Inverse equivalence of categories What is an "inverse equivalence" between two categories and how is it different from the regular equivalence notion?
My book says that two functors are an "inverse equivalence"... I thought this means they are one the inverse of the other, but isn't that a regular equivalence?
Note that the answer might be "it's the same thing, just stated in a weird way". Being kinda new to category theory I just wanted to know if such an "Inverse equivalence" notion exists. Thank you.
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Context: affine Hecke algebras, with $H_n$ being the algebra and $P_n$ being the Laurent polynomial subalgebra generated by $X_i$. Also $S_n$ is the symmetric group that acts via permutation of indexes. The sentence is
The functors ($H_n c^\tau_n \otimes_{P_n^{S_n}}$ − ) and $c^\tau_n H_n \otimes_{H_n} −$ are inverse equivalences of categories between the category of $P_n^{S_n}$-modules that are locally nilpotent for $n_n=(x_1,\dots,x_n)^{S_n}$ (notation meaning max.ideal generated by those $x_i$) and the $H_n$-modules that are locally nilpotent for $n_n$.

"Functors F and G are inverse equivalences" just means that not only do F and G have inverses, they are each other's inverses. 
