A concept in category theory. Other names are triple, monad and functor-algebra.
Let be a category. A standard construction is a functor equipped with natural transformations and such that the following diagrams commute:
The basic use of standard constructions in topology is in the construction of various classifying spaces and their algebraic analogues, the so-called bar-constructions.
|||J.M. Boardman, R.M. Vogt, "Homotopy invariant algebraic structures on topological spaces" , Springer (1973)|
|||J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)|
|||J.P. May, "The geometry of iterated loop spaces" , Lect. notes in math. , 271 , Springer (1972)|
|||S. MacLane, "Categories for the working mathematician" , Springer (1971)|
The term "standard construction" was introduced by R. Godement [a1] for want of a better name for this concept. It is now entirely obsolete, having been generally superseded by "monad" (although a minority of authors still use the term "triple" ). Monads have many other uses besides the one mentioned above, for example in the categorical approach to universal algebra (see [a2], [a3]).
|[a1]||R. Godement, "Théorie des faisceaux" , Hermann (1958)|
|[a2]||E.G. Manes, "Algebraic theories" , Springer (1976)|
|[a3]||M. Barr, C. Wells, "Toposes, triples and theories" , Springer (1985)|
Standard construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_construction&oldid=49442