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Added:
The original result is due to Lurie:
A considerably simplified presentation is available in
I have added an “Idea”-section header and one sentence to the very top, in the spirit of starting out with a single sentence or two that expresses the whole idea, before going into more details.
With this, the line “This is a subentry of…” could now be dropped, I think. But I leave that to you.
Probably being dim, but what is $C$ in
$(St_\phi\dashv Un_\phi) \;\colon\; sSet/S \stackrel{St}{\to} [C^{op}, sSet]$I’ve added in what $C$ and $\phi$ are.
Thanks. So $\phi$ is the simplicial functor $\mathfrak{C}[S] \to C$?
Dare I ask further what operation $\mathfrak{C}$ is?
It’s the left part of the Quillen equivalence between the model structures on quasicategories and simplicially enriched categories. I don’t know what page mentions it off hand. (too distracted to do a search atm)
Yes, we have a page for this (which would deserve improvement, as always): relation between quasi-categories and simplicial categories.
I am adding a pointer to this entry here.
added more references:
Danny Stevenson, Covariant Model Structures and Simplicial Localization, North-West. Eur. J. Math. 3 (2017) 141-202 [arXiv:1512.04815]
Gijs Heuts, Ieke Moerdijk, Left fibrations and homotopy colimits II [arXiv:1602.01274]
Alexander Campbell, A modular proof of the straightening theorem, talk at Macquarie University (2020) [pdf]
The page contains a justification for the “straightening” terminology:
These names have been chosen due to the fact that objects in the left hand category are defined by existential assertions and choices where on the right side these properties become coherence laws being part of the structure.
But this is not an explanation, since it is not obvious why coherence laws should be “straighter” than existential assertions. I looked in HTT, but could not see an explanation there.
I think it’s a general metaphor for converting things defined “up to homotopy” into things defined strictly.
For example, the fact that the category of simplicially enriched categories is a model for $(\infty,1)Cat$ I’ve heard described as saying “every $(\infty,1)$-category can be straightened into a simplicially enriched category”. Analogously to how every 2-category is equivalent to a strict 2-category.
The older term was ’rectification’ I believe and was used in categorical circles for years.
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