Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, _On topological cyclic homology_, arXiv:[1707.01799][1] (on p147).  

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I am still lost. But from Maxime's helpful comment and replies, let me  list out my concerns - which are listed as (X),(Y),(Z). 

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**The proof of B.4 spelt out in steps: (read the numericals for the main steps)** 

 1. We begin with a $1$ -category $\Lambda_\infty$ with a $B \Bbb Z $-action. We want to show 
 $$|\Lambda_1| \simeq K(\Bbb Z, 2)$$ 

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So I am trying to understand why this means. 

Firstly, which category does this take place in? $B\Bbb Z=A$ lives in com. alg. obj. of $Cat$. I shall use $A$ instead of $B\Bbb Z$ since what $B\Bbb Z$ is seems to be irrelevant for most of the argument.  

Hence we can consider $Mod_{A}(Cat)$ of left module objects over $B\Bbb Z$. So the claim is saying that 
$$\Lambda_\infty \in Mod_{A}(Cat)$$

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 2. We construct a new category, $\Lambda_1:= \Lambda_\infty/B\Bbb Z= \Lambda_\infty/A$. 

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Now I don't understand what $(-)/B\Bbb Z$ means. i.e. What kind of colimit are we taking? 


>(X) for each $A \in CAlg(Cat)$ some object $BA \in Cat$,  
**$$Mod_{A}(Cat) \simeq Fun(BA, Cat)  $$**

Hence 
$$\Lambda_1 \simeq colim _{BA} \Lambda_\infty$$ 

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 3. We wish to compute $|N\Lambda_1|$. Then as $|\quad|$ is left adjoint. 


$$ |\Lambda_1| \simeq colim_A |\Lambda_\infty| \simeq |BA| $$ 



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The second equivalence requires the fact that

> (Y) $Spc^{|BA|}   \rightarrow Spc$ is conservative. Does this follow from that $Mod_{|BA|}(Spc) \rightarrow Spc$ is conservative? .


>(Z) An explicit formula for $BA$. It doesn't seem clear to me why we would now have $BB\Bbb Z= K(\Bbb Z,2)$. 


  [1]: https://arxiv.org/abs/1707.01799