Gregory Maxwell "Max" Kelly (5 June 1930 – 26 January 2007) was an Australian mathematician who worked on category theory.

Kelly was born in Bondi, New South Wales, Australia, on 5 June 1930.^[1] He obtained his PhD at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory.^[1][2] He taught in the Pure Mathematics department at the University of Sydney from 1957 to 1966, rising from lecturer to reader.^[1] During 1963–1965 he was a visiting fellow at Tulane University and the University of Illinois,^[1] where with Samuel Eilenberg he formalized and developed the notion of an enriched category based on intuitions then in the air about making the homsets of a category just as abstract as the objects themselves.^{[citation needed]}

He subsequently developed the notion in considerably more detail in his 1982 monograph Basic Concepts of Enriched Category Theory.^[3] Let ${\displaystyle {\cal {V}}}$ be a monoidal category, and denote by ${\displaystyle {\cal {V}}}$-Cat the category of ${\displaystyle {\cal {V}}}$-enriched categories. Among other things, Kelly showed that ${\displaystyle {\cal {V}}}$-Cat has all weighted limits and colimits even when ${\displaystyle {\cal {V}}}$ does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extensions, density of the Yoneda embedding, and essentially algebraic theories.^{[citation needed]}

In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales.^[1] In 1972 he was elected a Fellow of the Australian Academy of Science.^{[citation needed]} He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1992.^[1] In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate^{[dubious – discuss]} in the department as professorial fellow and professor emeritus until his death at age 76 on 26 January 2007.^[1]

Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of collaborations.^{[citation needed]} His PhD students include Ross Street.^[2]