I've been trying to learn some of the basic language of infinity-category theory (in the sense of Lurie), and in particular, to understand which basic statements in (1-)categories have analogues in the infinity-categorical setting. Specifically, I'm trying to understand how equivalences behave in infinity categories, motivated by the following question.



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> **Vague Question:** in an infinity-category $\mathcal C$, to what extent is it true that a quasi-inverse to a morphism $f$, if it exists, is unique up to a contractible space of choices?
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Here is how I believe this question should be made precise. If we let $I$ denote the nerve of the undirected interval category (the category with two objects and exactly one morphism between each pair of objects), then there is an inclusion $\Delta^1\hookrightarrow I$ coming from the inclusion of the directed interval category in the undirected interval category. Thus we have a restriction map
$$\mathcal C^I\rightarrow\mathcal C^{\Delta^1}$$
of simplicial sets (in fact a functor of infinity-categories which is a fibration for the Joyal model structure), where informally the left-hand side is the infinity-category of quasi-inverse pairs $(g,h)$ of morphisms in $\mathcal C$ (with a family of homotopies realising their quasi-inverseness), the right-hand side is the infinity-category of arrows in $\mathcal C$, and the functor takes a pair $(g,h)$ to its first component $g$.



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> **Precise Question:** if $f$ is a morphism in an infinity-category $\mathcal C$, is it true that the fibre of the map $\mathcal C^I\rightarrow\mathcal C^{\Delta^1}$ over the point $f$ of $\mathcal C^{\Delta^1}$ is either empty or a contractible Kan complex?
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I am interested both in answers to the precise question and the vague questions above, especially if there are alternative precise formulations which make the answer clearer.




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**Remark:**


The condition that $f$ admit a quasi-inverse should be be equivalent to the assertion that it be an isomorphism in the homotopy category of $\mathcal C$. This is asserted in [this](https://ncatlab.org/nlab/show/equivalence+in+an+%28infinity%2C1%29-category) nlab page, and something similar in the language of topological categories is proved in the setting of topological categories in Proposition 1.2.4.1 of *Higher Topos Theory*.



Uniqueness of quasi-inverses in infinity categories
A possibly simpler way of proving what you are after is using *marked simplicial set*.


Recall that marked simplicial sets are pairs $(X,S)$ where $X$ is a simplicial set and $S\subseteq X\_1$ is a set of 1-simplices of $X$ containing all degenerate 1-simplices. If $X$ is a simplicial set we will denote the minimal and maximal marking by $X^\flat$ and $X^\sharp$ respectively


If $X,Y$ are two marked simplicial sets, we can form additional simplicial sets $\mathrm{Map}^\flat(X,Y)$ and $\mathrm{Map}^\sharp(X,Y)$ by
$$ \mathrm{Hom}\_{\mathrm{sSet}}(K,\mathrm{Map}^\flat(X,Y))=\mathrm{Hom}\_{\mathrm{sSet}^+}(K^\flat\times X,Y)\,,$$
$$ \mathrm{Hom}\_{\mathrm{sSet}}(K,\mathrm{Map}^\sharp(X,Y))=\mathrm{Hom}\_{\mathrm{sSet}^+}(K^\sharp\times X,Y)\,.$$


In HTT.3.1.3.7 a simplicial model structure is constructed on the category of marked simplicial sets such that the fibrant objects are precisely the $\infty$-categories with the equivalences marked.


The important part here will be that 


* For any anodyne morphism of simplicial sets $A\to B$ the map $A^\sharp\to B^\sharp$ is a marked trivial cofibration. This is because of the definition of marked trivial cofibration (HTT.3.1.3.3) and the fact that for any simplicial set $A$ and every $\infty$-category $C$
$$ \mathrm{Map}^\sharp(A^\sharp,C)=\mathrm{Map}(A,\mathrm{Core}(C))$$
* If $f:X\to Y$ is a marked trivial cofibration and $C$ is a fibrant object (i.e. an $\infty$-category with the equivalences marked), then the map
$$f^\*:\mathrm{Map}^\flat(B,C)\to \mathrm{Map}^\flat(A,C)$$
is a trivial fibration (HTT.3.1.3.3)


Then we can factorize the arrow you want to study as
$$C^J=\mathrm{Map}^\flat(J^\sharp,C)\to \mathrm{Map}^\flat((\Delta^1)^\sharp, C)\to \mathrm{Map}^\flat((\Delta^1)^\flat,C)=C^{\Delta^1}$$
The first arrow is a trivial fibration and the second is precisely the inclusion of the subcategory of $C^{\Delta^1}$ spanned by the equivalences.


