simplicial set with a unique non-generate ${{m}}$-simplex and no relations among its faces. The simplicial subset generated by the ${{i}}$-th face of ${\Delta^{{{m}}}}$ will be denoted by ${d}_{{i}}{\Delta^{{{m}}}}$. The boundary $\partial{\Delta^{{{m}}}}$ is the union of all the faces and the ${{i}}$-th horn ${\mbox{$\xy
<0pt,-\the\fontdimen22\textfont2>;p+<.1em,0em>:
{\ar@{-}(0,0.1);(3,7)},
{\ar@{-}(3,7);(6,0.1)},
{\ar@{-}(3.2,7);(6.2,0.1)},
{\ar@{-}(3.4,7);(6.4,0.1)}
\endxy\;\!{}^{{{m}}}_{{{i}}}$}}$ is generated by all faces ${d}_{{j}}{\Delta^{{{m}}}}$, ${{j}}\neq{{i}}$. Finally, we denote the vertices of ${\Delta^{{{m}}}}$ by $0,\ldots,{{m}}$.

Sergeraert et al. (see [@SergerGenova]) have developed an “effective version” of homological algebra, in which a central notion is an object (simplicial set or chain complex) with effective homology. Here we will discuss analogous notions in the equivariant setting, as well as some other extensions. For a key result, we rely on a separate paper [@Vokrinek] which shows, roughly speaking, that if the considered action is free, equivariant effective homology can be obtained from non-equivariant one.

We begin with a description of the basic computational objects, sometimes called *locally effective* objects. The underlying idea is that in every definition one replaces sets by computable sets and mappings by computable mappings. A *computable set* is a set whose elements have a finite encoding by bit strings, so that they can be represented in a computer, and 