| Title:
|
Internal object actions (English) |
| Author:
|
Borceux, F. |
| Author:
|
Janelidze, G. |
| Author:
|
Kelly, G. M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2005 |
| Pages:
|
235-255 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
monoidal category |
| Keyword:
|
monoidal functor |
| Keyword:
|
monoid |
| Keyword:
|
action |
| Keyword:
|
action of an object |
| Keyword:
|
semi-abelian category |
| Keyword:
|
semidirect product |
| Keyword:
|
groups |
| Keyword:
|
Lie algebras |
| Keyword:
|
crossed modules |
| Keyword:
|
actors |
| MSC:
|
18C15 |
| MSC:
|
18C20 |
| MSC:
|
18D10 |
| MSC:
|
18D15 |
| MSC:
|
18G50 |
| idZBL:
|
Zbl 1121.18004 |
| idMR:
|
MR2176890 |
| . |
| Date available:
|
2009-05-05T16:50:50Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119522 |
| . |
| Reference:
|
[BB] Borceux F., Bourn D.: Mal'cev, Protomodular, Homological and Semi-abelian Categories.Kluwer, Dordrecht, 2004. |
| Reference:
|
[B] Bourn D.: Normalization equivalence, kernel equivalence, and affine categories.Lecture Notes in Mathematics 1488, Springer, Berlin, 1991, pp.43-62. Zbl 0756.18007, MR 1173004 |
| Reference:
|
[BJ] Bourn D., Janelidze G.: Protomodularity, descent, and semidirect products.Theory Appl. Categ. 4 (1998), 37-46. Zbl 0890.18003, MR 1615341 |
| Reference:
|
[CL] Casas J.M., Ladra M.: The actor of a crossed module in Lie algebras.Comm. Algebra 26 7 (1998), 2065-2089. Zbl 0922.17013, MR 1626629 |
| Reference:
|
[JK] Janelidze G., Kelly G.M.: A note on actions of a monoidal category.Theory Appl. Categ. 9 (2001), 61-91. Zbl 1009.18005, MR 1897810 |
| Reference:
|
[JMT] Janelidze G., Márki L., Tholen W.: Semi-abelian categories.J. Pure Appl. Algebra 168 (2002), 367-386. Zbl 0993.18008, MR 1887164 |
| Reference:
|
[K1] Kelly G.M.: On Mac Lane's conditions for coherence of natural associativities, commutativities, etc..J. Algebra 1 (1964), 397-402. MR 0182649 |
| Reference:
|
[K2] Kelly G.M.: Basic Concepts of Enriched Category Theory.London Math. Soc. Lecture Notes Ser. 64, Cambridge University Press, Cambridge-New York, 1982. Zbl 1086.18001, MR 0651714 |
| Reference:
|
[KS] 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. Zbl 0334.18016, MR 0357542 |
| Reference:
|
[L] Lue A.S.-T.: Semi-complete crossed modules and holomorphs of groups.Bull. London Math. Soc. 11 (1979), 8-16. Zbl 0416.20030, MR 0535788 |
| Reference:
|
[M1] Mac Lane S.: Duality for groups.Bull. Amer. Math. Soc. 56 (1950), 485-516. MR 0049192 |
| Reference:
|
[M2] Mac Lane S.: Categories for the Working Mathematician.Springer, New York-Heidelberg-Berlin, 1971; Second Edition: 1998. Zbl 0906.18001, MR 0354798 |
| Reference:
|
[N] Norrie K.: Actions and automorphisms of crossed modules.Bull. Soc. Math. France 118 (1990), 129-146. Zbl 0719.20018, MR 1087375 |
| Reference:
|
[W] Whitehead J.H.C.: On operators in relative homotopy groups.Ann. of Math. 49 (1948), 610-640. Zbl 0041.10102, MR 0028026 |
| . |
