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In HTT.5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch the general idea that something like $\hom(P,|S|_{X^*}) \simeq |S|_{\hom(P,X^*)}$ should happen, if

  1. $X_*$ is a cosimplicial object in $\bf C$;
  2. $|-|_{\hom(P,X^*)}$ is the "realization" obtained via familiar abstraction from the cosimplicial object $X^*$: since $\hom(P,X^*)\in [\Delta, {\bf Set}]$ it can be Kan-extended to a functor ${\bf sSet}\to \bf C$, and this functor admits a right adjoint (the "nerve").
  3. $|-|_{X^*}$ is a realization similarly induced as a ($\infty$?)-functor $N(\boldsymbol \Delta)\to \bf C$.

But this "similarly" is kind of too vague, I strive for a more precise description. Unfortunately HTT seems to skip a precise definition of the paradigm "geometric realization"-"nerve".

Does it transport to the $(\infty,1)$-categorical case? Do I recover the precise form of that statement (i.e.: give a cosimplicial object in $\bf C\in QCat$, then there is a pair of $(\infty,1)$-adjoint functors obtained as Yoneda extension / "nerve")?

On the same vein, Remark 5.5.8.5 is totally obscure to me:

The formation of the geometric realizations of simplicial objects should be thought of as the $\infty$-categorical analogue of the formation of reflexive coequalizers.

Can you help me?

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  • $\begingroup$ A comment about Remark 5.5.8.5: if you look at Notation 6.1.2.12, he kinda gives a definition of geometric realization of a simplicial object, which might be a bit different from the one you were expecting. However, it should be clear why this generalizes the formation of reflexive coequalizers (and so why the notion of projective object is a generalization of the usual one that can be found for example in Quillen's Homotopical Algebra). $\endgroup$ – Mauro Porta Nov 6 '13 at 11:54
  • $\begingroup$ I think that Lure, as is more standard, uses the term "geometric realization" only for simplicial objects, not cosimplicial ones. (The standard term for a more or less similar thing for cosimplicial objects is "totalization".) $\endgroup$ – Omar Antolín-Camarena Nov 6 '13 at 15:40
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The term geometric realization is used in HTT to refer to colimits indexed by $\Delta^{op}$. So an object $P \in \mathcal{C}$ is projective if and only if, for every simplicial object $X_{\ast}$ in $\mathcal{C}$, the canonical map $$ \varinjlim \text{Map}(P, X_{\ast} ) \rightarrow \text{Map}(P, \varinjlim X_{\ast} )$$ is a homotopy equivalence of spaces.

If $\mathcal{C}$ is an ordinary category and $X_{\ast}$ is a simplicial object of $\mathcal{C}$, then a colimit of $X_{\ast}$ is a coequalizer of the pair of face maps $X_1 \rightarrow X_0$. Since there is also a degeneracy map $X_0 \rightarrow X_1$, this is a reflexive coequalizer. Many good properties of reflexive coequalizers in ordinary category theory generalize to statements about geometric realizations in $\infty$-categories.

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  • $\begingroup$ Thank you for your answer. My problem is that I would like to establish a link between the two definitions. To my eye, geometric realizations arise as Yoneda extension of certain dense functors $\Delta\to \bf C$ to a cocomplete category (cocompleteness implies this Kan extension has a right adjoint, density of $\iota$ that this right adjoint is a fully faithful embedding; this is well-known). Is it possible to generalize this pattern to the quasicategorical setting? If yes, is there a link with the definition of projective object you (& Joyal) gave? $\endgroup$ – Fosco Nov 6 '13 at 16:57
  • $\begingroup$ @tetrapharmakon This notion is entirely different. Rather it is based on the observation that the geometric realisation of a simplicial space is (homotopy equivalent to) its homotopy colimit. $\endgroup$ – Zhen Lin Nov 6 '13 at 18:00
  • $\begingroup$ What you describe works exactly the same way in the setting of quasi-categories: if C is a category which admits small colimits, then any cosimplicial object of C determines a pair of adjoint functors relating C to the quasicategory of simplicial spaces. If C is the quasi-category of spaces and your cosimplicial space is contractible in each degree, the resulting functor from simplicial spaces to spaces is given by taking the colimit. $\endgroup$ – Jacob Lurie Nov 6 '13 at 18:25
  • $\begingroup$ @JacobLurie I have to meditate more on this (especially because Zhen Lin and you seem to disagree). In the meanwhile, thank you again! $\endgroup$ – Fosco Nov 10 '13 at 22:38
  • $\begingroup$ @tetrapharmakon What Jacob Lurie has described is the $\infty$-analogue of the observation that the ordinary colimit for a diagram $\mathbf{\Delta}^\mathrm{op} \to \mathcal{C}$ is the same thing as the colimit for the same diagram weighted by the terminal presheaf on $\mathbf{\Delta}^\mathrm{op}$ (a.k.a. "functor tensor product"). The fact that geometric realisation (and more generally left Kan extension) can be expressed in terms of weighted colimits is not conceptually helpful here, in my view. $\endgroup$ – Zhen Lin Nov 11 '13 at 1:02

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