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

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 $\mathsf{Grpd}$.

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 $\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd}$ the functor defined as follows.

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

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

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

$$\array{ x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n} }$$

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

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

or of the following form.

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

ii) We may replace an entire zig-zag

$$\array{ x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n} }$$

with a zig-zag

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

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

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

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

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

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

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 $x$ of $\Pi^{\leq 2}_{1}(X)$ is the zig-zag with $n = 0$ consisting simply of $x$.

f) The inverse of an arrow

$$\array{ x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n} }$$

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

$$\array{ x_{2n} & \overset{g_{n}}{\rightarrow} & x_{2n-1} \overset{f_{n}}{\leftarrow} & x_{2n-2} & \cdots & x_{2} & \overset{g_{1}}{\rightarrow} & x_{1} \overset{f_{1}}{\leftarrow} & x_{0} }$$

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

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

b) To a zig-zag as follows

$$\array{ x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n} }$$

we associate the following zig-zag.

$$\array{ F(x_{0}) & \overset{F(f_{1})}{\rightarrow} & F(x_{1}) \overset{F(g_{1})}{\leftarrow} & F(x_{2}) & \cdots & F(x_{2n-2}) & \overset{F(f_{n})}{\rightarrow} & F(x_{2n-1}) \overset{F(g_{n})}{\leftarrow} & F(x_{2n}) }$$

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

Terminology

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

2-truncated nerve functor

Notation

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

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

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

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

c) The 2-cubes

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

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

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

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

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

or as follows.

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

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

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

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

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

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

of $\mathcal{A}$ to the commutative square

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

of $\mathcal{A}$.

Terminology

We refer to $N^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}}$ 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 $\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id$ the natural transformation which to a groupoid $\mathcal{A}$ associates the functor $\epsilon(\mathcal{A}) : \Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}) \rightarrow \mathcal{A}$ defined as follows.

1) On objects it is the identity.

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

$$\array{ a_{0} & \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}}{\leftarrow} & a_{2n} }$$

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

$$\array{ a_{0} & \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}^{-1}}{\rightarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}^{-1}}{\rightarrow} & a_{2n} }$$

of $\mathcal{A}$.

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

1) On objects it is the identity.

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

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

3) To a 2-cube

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

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

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

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

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

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

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

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

Proposition

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

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 $\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ the functor

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

Terminology

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

Nerve functor

Notation

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

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

Terminology

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

Adjunction between the fundamental groupoid functor and the nerve functor

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