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\begin{document}

%-------------------------------------------------------------------


\section*{invariant}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{ViaSectionsOfActionGroupoidProjections}{Via sections of action groupoid projections}\dotfill \pageref*{ViaSectionsOfActionGroupoidProjections} \linebreak
\noindent\hyperlink{ForInfinityGroupActions}{Invariants of $\infty$-group actions}\dotfill \pageref*{ForInfinityGroupActions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $A$ a [[monoid]] equipped with an [[action]] on an object $V$, an \textbf{invariant} of the action is an [[generalized element|element]] of $V$ which is taken by the action to itself, hence a [[fixed point]] for all the operations in the monoid.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A robust definition of invariants that generalizes to [[homotopy theory]] is via the expression of actions as [[action groupoids]] regarded as sitting over [[delooping]] [[groupoids]], as discussed at \emph{[[infinity-action]]} and at \emph{[[geometry of physics -- representations and associated bundles]]}.

We describe how the ordinary concept of invariants is recovered from this perspective and then consdider its immediate generalizations to [[(infinity,1)-topos theory]] and its formalization in [[homotopy type theory]].

\hypertarget{ViaSectionsOfActionGroupoidProjections}{}\subsubsection*{{Via sections of action groupoid projections}}\label{ViaSectionsOfActionGroupoidProjections}

\begin{prop}
\label{}\hypertarget{}{}
For $G$ a [[discrete group]], $\rho$ a $G$-[[action]] on some set $S$, then the set of [[invariants]] of that action is equivalent to the groupoid of [[sections]] of the [[action groupoid]] projection of \href{geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG}{this proposition}, corresponding to the action via \href{geometry+of+physics+--+representations+and+associated+bundles#EquivalenceOfPermutationRepresentationsWithActionGroupodsInSlice}{this proposition}.

\end{prop}
\begin{proof}
The sections in question are diagrams in [[Grpd]] of the form

\begin{displaymath}
\itexarray{
    \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{id}}\searrow 
    &\swArrow_{\mathrlap{\simeq}}& 
    \swarrow_{\mathrlap{p_\phi}}
    \\
    && \mathbf{B}G
  }
  \,,
\end{displaymath}
hence the groupoid which they form is equivalently the [[hom-groupoid]]

\begin{displaymath}
Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho)
  \in Grpd
\end{displaymath}
in the [[slice (infinity,1)-category|slice]] of [[Grpd]] over $\mathbf{B}G$. As in the proof of \href{geometry+of+physics+--+representations+and+associated+bundles#IntertwinersOfPermutationActionAsSliceHoms}{this proposition}, with the fibrant presentation $(p_\rho)_\bullet$ of \href{geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG}{this proposition}, this is equivalently given by strictly commuting diagrams of the form

\begin{displaymath}
\itexarray{
    (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet
    \\
    & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\phi)_\bullet}}
    \\
    && (\mathbf{B}G)_\bullet
  }
  \,.
\end{displaymath}
These $\sigma$ now are manifestly functors that are the identiy on the group labels of the morphisms

\begin{displaymath}
\sigma_\bullet
  \;\colon\;
  \left(
  \itexarray{
    \ast
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \ast
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \itexarray{
    \sigma(\ast)
    \\
    \downarrow^{\mathrlap{g}}
    \\
    \sigma(\ast) & = \rho(\sigma(\ast)(g))
  }
  \right)
  \,.
\end{displaymath}
This shows that they pick precisely those elements $\sigma(\ast) \in S$ which are fixed by the $G$-action $\rho$.

Moreover, since these functors are identity on the group labels, there are no non-trivial [[natural isomorphisms]] between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

\end{proof}
More generally, we may consider sections of these groupoid projections after pulling them back along some cocycle:

\begin{prop}
\label{}\hypertarget{}{}
Given an [[associated bundle]] $P \times_G V\to X$ [[modulating morphism|modulated]], as in \href{geometry+of+physics+--+representations+and+associated+bundles#Associated1BundleAsPullbackOfActionGroupoid}{this proposition}, by a morphism of [[smooth groupoids]] of the form $g \colon X \longrightarrow \mathbf{B}G$, then its set of [[sections]] is equivalently the groupoid of diagrams

\begin{displaymath}
\itexarray{
    X && \stackrel{\sigma}{\longrightarrow} && S//G
    \\
    & {}_{\mathllap{g}}\searrow 
    &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\phi}}
    \\
    && \mathbf{B}G
  }
  \,,
\end{displaymath}
hence the groupoid of sections is the slice [[hom-groupoid]]

\begin{displaymath}
\Gamma_X(P\times_G V)
  \simeq
  Grpd_{/\mathbf{B}G}(g, p_\rho)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the defining [[universal property]] of the [[homotopy pullback]] in \href{geometry+of+physics+--+representations+and+associated+bundles#Associated1BundleAsPullbackOfActionGroupoid}{this proposition}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Taken together this means that [[invariants]] of group actions are equivalently the sections of the corresponding [[universal principal bundle|universal]] [[associated bundle]].

\end{remark}
\hypertarget{ForInfinityGroupActions}{}\subsubsection*{{Invariants of $\infty$-group actions}}\label{ForInfinityGroupActions}

For $\mathbf{H}$ an [[(∞,1)-topos]], $G \in Grp(\mathbf{H})$ an [[∞-group]] and

\begin{displaymath}
* : \mathbf{B} G \vdash : V(*) : Type
\end{displaymath}
an [[∞-action]] of $G$ on $V \in \mathbf{H}$, the type of invariants is the absolute [[dependent product]]

\begin{displaymath}
\vdash \prod_{* : \mathbf{B}G} V(*) : Type
  \,.
\end{displaymath}
The connected components of this is equivalently the [[group cohomology]] of $G$ with [[coefficients]] in the [[infinity-module]] $V$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero}\hypertarget{TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero}{}
\textbf{(in [[characteristic zero]], [[invariants]] for [[finite group]] are compatible with [[chain homology]])}

Let $(V_\bullet, \partial)$ be a [[chain complex]] over a [[ground field]] of [[characteristic zero]], equipped with an [[action]] by a [[finite group]] $G$. Then taking $G$-invariants commutes with passing to [[chain homology]]:

\begin{displaymath}
H_\bullet((V_\bullet,\partial)^G)
  \;\simeq\;
  H_\bullet((V_\bullet,\partial))^G
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Since the [[ground field]] has [[characteristic zero]], [[group averaging]] exists and provides a [[linear map]]

\begin{displaymath}
\itexarray{
    V_\bullet & \overset{p}{\longrightarrow} & V_\bullet^G
    \\
    x &\mapsto& \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} g(x)
  }
\end{displaymath}
onto the $G$-invariants.

Now for a [[chain homology]]-class $[x] \in H_\bullet((V_\bullet,\partial))$ being $G$-invariant means that $g[x] \coloneqq [g(x)] = [x]$ for all $g \in G$, which implies that $[x] = [p(x)]$. This means that each invariant homology class has an invariant representative, hence that the map from invariant [[cycles]] to invariant [[chain homology]]-classes

\begin{displaymath}
Z((V_\bullet^G,\partial)) \longrightarrow H_\bullet((V_\bullet,\partial))
\end{displaymath}
is an [[epimorphism]].

Next consider the [[kernel]] of this map, which a priori is $Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))$. It is now sufficient to show that this coincides with the space of $G$-invariant [[boundaries]]:

\begin{displaymath}
Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
  \;\simeq\;
  B((V_\bullet^G, \partial))
  \,.
\end{displaymath}
It is clear that there is an [[injective map|inclusion]]

\begin{displaymath}
B((V_\bullet^G, \partial)) 
    \hookrightarrow 
  Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
\end{displaymath}
so it only remains to see that this is also a [[surjection]].

To that end, consider any

\begin{displaymath}
x \in Z((V_\bullet^G,\partial)) \cap B((V_\bullet,\partial))
  \,.
\end{displaymath}
Since in particular $x \in B((V_\bullet,\partial))$, there is $y \in V_\bullet$ with $x = \partial y$; and since moreover $x \in V_\bullet(G)$, the above implies that

\begin{displaymath}
x = p(x) = p(\partial y) = \partial(p y)
\end{displaymath}
and hence that

\begin{displaymath}
x \in B((V_\bullet^G,\partial)) 
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[group averaging]], [[norm map]]


\item [[stabilizer subgroup]]


\item [[invariant theory]], [[invariant polynomial]]


\item [[gauge invariance]]


\item [[topological invariant]]/[[homeomorphism invariant]]


\item [[homotopy invariant]]



\end{itemize}
[[!include homotopy type representation theory -- table]]

[[!redirects invariants]]



\end{document}