly a typo, proliferated by copy-and-pasting. I have fixed it now.