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

]]>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?

]]>Thanks! I’ve added it to the page.

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

]]>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.

]]>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?

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

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.

]]>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.

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