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edited Oct 30, 2014 at 12:37

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Here is a purely combinatorial and straightforward proof.

First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining one idempotent endomorphism to each object of $[m]$. These categories $[m]'$ are contractible since we can write down a natural transformation from the constant functor at $0$ to the identity functor.

Correction: these idempotents are not exactly freely adjoined. They are supposed to act as identities on the morphisms of $[m].$

Next, we use a general fact (verified by induction over skeleta) that if we have a functor $F$ from simplicial sets to simplicial sets and a natural transformation between $F$ and identity such that $F$ preserves colimits and cofibrations and sends simplices to contractible simplicial sets, then this natural transformation is a weak equivalence. The first two conditions are clear for $FS = S'$ and I have verified the third one above so the conclusion follows.

Here is a purely combinatorial and straightforward proof.

First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining one idempotent endomorphism to each object of $[m]$. These categories $[m]'$ are contractible since we can write down a natural transformation from the constant functor at $0$ to the identity functor.

Next, we use a general fact (verified by induction over skeleta) that if we have a functor $F$ from simplicial sets to simplicial sets and a natural transformation between $F$ and identity such that $F$ preserves colimits and cofibrations and sends simplices to contractible simplicial sets, then this natural transformation is a weak equivalence. The first two conditions are clear for $FS = S'$ and I have verified the third one above so the conclusion follows.

Here is a purely combinatorial and straightforward proof.

First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining one idempotent endomorphism to each object of $[m]$. These categories $[m]'$ are contractible since we can write down a natural transformation from the constant functor at $0$ to the identity functor.

Correction: these idempotents are not exactly freely adjoined. They are supposed to act as identities on the morphisms of $[m].$

Next, we use a general fact (verified by induction over skeleta) that if we have a functor $F$ from simplicial sets to simplicial sets and a natural transformation between $F$ and identity such that $F$ preserves colimits and cofibrations and sends simplices to contractible simplicial sets, then this natural transformation is a weak equivalence. The first two conditions are clear for $FS = S'$ and I have verified the third one above so the conclusion follows.

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created Oct 29, 2014 at 12:33

7.7k
27
38

Here is a purely combinatorial and straightforward proof.

First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining one idempotent endomorphism to each object of $[m]$. These categories $[m]'$ are contractible since we can write down a natural transformation from the constant functor at $0$ to the identity functor.

Next, we use a general fact (verified by induction over skeleta) that if we have a functor $F$ from simplicial sets to simplicial sets and a natural transformation between $F$ and identity such that $F$ preserves colimits and cofibrations and sends simplices to contractible simplicial sets, then this natural transformation is a weak equivalence. The first two conditions are clear for $FS = S'$ and I have verified the third one above so the conclusion follows.