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It is well known that any homotopy type can be obtained as the classifying space of a ($1$-)category. The classifying space of a category $\mathcal{C}$ can be interpreted in at least two ways:

  1. We can view $\mathcal{C}$ as an object in the $(\infty,1)$-category of $(\infty,1)$-categories, and localise $\mathcal{C}$ along all its morphisms.
  2. We can consider the constant functor $\mathcal{C} \to \mathbf{Type}, \; X \mapsto *$ (where $\mathbf{Type}$ denotes the $(\infty,1)$-category of homotopy types), and take its colimit. (Or thus equivalently the free homotopy colimit of the unique functor $\mathcal{C} \to 1$, where $1$ denotes the category with only one morphism; see Dugger).

Using various models these two views as well as their equivalence can be made precise as follows: View $\mathcal{C}$ as an object in $\mathbf{SSet}$ equipped with the Joyal model structure. Inverting all morphisms of $\mathcal{C}$ just corresponds to taking its fibrant replacement in the Kan model structure. Taking the colimit in 2. can be formalised by taking the geometric realisation of (the nerve of) $\mathcal{C}$: This is exactly the formula you get for computing the homotopy colimit of the constant functor $\mathcal{C} \to \mathbf{Top}, \; X \to *$ using the simplicial replacement of this functor (described e.g. here).

My question then is:

Is there a model independent way of seeing that these two constructions are equivalent?

So I'm looking for an argument which could be formalised in any model of $(\infty,1)$-categories.

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    $\begingroup$ A quick observation: by Yoneda this is equivalent to $\lim_C X = \mathrm{Map}(C,X)$. Proving this is very easy if you have at your disposal the theory of (co)cartesian fibrations, but of course this is not model independent. $\endgroup$ – Denis Nardin Nov 10 '16 at 21:23
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Here is an argument, which is basically Denis Nardin's comment.

To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them via adjunctions, but from a pragmatic point of view I am not sure this is so helpful. Ultimately to do any kind of calculation you will have to pick a model and a construction of the localization/hocolim and if your definition is the model independent one, then you have the additional task of proving that they satisfy these definitions.

Anyway...

There is an ($\infty$-)adjuction of $(\infty,1)$-categories: $$hocolim_C : Cat_{(\infty,0)}^C \leftrightarrows Cat_{(\infty,0)}: const$$ and this is a definition $hocolim_C$. Here $Cat_{(\infty,0)} \simeq Top$ is the usual $(\infty,1)$-category of spaces which the OP called Type.

There is another one: $$ ||-||: Cat_{(\infty,1)} \leftrightarrows Cat_{(\infty,0)}: i $$ which defines the localization at all morphisms $||C||$. Here $i$ is the inclusion of $(\infty,0)$-cats (= spaces) into all $(\infty,1)$-categories.

Then this means that for all spaces $X$ we have $$ \begin{align} Cat_{(\infty,0)}( hocolim_C const(*) , X) &\simeq Cat_{(\infty,0)}^C( const(*), const(X)) \\ &\simeq Fun(C, Cat_{(\infty,0)}(*, X)) \\ &\simeq Fun( C, iX) \\ & \simeq Cat_{(\infty,0)}( ||C||, X) \end{align} $$

And so by Yoneda you conclude that $hocolim_C const(*) \simeq ||C||$.

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  • $\begingroup$ Huh, this was much more straightforward than I thought. Thanks for the detailed answer! $\endgroup$ – Adrian Clough Nov 10 '16 at 22:26
  • $\begingroup$ Using $Top$ is a bit weird for what is a model-independent argument. Why not $\mathcal{S}$ ('spaces') or $\infty Gpd$ or the like? $\endgroup$ – David Roberts Nov 11 '16 at 0:56
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    $\begingroup$ @DavidRoberts That is a fair point. I edited to use $Cat_{(\infty,0)}$. $\endgroup$ – Chris Schommer-Pries Nov 11 '16 at 13:49

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