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In enriched category theory, we can endow functor categories with an enrichment such that the mapping object between two functors $F,G:\mathcal{C}\to \mathcal{D}$ is described as the end $\int_{c\in\mathcal{C}} \mathrm{Map}_{\mathcal{D}}(Fc,Gc)$. My questions are :

1) Is there a theory of $(\infty,1)$-ends (in the $\infty$-category of spaces) and correponding homotopy ends (in the Kan-Quillen model category of simplicial sets) ? I found out about a paper by Charles Weibel, Homotopy Ends and Thomason Model Categories, but I'm still quite new to the whole model category/$\infty$-category business and am struggling to see if it applies.

2) Do these ends coincide with a certain $(\infty,1)$-equalizer ?

3) Can we describe the mapping space between two $\infty$-functors as an end ?

For instance, if $\mathcal{C}$ is a simplicial category, $\mathbb{D}$ is a combinatorial simplicial model category modelling an $\infty$-category $\mathcal{D}$, then $[N(\mathcal{C}),\mathcal{D}]$ is modelled by the simplicial model category $[\mathcal{C},\mathbb{D}]$ with the projective model structure. Then if $F,G$ are bifibrant, $\mathrm{Map}_{[N(\mathcal{C}),\mathcal{D}]}(F,G)$ has the homotopy type of $\mathrm{Map}_{[\mathcal{C},\mathbb{D}]}(F,G)$, which is the end $\int_{c\in\mathcal{C}} \mathrm{Map}_{\mathbb{D}}(Fc,Gc)$. But $\mathrm{Map}_{\mathbb{D}}(Fc,Gd)$ has the homotopy type of $\mathrm{Map}_{\mathcal{D}}(Fc,Gd)$ for $c,d \in \mathcal{C}$. So the question is, is that end an "homotopy end" / does it respect weak equivalences ?

I have tried to write this end as an equalizer, and to show that the equalizer is actually a homotopy equalizer. For that I should show that the map $(f,g) : \prod_{c\in\mathcal{C}}\mathrm{Map}_{\mathbb{D}}(Fc,Gc) \to \left[\prod_{c,d\in\mathcal{C}}\mathrm{Map}_{\mathrm{sSet}}(\mathrm{Map}_{\mathcal{C}}(c,d),\mathrm{Map}_{\mathbb{D}}(Fc,Gd))\right]^2$ induced by the two natural maps is a Kan fibration. But I don't see how that would be linked with F,G being bifibrant.

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The end of an $\infty$-functor has been defined by several authors in terms of the limit over the twisted arrow category, and Saul Glasman proves your desired result-somewhat laboriously, in 2.3 of his paper here: https://arxiv.org/pdf/1408.3065.pdf

This is not just a certain equalizer, for the same reason that a general $\infty$-categorical limit can’t be turned into an equalizer. Instead it can be realized as the limit of a certain cosimplicial object over which a cone is, roughly, a family of maps $a\to F(c,c)$ together with homotopies between the two different routes to $F(c,c’)$ related to a map $f:c\to c’$ together with homotopies making coherent all the data so far given involving $f:c\to c’,f’:c’\to c’’,$ and their composite, and so on ad infinitum.

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  • $\begingroup$ Do you have a reference for the "totalization of a cosimplicial object" formula ? Also I'd like to add for reference the paper "Lax colimits and free fibrations" by Gepner, Hausgeng and Nikolaus, which contains another, less combinatorial, proof. $\endgroup$ Commented May 23, 2020 at 9:33

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