That was me.

]]>I added a reference to the second Exposé:

- Jay Shah,
*Parametrized higher category theory and higher algebra: Exposé II - Indexed homotopy limits and colimits*, (arXiv:1809.05892)

This is largely “indexed (∞,1)-category theory” in the sense Mike suggested.

Anonymous

]]>I have slightly expanded the text for clarification, added hyperlinks, gave the list of examples an item numbering, fixed the formatting of the references, undid the “due to Barwick and some unnamed others”-style

Looking again at the intro article (here), and remembering that Clark worries about the future of homotopy theory (here) I have the thought that it is unwise to refer to strict fixed point sets as the “honest” ones, thus leaving the reader to conclude that homotopy theory provides the dishonest version of concepts.

]]>Added Clark’s comment that many of their bases are EI (∞,1)-categories. I guess many of these are in addition inverse EI (∞,1)-categories.

Is a stratification always well-founded?

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