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am starting to work again on locally presentable (infinity,1)-category
not much new yet, apart from some polishing. But at least the title I have finally changed, following Mike’s suggestion.
That terminological issue should really bother me less than it does.
It’s locally-presentable-$(\infty,1)$-category-theory-day! :-)
So I am in the process of adding material. Some things I did so far
expanded definitions and properties at locally presentable (infinity,1)-category a bit more (and more to come);
expanded Pr(infinity,1)Cat a bit more, in particular added a brief section on how to think of it as something like $\infty Vect$.
created stub for adjoint (infinity,1)-functor theorem.
added the detailed proofs for why a locally presentable (oo,1)-category (defined as an accessible oo-cat with all colimits) also has all limits.
The proof is really general abstract and relies on the (oo,1)-Yoneda lemma. So it is actually verbatim the proof that would also work for 1-categories, or for any notion of higher category that satisfies abstract category theory.
I have been trying to tidy-up locally presentable (infinity,1)-category a bit. Now I will try to fill in more details…
Added the statement about transport of reflective sub-(oo,1)-categories along left adjoint $\infty$-functors.
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