Homotopy for Functors

We show that natural transformations play the role of homotopy for (covariant) functors. Homotopic functors are shown to induce identical maps between the homology groups of categories. For a space X, there is an associated category <!-- MATH $\Lambda S(X)$ --> . We show that the classifying space of <!-- MATH $\Lambda S(X)$ --> has the same homotopy type as X if X is a CW complex. Moreover, we prove that, for CW complexes X and Y, f and are homotopic if and only if <!-- MATH $\Lambda S(f)$ --> and <!-- MATH $\Lambda S(g)$ --> are.