In mathematics, algebraic homotopy is a research program on homotopy theory proposed by J.H.C. Whitehead in his 1950 ICM talk, where he described it as:Baues, Homotopy Types in Chapter 1 in Handbook of Algebraic TopologyNB: we need a citation to the original source.
In spirit, the program is somehow similar to Grothendieck's homotopy hypothesis. However, according to Ronnie Brown, "Looking again at Esquisses d'un Progamme, it seems that programme has currently little relation to Whitehead's."https://mathoverflow.net/questions/266738/current-status-of-grothendiecks-homotopy-hypothesis-and-whiteheads-algebraic-h
References
https://ncatlab.org/nlab/show/algebraic+homotopy Handbook of Algebraic Topology edited by I.M. James
Further reading https://ncatlab.org/nlab/show/Algebraic+Homotopy, an entry about a book