This abstract isomorphism with a product is not natural, as some isomorphisms of formula_183 do not preserve the product: the self-homeomorphism of formula_183 (thought of as the quotient space formula_185) given by formula_186 (geometrically a Dehn twist about one of the generating curves) acts as this matrix on formula_187 (it's in the general linear group formula_188 of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product formula_189 – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components".