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A concept in category theory. Other names are [[Triple|triple]], monad and functor-algebra.
 
A concept in category theory. Other names are [[Triple|triple]], monad and functor-algebra.
  
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Let $  \mathfrak S $
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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:
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be a [[Category|category]]. A standard construction is a functor $  T: \mathfrak S \rightarrow \mathfrak S $
 
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equipped with natural transformations $  \eta : 1 \rightarrow T $
 
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and $  \mu : T  ^ {2} \rightarrow T $
 
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such that the following diagrams commute:
 
  
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$$
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<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.
 
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.
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====References====
 
====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>
 
<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>
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====Comments====
 
====Comments====

Revision as of 14:53, 7 June 2020

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.

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)


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

[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)
How to Cite This Entry:
Standard construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_construction&oldid=48800
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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