American mathematician who was a cocreator of category theory (mathematics, foundations of), an architect of homological algebra, and an advocate of categorical foundations for mathematics (mathematics, foundations of).

Mac Lane graduated from Yale University in 1930 and then began graduate work at the University of Chicago. He soon moved to Germany, where he, with a dissertation on mathematical logic (logic), received a doctorate degree in 1933 from the University of Göttingen. While in Germany, he stayed in the homes of Hermann Weyl (Weyl, Hermann) and Richard Courant (Courant, Richard), and he saw his dissertation adviser Paul Bernays (Bernays, Paul Isaak) barred from teaching by the Nazis. Mac Lane returned home and taught at various universities before settling permanently at the University of Chicago in 1947.

About 1940 Mac Lane made some purely algebraic calculations in group theory, and the Polish American mathematician Samuel Eilenberg noticed that they applied to the topology of infinitely coiled curves called solenoids. To understand and generalize this link between algebra and topology, the two men created category theory, the general cohomology of groups, and the basis for the Eilenberg-Steenrod axioms for homology of topological spaces (topological space). Mac Lane worked with categorical duality and defined categorical universal properties. He defined and named Abelian categories, further developed by Alexandre Grothendieck (Grothendieck, Alexandre) to become central to homological algebra.

From the 1960s Mac Lane pursued aspects of category theory, including the work of the American mathematician F. William Lawvere on categorical foundations for mathematics. Mac Lane served as president of the Mathematical Association of America (1951–52), the American Philosophical Society (1968–71), and the American Mathematical Society (1973–74). He served as vice president of the National Academy of Sciences (1973–81). His works include A Survey of Modern Algebra (1941; with Garrett Birkhoff), Homology (1963), Categories for the Working Mathematician (1971), and Sheaves in Geometry and Logic: A First Introduction to Topos Theory (1992; with Ieke Moerdijk).