nLab
automorphism infinity-group

Definition

Externally

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

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

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

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

This is an ∞-group in ∞Grpd,

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

Internally

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

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

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

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

of the internal hom on those morphism that are equivalences.

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

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

of the morphism

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

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

In the syntax of homotopy type theory

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

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

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

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

where on the right we have dependent sum over one argument of the bracket type/(-1)-truncation $[X=Y]=\mathrm{isInhab}(X=Y)$ 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 $𝒞$ happens to be a 1-category then the external automorphism $\mathrm{\infty }$-group of an object is the ordinary automorphism group of that object.

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

Of $\mathrm{\infty }$-groups

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

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

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

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

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

Related concepts

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

• group, ∞-group,

• automorphism group, automorphism 2-group, automorphism $\mathrm{\infty }$-group,

• center, center of an ∞-group

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

• rigidification of a stack