nLab automorphism infinity-group

Definition

Externally

Let $\mathcal{C}$ be an (∞,1)-category. Let $X \in \mathcal{C}$ be an object.

As a discrete ∞-group the automorphism $\infty$-group of $X$ is the sub-∞-groupoid

$$Aut(X) \hookrightarrow \mathcal{C}(X,X)$$

of the derived hom space of morphisms in $\mathcal{C}$ from $X$ to itself, on those that are equivalences.

This is an ∞-group in ∞Grpd,

$$Aut(X)\in Grp(\infty Grpd) \,.$$

Internally

Let $\mathcal{C}$ be a cartesian closed (∞,1)-category (for instance an (∞,1)-topos). Write

$$[-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$$

for the internal hom. Then for $X \in \mathcal{C}$ an object, the internal automorphism $\infty$-group is the subobject

$$\mathbf{Aut}(X) \hookrightarrow [X,X]$$

of the internal hom on those morphism that are equivalences.

In the special case that $\mathcal{C}$ is an ∞-topos, the delooping $\mathbf{B}\mathbf{Aut}(X)$ of the internal automorphism $\infty$-group is equivalently the ∞-image

$$* \to \mathbf{B}\mathbf{Aut}(X) \hookrightarrow Obj$$

of the morphism

$$* \stackrel{\vdash X}{\to} Obj$$

to the object classifier, that modulates $X$ (the “name” of $X$).

In homotopy type theory

Let $\mathcal{C}$ be an (∞,1)-topos. Then its internal language is homotopy type theory. In terms of this the object $X \in \mathcal{C}$ is a type (homotopy type). In the type theory syntax the internal automorphism $\infty$-group $\mathbf{Aut}(X)$ then is (as a type, without yet the group structure)

$$\vdash (X \stackrel{\simeq}{\to} X) : Type \,,$$

the subtype of the function type on the equivalences. Its delooping $\mathbf{B}\mathbf{Aut}(X)$ is

$$\vdash \; \left(\sum_{Y : Type} [Y = X]\right) \colon Type \,,$$

where on the right we have the dependent sum over one argument of the bracket type/(-1)-truncation $[X = Y] = isInhab(X = Y)$ of the identity type $(X = Y)$.

The equivalence of this definition to the previous one is essentially equivalent to the univalence axiom.

Examples

In a 1-category

If $\mathcal{C}$ happens to be a 1-category then the external automorphism $\infty$-group of an object is the ordinary automorphism group of that object.

If $\mathcal{C}$ happens to be a 1-topos, then the internal automorphism $\infty$-group is the traditional automorphism group object in the topos. Etc.

Of $\infty$-groups

For $G \in \infty Grp(\mathcal{X})$ an ∞-group there is the direct automorphism $\infty$-group $Aut(G)$. But there is also the delooping $\mathbf{B}G \in \mathcal{X}$ and its automorphism $\infty$-group.

Sometimes (for instance in the discussion of ∞-gerbes) one considers

$$AUT(G) := Aut(\mathbf{B}G)$$

and calls this the automorphism $\infty$-group of $G$.

For instance when $G$ is an ordinary group, $AUT(G)$ is the 2-group discussed at automorphism 2-group.

Related concepts

• There may be the structure of an ∞-Lie group on $Aut(F)$. The corresponding ∞-Lie algebra is an automorphism ∞-Lie algebra.

• group, ∞-group,

• automorphism group, automorphism 2-group

• autoequivalence type

• center, center of an ∞-group

• outer automorphism group, automorphism 2-group, outer automorphism ∞-group

• rigidification of a stack