This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion.

We have the categories $\Lambda:=\Lambda_\infty/B\Bbb Z, \Lambda_\infty$ explained in my previous question.

In B.5, the authors describes a functor given by composition $$ Fun(\Lambda^{op}, C) \rightarrow Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C) \rightarrow Fun^{B\Bbb Z}(pt, C) = C^{BB\Bbb Z} = C^{B\Bbb T} $$

I have fundamental confusion in the first 2 arrows. Edit:11/24/20.

Q1: What exactly is the cateogry $Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C)$. I understand it is to be understood as $B\Bbb Z$ equivariant maps.

But how is this made precise? Irregardless, I'd expect $$ Fun^{B\Bbb Z}(\Lambda_\infty, C) \subset Fun(\Lambda_\infty^{op}, C )$$

But without a concrete meaning, I could not make sense of the following two.

Q1': What is the relationship $Map_{Fun(B\Bbb Z, Cat)}(\Lambda_\infty,C)$ and $Fun^{B\Bbb Z}(\Lambda_\infty, C)$?

In fact, are there general result about symmetric monoidal cateogires $C$ which is enriched over itself and the mapping spaces of its objects?

Q2 why does taking collimit preserve $B\Bbb Z$-equivariance?

Q3: How do we show $Fun^{B\Bbb Z}(pt, C)=C^{BB\Bbb Z}$?

This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion.

We have the categories $\Lambda:=\Lambda_\infty/B\Bbb Z, \Lambda_\infty$ explained in my previous question.

In B.5, the authors describes a functor given by composition $$ Fun(\Lambda^{op}, C) \rightarrow Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C) \rightarrow Fun^{B\Bbb Z}(pt, C) = C^{BB\Bbb Z} = C^{B\Bbb T} $$

I have fundamental confusion in the first 2 arrows. Edit:11/24/20.

Q1: What exactly is the cateogry $Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C)$. I understand it is to be understood as $B\Bbb Z$ equivariant maps.

But how is this made precise? Irregardless, I'd expect $$ Fun^{B\Bbb Z}(\Lambda_\infty, C) \subset Fun(\Lambda_\infty^{op}, C )$$

But without a concrete meaning, I could not make sense of the following two.

Q1a: What is the relationship $Map_{Fun(B\Bbb Z, Cat)}(\Lambda_\infty,C)$ and $Fun^{B\Bbb Z}(\Lambda_\infty, C)$?

In fact, are there general result about symmetric monoidal cateogires $C$ which is enriched over itself and the mapping spaces of its objects?

Q2 why does taking collimit preserve $B\Bbb Z$-equivariance?

Q3: How do we show $Fun^{B\Bbb Z}(pt, C)=C^{BB\Bbb Z}$?

This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion.

We have the categories $\Lambda:=\Lambda_\infty/B\Bbb Z, \Lambda_\infty$ explained in my previous question.

In B.5, the authors describes a functor given by composition $$ Fun(\Lambda^{op}, C) \rightarrow Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C) \rightarrow Fun^{B\Bbb Z}(pt, C) = C^{BB\Bbb Z} = C^{B\Bbb T} $$

I have fundamental confusion in the first 2 arrows.

Q : What exactly is the cateogry $Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C)$. I understand it is to be understood as $B\Bbb Z$ equivariant maps.

But how is this made precise?

  Irregardless, any definition should be that we have a subcat. $$ Fun^{B\Bbb Z}(\Lambda_\infty, C) \subset Fun(\Lambda_\infty^{op}, C )$$

But without a concrete meaning, I could not make sense of the following two.

Q2 why does taking collimit preserve $B\Bbb Z$-equivariance?

Q3: How do we show $Fun^{B\Bbb Z}(pt, C)=C^{BB\Bbb Z}$?

I suppose we could replace $\Bbb Z$ by any topological group $G$.

This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion.

We have the categories $\Lambda:=\Lambda_\infty/B\Bbb Z, \Lambda_\infty$ explained in my previous question.

In B.5, the authors describes a functor given by composition $$ Fun(\Lambda^{op}, C) \rightarrow Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C) \rightarrow Fun^{B\Bbb Z}(pt, C) = C^{BB\Bbb Z} = C^{B\Bbb T} $$

I have fundamental confusion in the first 2 arrows. Edit:11/24/20.

Q1: What exactly is the cateogry $Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C)$. I understand it is to be understood as $B\Bbb Z$ equivariant maps.

But how is this made precise? Irregardless, I'd expect $$ Fun^{B\Bbb Z}(\Lambda_\infty, C) \subset Fun(\Lambda_\infty^{op}, C )$$

But without a concrete meaning, I could not make sense of the following two.

Q1': What is the relationship $Map_{Fun(B\Bbb Z, Cat)}(\Lambda_\infty,C)$ and $Fun^{B\Bbb Z}(\Lambda_\infty, C)$?

In fact, are there general result about symmetric monoidal cateogires $C$ which is enriched over itself and the mapping spaces of its objects?

Q2 why does taking collimit preserve $B\Bbb Z$-equivariance?

Q3: How do we show $Fun^{B\Bbb Z}(pt, C)=C^{BB\Bbb Z}$?