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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2013
   • (edited Nov 15th 2013)
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   Author: Urs
   Format: MarkdownItexI noticed by accident that we have an entry _[[coinvariant]]_. Then I noticed that we also have an entry _[[homotopy coinvariant functor]]_. I have now added cross-links between these entries and with _[[invariant]]_ and _[[orbit]]_, so that they no longer remain hidden. I also edited the first case of group representation coinvariants at _[[coinvariant]]_ a little.

   I noticed by accident that we have an entry coinvariant. Then I noticed that we also have an entry homotopy coinvariant functor.

   I have now added cross-links between these entries and with invariant and orbit, so that they no longer remain hidden.

   I also edited the first case of group representation coinvariants at coinvariant a little.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 15th 2013
   • (edited Nov 15th 2013)
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   Author: David_Corfield
   Format: MarkdownItexIs there a HoTT formulation as for [invariants](http://ncatlab.org/nlab/show/infinity-action#Invariants)?

   Is there a HoTT formulation as for invariants?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo the coinvariant should just be the dependent sum, where the invariants is the dependent poduct. In this sense: for $G$ a group object, types $V$ with $G$-action $\rho_V$ are $\mathbf{B}G$-dependent types. The homotopy invariants are $\underset{\mathbf{B}G}{\prod} \rho_V \simeq H_{Grp}(G,V)$. The homotopy co-invariants are $\underset{\mathbf{B}G}{\sum} \rho_V \simeq V//G$.

   So the coinvariant should just be the dependent sum, where the invariants is the dependent poduct.

   In this sense: for $G$ a group object, types $V$ with $G$-action $\rho_V$ are $\mathbf{B}G$-dependent types.

   The homotopy invariants are $\underset{\mathbf{B}G}{\prod} \rho_V \simeq H_{Grp}(G,V)$.

   The homotopy co-invariants are $\underset{\mathbf{B}G}{\sum} \rho_V \simeq V//G$.