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 .
Fundamental groupoid for 2-truncated cubical sets
Notation
We denote by the functor defined as follows.
1) To a 2-truncated cubical set , we associate the groupoid defined as follows.
a) The objects of are the 0-cubes of .
b) The arrows of are zig-zags of 1-cubes of up to the notion of equivalence defined below, where by a zig-zag of 1-cubes of we mean, for some integer , a set of 1-cubes of whose faces match up as follows.
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
or of the following form.
ii) We may replace an entire zig-zag
with a zig-zag
if there is, for every , a 2-cube of whose horizontal 1-cubes are as follows
and there is, for every , a 2-cube of whose horizontal 1-cubes are as follows.
c) The source of a zig-zag as at the beginning of b) is , and the target is .
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 of is the zig-zag with consisting simply of .
f) The inverse of an arrow
of is the following arrow (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).
2) To a morphism of 2-truncated cubical sets , we associate the functor defined as follows.
a) On objects, is the same as .
b) To a zig-zag as follows
we associate the following zig-zag.
It is immediately verified that this is well-defined with respect to the equivalence relation of 1) b).
Terminology
We refer to as the fundamental groupoid functor for 2-truncated cubical sets.
2-truncated nerve functor
Notation
We denote by the functor defined as follows.
1) To a groupoid , we associate the 2-truncated cubical set defined as follows.
a) The 0-cubes of are the objects of .
b) The 1-cubes of are the arrows of with source and target .
c) The 2-cubes
of are the commutative squares of whose boundary looks the same as this.
d) The degenerate 1-cubes of are the identity arrows of .
e) The degenerate 2-cubes of are the commutative squares of which look as follows
or as follows.
2) To a functor , we associate the morphism of 2-truncated cubical sets defined as follows.
a) On 0-cubes, is the same as .
b) On 1-cubes, is the same as .
c) On 2-cubes, sends a commutative square
of to the commutative square
of .
Terminology
We refer to 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 the natural transformation which to a groupoid associates the functor defined as follows.
1) On objects it is the identity.
2) To an arrow of , given by a zig-zag of arrows
of , we associate the arrow of given by the composition in of the arrows
of .
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 the natural transformation which to a 2-truncated cubical set associates the morphism of 2-truncated cubical sets defined as follows.
1) On objects it is the identity.
2) To a 1-cube of we associate the following zig-zag of 1-cubes of , where the right arrow is the degeneracy on .
3) To a 2-cube
of we associate the commutative square in given as follows.
The following diagram of commutative squares in illustrates that the above square does indeed commute in .
Here A is . That the square
commutes follows easily from the commutativity of the square arising from the 2-cube of above.
Proposition
The natural transformations and define an adjunction between and .
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 the functor
Terminology
We refer to the functor as the fundamental groupoid functor.
Nerve functor
Notation
Adopting the notation of cubical truncation, skeleton, and co-skeleton, we denote by the functor
Terminology
We refer to the functor as the nerve functor.
Adjunction between the fundamental groupoid functor and the nerve functor
Since is left adjoint to by Proposition above, and since is left adjoint to , we have that is left adjoint to .