Standard construction

A concept in category theory. Other names are triple, monad and functor-algebra.

Let $\mathfrak{S}$ be a category. A standard construction is a functor $T : \mathfrak{S} \to \mathfrak{S}$ equipped with natural transformations $\eta : \operatorname{Id} \to T$ and $\mu : T^2 \to T$ 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.

Comments

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 It is now entirely obsolete, having been generally superseded by "monad" (although a minority of authors still use the term "triple" ).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]).

References

[1]	J.M. Boardman, R.M. Vogt, "Homotopy invariant algebraic structures on topological spaces" , Springer (1973)
[2]	J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
[3]	J.P. May, "The geometry of iterated loop spaces" , Lect. notes in math. , 271 , Springer (1972)
[4]	S. MacLane, "Categories for the working mathematician" , Springer (1971)
[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)