Max Kelly
Encyclopedia
Gregory Maxwell Kelly, 1930-2007, mathematician, founded the thriving Australian school of category theory
.
A native of Australia
, Kelly obtained his Ph.D. at Cambridge University in homological algebra
in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory. He taught in the Pure Mathematics department at Sydney University from 1957 to 1966, rising from lecturer to reader. During 1963-1965 he was a visiting fellow at Tulane University and the University of Illinois
, 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.
He subsequently developed the notion in considerably more detail in his 1981 monograph Basic Concepts of Enriched Category Theory. Let V be a monoidal category
, and denote by V-Cat the category of V-enriched categories. Among other things, Kelly showed that V-Cat has all weighted limits and colimits even when V does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extension
s, density of the Yoneda embedding, and essentially algebraic theories. The explicitly foundational role of the category Set
in his treatment is noteworthy in view of the folk intuition that enriched categories liberate category theory from the last vestiges of Set as the codomain of the ordinary external hom-functor.
In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales
. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1994. In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate in the department as Professorial Fellow and Professor Emeritus until his death at age 76 on 26 January 2007.
Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of fruitful collaborations. His Ph.D. student Ross Street is himself a noted category theorist and early contributor to the Australian category theory school.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.
A native of Australia
Australia
Australia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...
, Kelly obtained his Ph.D. at Cambridge University in homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory. He taught in the Pure Mathematics department at Sydney University from 1957 to 1966, rising from lecturer to reader. During 1963-1965 he was a visiting fellow at Tulane University and the University of Illinois
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...
, where with Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
he formalized and developed the notion of an enriched category
Enriched category
In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely...
based on intuitions then in the air about making the homsets
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of a category just as abstract as the objects themselves.
He subsequently developed the notion in considerably more detail in his 1981 monograph Basic Concepts of Enriched Category Theory. Let V be a monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
, and denote by V-Cat the category of V-enriched categories. Among other things, Kelly showed that V-Cat has all weighted limits and colimits even when V does not have all ordinary limits and colimits. He also developed the enriched counterparts of Kan extension
Kan extension
Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M...
s, density of the Yoneda embedding, and essentially algebraic theories. The explicitly foundational role of the category Set
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
in his treatment is noteworthy in view of the folk intuition that enriched categories liberate category theory from the last vestiges of Set as the codomain of the ordinary external hom-functor.
In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales
University of New South Wales
The University of New South Wales , is a research-focused university based in Kensington, a suburb in Sydney, New South Wales, Australia...
. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1994. In 2001 he was awarded the Australian government's Centenary Medal. He continued to participate in the department as Professorial Fellow and Professor Emeritus until his death at age 76 on 26 January 2007.
Kelly worked on many other aspects of category theory besides enriched categories, both individually and in a number of fruitful collaborations. His Ph.D. student Ross Street is himself a noted category theorist and early contributor to the Australian category theory school.
External links
- Max Kelly's Perpetual Web Page: a memorial page set up by Kelly's son Simon Kelly.
- Obituary published in The Sydney Morning Herald
- Reprint of "Basic Concepts of Enriched Categories," 1982