# Mapping space in functor $\infty$-category as an end

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.

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.