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fundamental groupoid of a cubical set and the cubical nerve of a groupoid

Contents

Contents

Introduction

We construct an adjunction between the category of groupoids and the category of cubical sets, the left adjoint of which is the fundamental groupoid of a cubical set, and the right adjoint of which is the (cubical) nerve of a groupoid.

Preliminaries

Notation

We denote the category of groupoids by Grpd\mathsf{Grpd}.

Notation

We make use throughout of the notation of category of cubes, cubical set, and cubical truncation, skeleton, and co-skeleton.

Fundamental groupoid for 2-truncated cubical sets

Notation

We denote by Π 1 2:Set 2 opGrpd\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd} the functor defined as follows.

1) To a 2-truncated cubical set XX, we associate the groupoid Π 1 2(X)\Pi^{\leq 2}_{1}(X) defined as follows.

a) The objects of Π 1 2(X)\Pi^{\leq 2}_{1}(X) are the 0-cubes of XX.

b) The arrows of Π 1 2(X)\Pi^{\leq 2}_{1}(X) are zig-zags of 1-cubes of XX up to the notion of equivalence defined below, where by a zig-zag of 1-cubes of XX we mean, for some integer n0n \geq 0, a set of 1-cubes of XX whose faces match up as follows.

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

We identify a pair of zig-zags if one can be obtained from the other by a sequence of the following manipulations.

i) We may remove or add a pair of arrows (anywhere in the zig-zag) of the form

x 0 f x 1 f x 0 \array{ x_{0} & \overset{f}{\rightarrow} & x_{1} & \overset{f}{\leftarrow} & x_{0} }

or of the following form.

x 0 f x 1 f x 0 \array{ x_{0} & \overset{f}{\leftarrow} & x_{1} & \overset{f}{\rightarrow} & x_{0} }

ii) We may replace an entire zig-zag

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

with a zig-zag

x 0 f 1 x 1g 1 x 2 x 2n2 f n x 2n1g n x 2n \array{ x'_{0} & \overset{f'_{1}}{\rightarrow} & x'_{1} \overset{g'_{1}}{\leftarrow} & x'_{2} & \cdots & x'_{2n-2} & \overset{f'_{n}}{\rightarrow} & x'_{2n-1} \overset{g'_{n}}{\leftarrow} & x'_{2n} }

if there is, for every 1in1 \leq i \leq n, a 2-cube of XX whose horizontal 1-cubes are as follows

x 2i2 f i x 2i1 x 2i2 f i x 2i1 \array{ x_{2i-2} & \overset{f_{i}}{\to} & x'_{2i-1} \\ \downarrow & & \downarrow \\ x'_{2i-2} & \underset{f'_{i}}{\to} & x'_{2i-1} }

and there is, for every 1in1 \leq i \leq n, a 2-cube of XX whose horizontal 1-cubes are as follows.

x 2i1 g i x 2i x 2i1 g i x 2i \array{ x_{2i-1} & \overset{g_{i}}{\leftarrow} & x'_{2i} \\ \downarrow & & \downarrow \\ x'_{2i-1} & \underset{g'_{i}}{\leftarrow} & x'_{2i} }

c) The source of a zig-zag as at the beginning of b) is x 0x_{0}, and the target is x 2nx_{2n}.

d) Composition of arrows is given by concatenation of zig-zags (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

e) The identity arrow on an object xx of Π 1 2(X)\Pi^{\leq 2}_{1}(X) is the zig-zag with n=0n = 0 consisting simply of xx.

f) The inverse of an arrow

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

of Π 1(X)\Pi_{1}(X) is the following arrow (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

x 2n g n x 2n1f n x 2n2 x 2 g 1 x 1f 1 x 0 \array{ x_{2n} & \overset{g_{n}}{\rightarrow} & x_{2n-1} \overset{f_{n}}{\leftarrow} & x_{2n-2} & \cdots & x_{2} & \overset{g_{1}}{\rightarrow} & x_{1} \overset{f_{1}}{\leftarrow} & x_{0} }

2) To a morphism of 2-truncated cubical sets F:XYF : X \rightarrow Y, we associate the functor Π 1 2(F):Π 1 2(X)Π 1 2(Y)\Pi^{\leq 2}_{1}(F) : \Pi^{\leq 2}_{1}(X) \rightarrow \Pi^{\leq 2}_{1}(Y) defined as follows.

a) On objects, Π 1 2(F)\Pi^{\leq 2}_{1}(F) is the same as FF.

b) To a zig-zag as follows

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

we associate the following zig-zag.

F(x 0) F(f 1) F(x 1)F(g 1) F(x 2) F(x 2n2) F(f n) F(x 2n1)F(g n) F(x 2n) \array{ F(x_{0}) & \overset{F(f_{1})}{\rightarrow} & F(x_{1}) \overset{F(g_{1})}{\leftarrow} & F(x_{2}) & \cdots & F(x_{2n-2}) & \overset{F(f_{n})}{\rightarrow} & F(x_{2n-1}) \overset{F(g_{n})}{\leftarrow} & F(x_{2n}) }

It is immediately verified that this is well-defined with respect to the equivalence relation of 1) b).

Terminology

We refer to Π 1 2:Set 2 opGrpd\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd} as the fundamental groupoid functor for 2-truncated cubical sets.

2-truncated nerve functor

Notation

We denote by N 2:GrpdSet 2 opN^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}} the functor defined as follows.

1) To a groupoid 𝒜\mathcal{A}, we associate the 2-truncated cubical set N 2(𝒜)N^{\leq 2}(\mathcal{A}) defined as follows.

a) The 0-cubes of N 2(𝒜)N^{\leq 2}(\mathcal{A}) are the objects of 𝒜\mathcal{A}.

b) The 1-cubes f:a 0a 1f : a_{0} \rightarrow a_{1} of N 2(𝒜)N^{\leq 2}(\mathcal{A}) are the arrows ff of 𝒜\mathcal{A} with source a 0a_{0} and target a 1a_{1}.

c) The 2-cubes

a 0 f 0 a 1 f 2 σ f 1 a 2 f 3 a 3 \array{ a_{0} & \overset{f_{0}}{\to} & a_{1} \\ f_{2} \downarrow & \sigma & \downarrow f_{1} \\ a_{2} & \underset{f_{3}}{\to} & a_{3} }

of N 2(𝒜)N^{\leq 2}(\mathcal{A}) are the commutative squares σ\sigma of 𝒜\mathcal{A} whose boundary looks the same as this.

d) The degenerate 1-cubes of N 2(𝒜)N^{\leq 2}(\mathcal{A}) are the identity arrows of 𝒜\mathcal{A}.

e) The degenerate 2-cubes of N 2(𝒜)N^{\leq 2}(\mathcal{A}) are the commutative squares of Π 1 2(𝒜)\Pi^{\leq 2}_{1}(\mathcal{A}) which look as follows

x 0 f x 1 id σ id x 0 f x 1 \array{ x_{0} & \overset{f}{\to} & x_{1} \\ id \downarrow & \sigma & \downarrow id \\ x_{0} & \underset{f}{\to} & x_{1} }

or as follows.

x 0 id x 0 f σ f x 1 id x 1 \array{ x_{0} & \overset{id}{\to} & x_{0} \\ f \downarrow & \sigma & \downarrow f \\ x_{1} & \underset{id}{\to} & x_{1} }

2) To a functor F:𝒜F : \mathcal{A} \rightarrow \mathcal{B}, we associate the morphism of 2-truncated cubical sets N 2(F):N 2(𝒜)N 2()N^{\leq 2}(F) : N^{\leq 2}(\mathcal{A}) \rightarrow N^{\leq 2}(\mathcal{B}) defined as follows.

a) On 0-cubes, N 2(𝒜)N^{\leq 2}(\mathcal{A}) is the same as FF.

b) On 1-cubes, N 2(F)N^{\leq 2}(F) is the same as FF.

c) On 2-cubes, N 2(F)N^{\leq 2}(F) sends a commutative square

a 0 f 0 a 1 f 2 σ f 1 a 2 f 3 a 3 \array{ a_{0} & \overset{f_{0}}{\to} & a_{1} \\ f_{2} \downarrow & \sigma & \downarrow f_{1} \\ a_{2} & \underset{f_{3}}{\to} & a_{3} }

of 𝒜\mathcal{A} to the commutative square

F(a 0) F(f 0) F(a 1) F(f 2) F(σ) F(f 1) F(a 2) F(f 3) F(a 3) \array{ F(a_{0}) & \overset{F(f_{0})}{\to} & F(a_{1}) \\ F(f_{2}) \downarrow & F(\sigma) & \downarrow F(f_{1}) \\ F(a_{2}) & \underset{F(f_{3})}{\to} & F(a_{3}) }

of 𝒜\mathcal{A}.

Terminology

We refer to N 2:GrpdSet 2 opN^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}} as the 2-truncated nerve functor.

Adjunction between the fundamental groupoid functor for 2-truncated cubical sets and the 2-truncated cubical nerve functor

Notation

We denote by ϵ:Π 1 2N 2id\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id the natural transformation which to a groupoid 𝒜\mathcal{A} associates the functor ϵ(𝒜):Π 1 2N 2(𝒜)𝒜\epsilon(\mathcal{A}) : \Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}) \rightarrow \mathcal{A} defined as follows.

1) On objects it is the identity.

2) To an arrow of Π 1 2N 2(𝒜)\Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}), given by a zig-zag of arrows

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} }

of 𝒜\mathcal{A}, we associate the arrow of 𝒜\mathcal{A} given by the composition in 𝒜\mathcal{A} of the arrows

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

of 𝒜\mathcal{A}.

It is straightforward to check that this is well-defined with respect to the equivalence relation of 1 b) of Notation , and that we indeed have a functor.

Notation

We denote by η:idN 2Π 1 2\eta : id \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1} the natural transformation which to a 2-truncated cubical set XX associates the morphism of 2-truncated cubical sets η(X):XN 2Π 1 2(X)\eta(X) : X \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1}(X) defined as follows.

1) On objects it is the identity.

2) To a 1-cube f:x 0x 1f : x_{0} \rightarrow x_{1} of XX we associate the following zig-zag of 1-cubes of XX, where the right arrow is the degeneracy on x 1x_{1}.

x 0 f x 1id x 1 \array{ x_{0} & \overset{f}{\rightarrow} & x_{1} \overset{id}{\leftarrow} & x_{1} }

3) To a 2-cube

x 0 f 0 x 1 f 2 σ f 1 x 2 f 3 x 3 \array{ x_{0} & \overset{f_{0}}{\to} & x_{1} \\ f_{2} \downarrow & \sigma & \downarrow f_{1} \\ x_{2} & \underset{f_{3}}{\to} & x_{3} }

of XX we associate the commutative square in Π 1 2(X)\Pi^{\leq 2}_{1}(X) given as follows.

x 0 f 0 x 1 id x 1 f 2 f 1 x 2 x 3 id id x 2 f 3 x 2 id x 3 \array{ x_{0} & \overset{f_{0}}{\to} & x_{1} & \overset{id}{\leftarrow} & x_{1} \\ f_{2} \downarrow & & & & \downarrow f_{1} \\ x_{2} & & & & x_{3} \\ id \downarrow & & & & \downarrow id \\ x_{2} & \underset{f_{3}}{\to} & x_{2} & \underset{id}{\leftarrow} & x_{3} \\ }

The following diagram of commutative squares in Π 1 2(X)\Pi^{\leq 2}_{1}(X) illustrates that the above square does indeed commute in Π 1 2(X)\Pi^{\leq 2}_{1}(X).

x 0 f 0 x 1 id x 1 f 1 x 2 id x 2 id A A id id x 0 f 2 x 2 id x 2 f 3 x 3 id x 3 \array{ x_{0} & \overset{f_{0}}{\rightarrow} & x_{1} & \overset{id}{\leftarrow} & x_{1} & \overset{f_{1}}{\rightarrow} & x_{2} & \overset{id}{\leftarrow} & x_{2} \\ id \downarrow & & \downarrow A & & \downarrow A & & \downarrow id & & \downarrow id \\ x_{0} & \underset{f_{2}}{\rightarrow} & x_{2} & \underset{id}{\leftarrow} & x_{2} & \underset{f_{3}}{\rightarrow} & x_{3} & \underset{id}{\leftarrow} & x_{3} }

Here A is f 2f 0 1f_{2} \circ f_{0}^{-1}. That the square

x 1 f 1 x 3 A id x 2 f 3 x 3 \array{ x_{1} & \overset{f_{1}}{\to} & x_{3} \\ A \downarrow & & \downarrow id \\ x_{2} & \underset{f_{3}}{\to} & x_{3} }

commutes follows easily from the commutativity of the square arising from the 2-cube σ\sigma of XX above.

Proposition

The natural transformations η\eta and ϵ\epsilon define an adjunction between Π 1 2\Pi^{\leq 2}_{1} and N 2N^{\leq 2}.

Proof

Straightforward verification that the triangle identities hold.

Fundamental groupoid functor

Notation

Adopting the notation of cubical truncation, skeleton, and co-skeleton, we denote by Π 1:Set opGrpd\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd} the functor

Set op tr 2 Set 2 op Π 1 2 Grpd. \array{ \mathsf{Set}^{\square^{op}} & \overset{tr_{2}}{\rightarrow} & \mathsf{Set}^{\square_{2}^{op}} & \overset{\Pi_{1}^{\leq 2}}{\rightarrow} & \mathsf{Grpd}. }
Terminology

We refer to the functor Π 1:Set opGrpd\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd} as the fundamental groupoid functor.

Nerve functor

Notation

Adopting the notation of cubical truncation, skeleton, and co-skeleton, we denote by N:GrpdSet opN : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square^{op}} the functor

Grpd N 2 Set 2 op cosk 2 Set op. \array{ \mathsf{Grpd} & \overset{N^{\leq 2}}{\rightarrow} & \mathsf{Set}^{\square_{\leq 2} ^{op}} & \overset{cosk_{2}}{\rightarrow} & \mathsf{Set}^{\square^{op}}. }
Terminology

We refer to the functor N:Set opGrpdN : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd} as the nerve functor.

Adjunction between the fundamental groupoid functor and the nerve functor

Since Π 1 2\Pi^{\leq 2}_{1} is left adjoint to N 2N^{\leq 2} by Proposition above, and since tr 2tr_{2} is left adjoint to cosk 2cosk_{2}, we have that Π 1\Pi_{1} is left adjoint to NN.

Last revised on April 27, 2016 at 15:35:10. See the history of this page for a list of all contributions to it.