View source for Standard construction

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871401.png" /> be a [[Category|category]]. A standard construction is a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871402.png" /> equipped with natural transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871404.png" /> such that the following diagrams commute:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087140/s0871405.png" /></td> </tr></table>

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.

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



====Comments====
The term  "standard construction"  was introduced by R. Godement [[#References|[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 It is now entirely obsolete, having been generally superseded by  "monad"  (although a minority of authors still use the term  "triple" ).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|universal algebra]] (see [[#References|[a2]]], [[#References|[a3]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Godement,   "Théorie des faisceaux" , Hermann  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.G. Manes,   "Algebraic theories" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Barr,   C. Wells,   "Toposes, triples and theories" , Springer  (1985)</TD></TR></table>