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LSpice
LSpice
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$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas (simplicial $\infty$-groupoids if you like) $\sAn$, we have the evaluation functor $\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\mathrm{const}_n$$\operatorname{const}_n$ and the realization functor $|\ |:\sAn\rightarrow \An$$\lvert\ \rvert:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\operatorname{const}_1 X|$$\Omega \lvert\operatorname{const}_1 X\rvert$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces...spaces…?

$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas (simplicial $\infty$-groupoids if you like) $\sAn$, we have the evaluation functor $\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\mathrm{const}_n$ and the realization functor $|\ |:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\operatorname{const}_1 X|$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces...?

$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas (simplicial $\infty$-groupoids if you like) $\sAn$, we have the evaluation functor $\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\operatorname{const}_n$ and the realization functor $\lvert\ \rvert:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega \lvert\operatorname{const}_1 X\rvert$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces…?

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YCor
YCor
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Realization of a Constant Simplicial Animaconstant simplicial anima

For$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas  (simplicial $\infty$-groupoids if you like) $sAn$$\sAn$, we have the evaluation functor $\text{ev}_n:sAn\rightarrow An$$\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\text{const}_n$$\mathrm{const}_n$ and the realization functor $|\ |:sAn\rightarrow An$$|\ |:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\text{const}_1 X|$$\Omega |\operatorname{const}_1 X|$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\text{const}_1X$$\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces...?

Realization of a Constant Simplicial Anima

For the category of simpicial animas(simplicial $\infty$-groupoids if you like) $sAn$, we have the evaluation functor $\text{ev}_n:sAn\rightarrow An$ with a left adjoint $\text{const}_n$ and the realization functor $|\ |:sAn\rightarrow An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\text{const}_1 X|$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\text{const}_1X$. Is this something concerning the theory of Segal spaces...?

Realization of a constant simplicial anima

$\DeclareMathOperator\sAn{\mathit{sAn}}\DeclareMathOperator\An{\mathit{An}}$For the category of simpicial animas  (simplicial $\infty$-groupoids if you like) $\sAn$, we have the evaluation functor $\mathrm{ev}_n:\sAn\rightarrow \An$ with a left adjoint $\mathrm{const}_n$ and the realization functor $|\ |:\sAn\rightarrow \An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\operatorname{const}_1 X|$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\operatorname{const}_1X$. Is this something concerning the theory of Segal spaces...?

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XiaYu
XiaYu
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Realization of a Constant Simplicial Anima

For the category of simpicial animas(simplicial $\infty$-groupoids if you like) $sAn$, we have the evaluation functor $\text{ev}_n:sAn\rightarrow An$ with a left adjoint $\text{const}_n$ and the realization functor $|\ |:sAn\rightarrow An$ with a right adjoint being the Rezk nerve.

I wonder why $\Omega |\text{const}_1 X|$ is $X$. This is used in the (4.1.27) in the Lecture notes: Lecture Notes on Algebraic K-Theory.

I can't even give a direct description for $\text{const}_1X$. Is this something concerning the theory of Segal spaces...?