Group Actions as Functors

Let G be a group, and X an object of the category C. The group action of G on A is by definition a functor from CG to C that maps o to X, where we have used o to denote the unique object of G. This is just a formal way of saying that the elements of G give rise to the structure-preserving maps (morphisms in our language) from X to itself. 

For example, C could be the category of topological spaces, or abstract simplicial complexes, or posets, in which case the structure-preserving maps would respectively be continuous maps, or simplicial maps, or order-preserving maps (see Definition 10.3). 

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