Daniel Kan
Encyclopedia
Daniel Marinus Kan is a mathematician
working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.
He is an emeritus professor at MIT
, where he has taught since the early 1960s. He received his Ph.D.
at Hebrew University in 1955, under the direction of Samuel Eilenberg
. His students include Aldridge K. Bousfield, William Dwyer, and Jeffrey H. Smith.
He played a role in the beginnings of modern homotopy theory perhaps analogous to that of Saunders Mac Lane
in homological algebra
, namely the adroit and persistent application of categorical
methods. His most famous work is the abstract formulation of the discovery of adjoint functors
, which dates from 1958. The Kan extension
is one of the broadest descriptions of a useful general class of adjunctions.
He also has made contributions to the theory of simplicial set
s and simplicial methods in topology in general: fibrations in the usual closed model category structure on the category of simplicial sets are known as Kan fibration
s, and the fibrant objects are known as Kan complexes.
Some of Kan's more recent work concerns model categories
and other homotopical categories.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.
He is an emeritus professor at MIT
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
, where he has taught since the early 1960s. He received his Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
at Hebrew University in 1955, under the direction of 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...
. His students include Aldridge K. Bousfield, William Dwyer, and Jeffrey H. Smith.
He played a role in the beginnings of modern homotopy theory perhaps analogous to that of Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
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...
, namely the adroit and persistent application of categorical
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...
methods. His most famous work is the abstract formulation of the discovery of adjoint functors
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
, which dates from 1958. The 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...
is one of the broadest descriptions of a useful general class of adjunctions.
He also has made contributions to the theory of simplicial set
Simplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
s and simplicial methods in topology in general: fibrations in the usual closed model category structure on the category of simplicial sets are known as Kan fibration
Kan fibration
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category...
s, and the fibrant objects are known as Kan complexes.
Some of Kan's more recent work concerns model categories
Model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
and other homotopical categories.