Groupoid actions and principal bundles | Higher Topos Theory notes

6.2. Groupoid actions and principal bundles

In this section, we discuss actions of groupoid objects in a topos $$T$$ and prove that the category of such actions is equivalent to the slice over the classifying object. We then characterize principal bundles as regular actions with effective-epimorphic atlas and show that they are classified by maps into the classifying object. Our treatment is based on [Nikolaus et al. 2015] and [Sati and Schreiber 2021, Section 3.2], where the analogous situation for group actions is discussed.

Throughout this section, we fix a topos $$T$$ and a groupoid object $\Gg\colon \simp\catop \to T$ .

6.2.1. Actions by groupoid objects

We introduce the notion of an action of a groupoid object, generalizing the treatment of group actions in [Sati and Schreiber 2021].

Definition 6.27.

Let $\Gg$ be a groupoid object in $$T$$ , and let $X \in T$ be an object. An action of $\Gg$ on $$X$$ consists of the following data:

A groupoid object $\Gg \ltimes X$ in $$T$$ ;
An isomorphism $(\Gg \ltimes X)_0 \simeq X$ ;
A cartesian natural transformation $c\colon \Gg \ltimes X \to \Gg$ of simplicial objects in $$T$$ .

We let $\Act_{\Gg}(T) \subseteq \Grpd(T)_{/\Gg}$ denote the full subcategory spanned by the $\Gg$ -actions in $$T$$ .

Remark 6.28.

Let $\Gg \ltimes X$ be an action of $\Gg$ on $$X$$ . Since $(\Gg \ltimes X)_0 \simeq X$ , the map $c\colon \Gg \ltimes X \to \Gg$ induces a structure map $c_0\colon X \to \Gg_0$ . Since $$c$$ is a cartesian natural transformation, there are cartesian squares

and in particular we obtain isomorphisms \begin{align*} (\Gg \ltimes X)_1 &\simeq \Gg_1 \times_{\Gg_0} X, \\ (\Gg \ltimes X)_2 &\simeq \Gg_2 \times_{\Gg_0} X \simeq \Gg_1 \times_{\Gg_0} \Gg_1 \times_{\Gg_0} X, \end{align*} and so on. Under these isomorphisms, the map $d_1\colon (\Gg \ltimes X)_1 \to (\Gg \ltimes X)_0$ corresponds to a map $a\colon \Gg_1 \times_{\Gg_0} X \to X$ , which we think of as the action map. The rest of the simplicial diagram $\Gg \ltimes X$ encodes the data witnessing that this action is unital and associative up to coherent homotopy.

Definition 6.29.

For a groupoid object $\Gg$ , we define its classifying object as the geometric realization $\bB \Gg := \abs{\Gg}$ .

If $\Gg \ltimes X$ is an action of a groupoid object $\Gg$ on an object $X \in T$ , we define the quotient $X \quot \Gg$ as the geometric realization of $\Gg \ltimes X$ :

X \quot \Gg \quad:=\quad \abs{\Gg \ltimes X} \quad\simeq\quad \colim_{[n] \in \simp\catop} \Gg_n \times_{\Gg_0} X.

The cartesian transformation $\Gg \ltimes X \to \Gg$ induces a map $X \quot \Gg \to \abs{\Gg} = \bB\Gg$ on geometric realizations, and this defines a functor

-\quot\Gg \colon \Act_{\Gg}(T) \to T_{/\bB\Gg}.

The following result shows that groupoid actions are classified by the classifying object:

Proposition 6.30. (Classification of groupoid actions)

The quotient functor is an equivalence of categories:

-\quot\Gg \colon \Act_{\Gg}(T) \iso T_{/\bB\Gg}.

Proof

Recall from Lemma 2.33 that sending a groupoid

\Gg

to the map

\Gg_0 \to \bB \Gg

defines an equivalence

\Grpd(T) \; \iso \; \EffEpi(T)

between the categories of groupoid objects and effective epimorphisms in

\(T\)

. Moreover, it follows from descent that a morphism of groupoids

f\colon \Hh \to \Gg

is cartesian if and only if the induced square

is a pullback square, giving an equivalence of categories

\Grpd(T)^{\cart} \; \iso \; \EffEpi(T)^{\cart}

. Passing to slices over

\Gg

, this results in an equivalence

\Act_{\Gg}(T) \quad = \quad (\Grpd(T)^{\cart})_{/\Gg} \quad \iso \quad (\EffEpi(T)^{\cart})_{/(\Gg_0 \to \bB \Gg)} \quad \iso \quad T_{/\bB \Gg}.

Here the last equivalence is induced by the target map. Restricted to cartesian morphisms over the fixed effective epimorphism

\Gg_0\to\bB\Gg

, this functor is an equivalence: its inverse sends a morphism

B \to \bB\Gg

to the pullback square

6.2.2. Free and transitive actions

Given a $\Gg$ -action in $$T$$ , one can define when the action is free or transitive. Actions which are both free and transitive will correspond to principal $\Gg$ -bundles.

Definition 6.31. (Shear maps)

Let $\Gg \ltimes X$ be an action of a groupoid object $\Gg$ on an object $$X$$ in $$T$$ . We define the shear map of $\Gg \ltimes X$ as the map

\shear_1 = (a, \pr_X)\colon \Gg_1 \times_{\Gg_0} X \to X \times X,

where $a\colon \Gg_1 \times_{\Gg_0} X \to X$ is the action map and $\pr_X$ is the projection onto $$X$$ .

More generally, we define the higher shear maps as follows. The atlas $X\to X\quot\Gg$ is an effective epimorphism, and the equivalence of Lemma 2.33 identifies the action groupoid $\Gg\ltimes X$ with its Čech nerve $\check{C}(X\to X\quot\Gg)$ . Consider now the commutative square

Regarding this as a morphism from $X \twoheadrightarrow X\quot\Gg$ to $X \to *$ in $\Ar(T)$ and passing to Čech nerves, we obtain a morphism of groupoid objects

\shear\colon (\Gg \ltimes X) \to \check{C}(X \to *).

Under the identifications of Remark 6.28, this may be displayed as follows:

Definition 6.32. (Free, transitive and regular actions)

Let $\Gg \ltimes X$ be an action of a groupoid object $\Gg$ on an object $X \in T$ . We say that this action is

Free if its shear map $\shear_1$ is a monomorphism;
Transitive if its shear map $\shear_1$ is an effective epimorphism;
Regular if its shear map $\shear_1$ is an isomorphism.

Note that $\Gg \ltimes X$ is regular if and only if it is both free and transitive.

In case the morphism $X \to *$ is an effective epimorphism in $$T$$ , regularity of $\Gg \ltimes X$ can be expressed as the triviality of the quotient:

Lemma 6.33.

Let $X \in T$ be an object such that $X \to *$ is an effective epimorphism. Let $\Gg \ltimes X$ be an action of a groupoid object on $$X$$ . Then the action is regular if and only if the quotient $X\quot\Gg$ is the terminal object of $$T$$ .

Proof

First assume that

X\quot\Gg \simeq *

. Then the map

X \twoheadrightarrow X\quot\Gg

is equivalent to the map

X \to *

. It follows that the shear map

\shear\colon (\Gg \ltimes X) \to \check{C}(X \to *)

is an isomorphism, being defined by applying Čech nerves to these two equivalent morphisms.Conversely, assume that

\Gg \ltimes X

is regular and that

X \twoheadrightarrow *

is an effective epimorphism. The map

X\quot\Gg \to *

is obtained from the higher shear map

\shear\colon (\Gg \ltimes X) \to \check{C}(X \to *)

by passing to colimits. It thus suffices to show that each higher shear map

\shear_n\colon \Gg_n \times_{\Gg_0} X \to X^{n+1}

is an isomorphism. Both its source and target are groupoid objects with object of objects

\(X\)

. By the Segal condition, their terms in degree

\(n\)

are the

\(n\)

-fold iterated fiber products of their terms in degree

\(1\)

over

\(X\)

, and

\shear_n

is the corresponding iterated pullback of

\shear_1

. Since

\shear_1

is an isomorphism by regularity, every

\shear_n

is an isomorphism.

6.2.3. Principal bundles

Given a groupoid object $\Gg$ in $$T$$ and an object $B \in T$ , we obtain a notion of principal $\Gg$ -bundles over $$B$$ by applying the definition of regular actions to the slice topos $T_{/B}$ . Observe that $\Gg$ gives rise to a groupoid $\Gg \times B$ in the slice $T_{/B}$ by applying the pullback functor $B \times -\colon T \to T_{/B}$ . For every object $P \in T_{/B}$ , there is an isomorphism

(\Gg_n \times B) \times_{\Gg_0 \times B} P \;\simeq\; \Gg_{n} \times_{\Gg_0} P \qquad\text{in } T_{/B},

and thus a $(\Gg \times B)$ -action on $$P$$ is the same as a $\Gg$ -action on the underlying object of $$P$$ for which the map $P\to B$ is invariant. For this reason, we will refer to a $(\Gg \times B)$ -action in the slice $T_{/B}$ simply as a $\Gg$ -action over $$B$$ .

Definition 6.34. (Principal bundles)

Let $B \in T$ be an object and let $\Gg$ be a groupoid object in $$T$$ .

Let $p\colon P \to B$ be an object of $T_{/B}$ equipped with a $\Gg$ -action over $$B$$ . We say that $$p$$ is a formally principal $\Gg$ -bundle over $$B$$ if this action is regular as a $(\Gg \times B)$ -action in the slice $T_{/B}$ , i.e. if the shear map
$\shear_{1}\colon \Gg_1 \times_{\Gg_0} P \iso P \times_B P$
is an isomorphism.
A formally principal $\Gg$ -bundle $p\colon P \to B$ is called a principal $\Gg$ -bundle if $$p$$ is an effective epimorphism.
We write
$\Bun_{\Gg}(B) \quad \subseteq \quad \Act_{\Gg\times B}(T_{/B})$
for the full subcategory of principal $\Gg$ -bundles over $$B$$ .

Remark 6.35.

For any morphism $f\colon B' \to B$ in $$T$$ , the pullback functor $f^*\colon T_{/B} \to T_{/B'}$ preserves principal $\Gg$ -bundles.

Lemma 6.36.

Let $p\colon P \twoheadrightarrow B$ be an effective epimorphism equipped with a $\Gg$ -action over $$B$$ , and let $P\quot\Gg \in T_{/B}$ denote the quotient. Then $$p$$ is a principal $\Gg$ -bundle if and only if the canonical map $P\quot\Gg \to B$ is an isomorphism.

Proof

This is an instance of Lemma 6.33 applied in the slice topos

T_{/B}

It follows from Lemma 6.36 that the forgetful functor from principal $\Gg$ -bundles over $$B$$ to $\Gg$ -actions in $$T$$ lands in the fiber over $$B$$ of the composite

\Act_{\Gg}(T)\xrightarrow{-\quot\Gg}T_{/\bB\Gg}\longrightarrow T.

Proposition 6.37.

The forgetful functor induces an equivalence of categories

\Bun_{\Gg}(B) \iso \Act_{\Gg}(T) \times_{T} \{B\}.

Proof

We construct an explicit inverse. Consider a

\Gg

-action

c\colon\Gg \ltimes P\to\Gg

\(T\)

equipped with an isomorphism

B \iso P\quot\Gg

. The colimit cocone exhibiting

\(B\)

as the realization of

\Gg\ltimes P

supplies compatible maps

(\Gg\ltimes P)_n\to B

and hence lifts the action groupoid to a groupoid object in

T_{/B}

. Pairing these structure maps with

\(c\)

gives a morphism of groupoid objects

\Gg\ltimes P\longrightarrow\Gg\times B

T_{/B}

. Its underlying morphism in

\(T\)

\(c\)

, so it is cartesian and therefore defines a

(\Gg\times B)

-action in the slice. This construction defines a functor

\Act_{\Gg}(T) \times_{T} \{B\} \to \Act_{\Gg\times B}(T_{/B}).

The map

P\to B

is an effective epimorphism because it is the atlas of the realization of the groupoid object

\Gg\ltimes P

, and its quotient in

T_{/B}

is the terminal object

B\to B

. Thus Lemma 6.36 shows that the resulting action is principal. Conversely, starting from a principal bundle, the same lemma identifies its quotient with

\(B\)

, and the colimit cocone recovers its original structure maps to

\(B\)

. Hence the two constructions are inverse.

We are now ready to state the main classification result:

Theorem 6.38. (Classification of principal bundles)

Let $\Gg$ be a groupoid object in $$T$$ and let $B \in T$ be an object. There is an equivalence of categories

\Hom_{T}(B,\bB\Gg) \iso \Bun_{\Gg}(B),

given on objects by sending a morphism $c\colon B \to \bB\Gg$ to the principal $\Gg$ -bundle $P \to B$ defined by the pullback square

Proof

Recall from Proposition 6.30 the equivalence

-\quot\Gg \colon \Act_{\Gg}(T) \iso T_{/\bB\Gg}

, which induces an equivalence on fibers over

\(T\)

\Act_{\Gg}(T) \times_{T} \{B\} \;\simeq\; T_{/\bB\Gg} \times_{T} \{B\} \;\simeq\; \Hom_{T}(B,\bB\Gg).

The inverse sends a morphism

c\colon B \to \bB\Gg

to the base change

P = B \times_{\bB\Gg} \Gg_0 \twoheadrightarrow B

of the atlas

\Gg_0 \twoheadrightarrow \bB\Gg

along

\(c\)

. Combining with the equivalence from Proposition 6.37 finishes the proof.

Corollary 6.39.

For every groupoid object $\Gg$ in $$T$$ and every object $B \in T$ , the category $\Bun_{\Gg}(B)$ is an anima.

References

Thomas Nikolaus, Urs Schreiber, Danny Stevenson. Principal $\infty $-bundles: general theory. J. Homotopy Relat. Struct., 10 (4), 749–801. 2015.
Hisham Sati, Urs Schreiber. Equivariant principal infinity-bundles. arXiv preprint arXiv:2112.13654. 2021.