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homotopy hypothesis for 1-types

Contents

Contents

Introduction

We give a proof that groupoids model homotopy 1-types. More precisely, we construct a Quillen equivalence between a model structure on the category of groupoids, and a certain model structure on the category of cubical sets.

Preliminaries

One direction of the equivalence

Proposition

The adjunction natural transformation ϵ:Π 1 2N 2id\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id is a natural isomorphism.

Proof

Let 𝒜\mathcal{A} be a groupoid. Let r(𝒜):𝒜Π 1 2N 2r(\mathcal{A}) : \mathcal{A} \rightarrow \Pi^{\leq 2}_{1} \circ N^{\leq 2} be the functor defined as follows.

1) On objects it is the identity.

2) To an arrow f:a 0a 1f : a_{0} \rightarrow a_{1} of 𝒜\mathcal{A}, we associate the arrow

a 0 f a 1id a 1 \array{ a_{0} & \overset{f}{\rightarrow} & a_{1} \overset{id}{\leftarrow} & a_{1} }

of Π 1 2N 2(𝒜)\Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}).

Because of 1 b) ii) of Notation 3 of fundamental groupoid of a cubical set and the cubical nerve of a groupoid, the following diagram of commutative squares illustrates that r(gf)=r(g)r(f)r(g \circ f) = r(g) \circ r(f), where f:a 0a 1f : a_{0} \rightarrow a_{1} and g:a 1a 2g : a_{1} \rightarrow a_{2} are arrows of 𝒜\mathcal{A}.

a 0 f a 1 id a 1 g a 2 id a 2 id g g id id a 0 gf a 2 id a 2 id a 2 id a 2 \array{ a_{0} & \overset{f}{\rightarrow} & a_{1} & \overset{id}{\rightarrow} & a_{1} & \overset{g}{\rightarrow} & a_{2} & \overset{id}{\leftarrow} & a_{2} \\ id \downarrow & & \downarrow g & & \downarrow g & & \downarrow id & & \downarrow id \\ a_{0} & \underset{g \circ f}{\rightarrow} & a_{2} & \underset{id}{\leftarrow} & a_{2} & \underset{id}{\rightarrow} & a_{2} & \underset{id}{\leftarrow} & a_{2} }

That r(id)=idr(id) = id follows immediately from 1 b) i) of Notation 3 of fundamental groupoid of a cubical set and the cubical nerve of a groupoid.

It is clear that ϵ(𝒜)r(𝒜)\epsilon(\mathcal{A}) \circ r(\mathcal{A}) is id(𝒜)id(\mathcal{A}).

Moreover, because of 1 b) ii) of Notation 3 of fundamental groupoid of a cubical set and the cubical nerve of a groupoid, the following diagram of commutative squares illustrates that r(𝒜)ϵ(𝒜)r(\mathcal{A}) \circ \epsilon(\mathcal{A}) is id(𝒜)id(\mathcal{A}).

a 0 f 1 a 1 g 1 a 2 a 2n2 f n a 2n1 g n a 2n id A B C g n 1 id a 0 D a 2n id a 2n a 2n id a 2n id a 2n \array{ a_{0} & \overset{f_{1}}{\rightarrow} & a_{1} & \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} & \overset{g_{n}}{\leftarrow} & a_{2n} \\ id \downarrow & & \downarrow A & & \downarrow B & & \downarrow C & & \downarrow g_{n}^{-1} & & \downarrow id \\ a_{0} & \underset{D}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} & a_{2n} & \cdots & a_{2n} & \underset{id}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} & a_{2n} }

Here

a 0 f 1 a 1g 1 a 2 a 2n2 f n a 2n1g n a 2n \array{ a_{0} & \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}}{\leftarrow} & a_{2n} }

is any zig-zag of arrows of 𝒜\mathcal{A}, and the arrows A, B, C, and D are given as follows.

A) g n 1f ng 2 1f 2g 1 1g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2} \circ g_{1}^{-1}

B) g n 1f ng 2 1f 2g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2}

C) g n 1f ng_{n}^{-1} \circ f_{n}

D) g n 1f ng 2 1f 2g 1 1f 1g_{n}^{-1} \circ f_{n} \circ \cdots g_{2}^{-1} \circ f_{2} \circ g_{1}^{-1} \circ f_{1}

We also make use of 1 b) i) of Notation 3 of fundamental groupoid of a cubical set and the cubical nerve of a groupoid in identifying

a 0 D a 2n id a 2n a 2n id x 2n id a 2n \array{a_{0} & \underset{D}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} & a_{2n} & \cdots & a_{2n} & \underset{id}{\rightarrow} & x_{2n} & \underset{id}{\leftarrow} & a_{2n}}

with

a 0 D a 2n ida 2n \array{a_{0} & \underset{D}{\rightarrow} & a_{2n} & \underset{id}{\leftarrow} a_{2n}}

as required.

Corollary

The adjunction natural transformation Π 1Nid\Pi_{1} \circ N \rightarrow id is a natural isomorphism.

Proof

Since tr 2tr_{2} is fully faithful, the adjunction natural transformation idtr 2cosk 2id \rightarrow tr_{2} cosk_{2} is a natural isomorphism. The corollary follows from this and Proposition .

Other direction of the equivalence

To be written.

Last revised on April 27, 2016 at 16:10:50. See the history of this page for a list of all contributions to it.