Internal object actions

Francis Borceux; George Z. Janelidze; Gregory Maxwell Kelly

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 2, page 235-255
  • ISSN: 0010-2628

Abstract

top
We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module.

How to cite

top

Borceux, Francis, Janelidze, George Z., and Kelly, Gregory Maxwell. "Internal object actions." Commentationes Mathematicae Universitatis Carolinae 46.2 (2005): 235-255. <http://eudml.org/doc/249553>.

@article{Borceux2005,
abstract = {We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module.},
author = {Borceux, Francis, Janelidze, George Z., Kelly, Gregory Maxwell},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {monoidal category; monoidal functor; monoid; action; action of an object; semi-abelian category; semidirect product; groups; Lie algebras; crossed modules; actors; monoidal category; monoidal functor; monoid; action; semi-abelian category; semidirect product},
language = {eng},
number = {2},
pages = {235-255},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Internal object actions},
url = {http://eudml.org/doc/249553},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Borceux, Francis
AU - Janelidze, George Z.
AU - Kelly, Gregory Maxwell
TI - Internal object actions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 235
EP - 255
AB - We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module.
LA - eng
KW - monoidal category; monoidal functor; monoid; action; action of an object; semi-abelian category; semidirect product; groups; Lie algebras; crossed modules; actors; monoidal category; monoidal functor; monoid; action; semi-abelian category; semidirect product
UR - http://eudml.org/doc/249553
ER -

References

top
  1. Borceux F., Bourn D., Mal'cev, Protomodular, Homological and Semi-abelian Categories, Kluwer, Dordrecht, 2004. 
  2. Bourn D., Normalization equivalence, kernel equivalence, and affine categories, Lecture Notes in Mathematics 1488, Springer, Berlin, 1991, pp.43-62. Zbl0756.18007MR1173004
  3. Bourn D., Janelidze G., Protomodularity, descent, and semidirect products, Theory Appl. Categ. 4 (1998), 37-46. (1998) Zbl0890.18003MR1615341
  4. Casas J.M., Ladra M., The actor of a crossed module in Lie algebras, Comm. Algebra 26 7 (1998), 2065-2089. (1998) Zbl0922.17013MR1626629
  5. Janelidze G., Kelly G.M., A note on actions of a monoidal category, Theory Appl. Categ. 9 (2001), 61-91. (2001) Zbl1009.18005MR1897810
  6. Janelidze G., Márki L., Tholen W., Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), 367-386. (2002) Zbl0993.18008MR1887164
  7. Kelly G.M., On Mac Lane's conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964), 397-402. (1964) MR0182649
  8. Kelly G.M., Basic Concepts of Enriched Category Theory, London Math. Soc. Lecture Notes Ser. 64, Cambridge University Press, Cambridge-New York, 1982. Zbl1086.18001MR0651714
  9. Kelly G.M., Street R.H., Review of the elements of 2 -categories, Sydney Category Seminar, Lecture Notes in Mathematics 420, Springer, Berlin, 1974, pp.75-103. Zbl0334.18016MR0357542
  10. Lue A.S.-T., Semi-complete crossed modules and holomorphs of groups, Bull. London Math. Soc. 11 (1979), 8-16. (1979) Zbl0416.20030MR0535788
  11. Mac Lane S., Duality for groups, Bull. Amer. Math. Soc. 56 (1950), 485-516. (1950) MR0049192
  12. Mac Lane S., Categories for the Working Mathematician, Springer, New York-Heidelberg-Berlin, 1971; Second Edition: 1998. Zbl0906.18001MR0354798
  13. Norrie K., Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), 129-146. (1990) Zbl0719.20018MR1087375
  14. Whitehead J.H.C., On operators in relative homotopy groups, Ann. of Math. 49 (1948), 610-640. (1948) Zbl0041.10102MR0028026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.