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I worked on brushing up (infinity,1)-category a little
mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.
in this context I also rearranged the order of the subsections
I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.
added a paragraph to the beginning of the subsection on model categories
added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)
Looking back at (infinity,1)-category I found that lots of context was missing there.
As a first step in an attempt to correct this, I created a subsection “Properties” with some pointers to relevant other entries.
I added the reference
This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.
In the entry on (infinity,1)-category there is the phrase:an (∞,1)-category is an internal to in ∞-groupoids/basic homotopy theory.
I tried to see how to clear up the grammar, but it was not clear to me what the wording was intended to be. There was a previous version:
To some extent an (∞,1)-category can be thought of as a category enriched in (∞,0)-categories, namely in ∞-groupoids.
That is vague, so needed changing, but there seem to be ’typos’ in the current version.
It was probably supposed to be “an internal category in …” or “a category internal to …”.
Yes, but is that completely correct? The ‘enriched’ version to me was clearer. Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid, or am I mistaken?
Thanks for catching that, I have fixed the sentence now and expanded it such as to read as follows:
More precisely, this is the notion of category up to coherent homotopy: an $(\infty,1)$-category is equivalently
an internal category in ∞-groupoids/basic homotopy theory (as such usually modeled as a complete Segal space).
a category homotopy enriched over ∞Grpd (as such usually modeled as a Segal category).
Is an (∞,1)-category really an internal category in ∞-groupoids, as that would mean the object of objects would be an ∞-groupoid,
Yes, it’s the completeness condition of complete Segal spaces that takes care of this issue. Details are at internal category in an (∞,1)-category.
Added pointer to the new preprint by Ayala and Rozenblyum. Though it doesn’t seem to have the previously announced statement about $(\infty,n)$-categories with duals yet.
Has anything more been made of the other approach to stratified spaces where one moves up through strata and back down again? You may remember that discussion at the Cafe here. It gave rise to Transversal homotopy theory by Jon Woolf, who also wrote a paper mentioned by Ayala and Rozenblyum, The fundamental category of a stratified space.
The idea was to give fundamental categories with duals.
have added pointer to
added pointer to
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