# nLab automorphism infinity-group

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## Definition

### Externally

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

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

$\mathrm{Aut}\left(X\right)↪𝒞\left(X,X\right)$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,

$\mathrm{Aut}\left(X\right)\in \mathrm{Grp}\left(\infty \mathrm{Grpd}\right)\phantom{\rule{thinmathspace}{0ex}}.$Aut(X)\in Grp(\infty Grpd) \,.

### Internally

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

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

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

$\mathrm{Aut}\left(X\right)↪\left[X,X\right]$\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 $B\mathrm{Aut}\left(X\right)$ of the internal automorphism $\infty$-group is equivalently the ∞-image

$*\to B\mathrm{Aut}\left(X\right)↪\mathrm{Obj}$* \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 $\infty$-group $\mathrm{Aut}\left(X\right)$ then is (as a type, without yet the group structure)

$⊢\left(X\stackrel{\simeq }{\to }X\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}},$\vdash (X \stackrel{\simeq}{\to} X) : Type \,,

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

$⊢\phantom{\rule{thickmathspace}{0ex}}\left(\sum _{Y:\mathrm{Type}}\left[Y=X\right]\right):\mathrm{Type}\phantom{\rule{thinmathspace}{0ex}},$\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 $\left[X=Y\right]=\mathrm{isInhab}\left(X=Y\right)$ of the identity type $\left(X=Y\right)$.

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 $\infty$-group of an object is the ordinary automorphism group of that object.

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

### Of $\infty$-groups

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

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

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

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

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

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

Revised on October 17, 2013 04:16:24 by Urs Schreiber (80.237.234.148)