Year 2020, Volume 8 , Issue 2, Pages 86 - 95 2020-10-15

Internal Categories in Crossed Semimodules and Schreier Internal Categories

Sedat TEMEL [1]


In this paper, we characterize internal categories in the category of crossed semimodules and the category of Schreier internal categories within monoids. Then we prove a natural equivalence between their categories. This allows us to produce various examples of double categories.                                                                                                                                                                                                                                                                                                          .
Crossed module, Crossed semimodule, Schreier internal category, Double category.
  • [1] Baez, J.C., Baratin, A., Freidel, L. and Wise, D.K.: Infinite-Dimensional Representations of 2-Groups. Mem. Am. Math. Soc. 219, (1032) (2012).
  • [2] Baez, J.C., Lauda, A.D.: Higher Dimensional Algebra V: 2-Groups. Theory Appl. Categ. 12, 423–491 (2004).
  • [3] Brown, R.: Topology and Groupoids. BookSurge LLC, North Carolina (2006).
  • [4] Brown, R, Spencer, C.B.: Double groupoids and crossed modules. Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (4), 343-362 (1976).
  • [5] Brown, R., Spencer, C.B.: G-groupoids, crossed modules and the fundamental groupoid of a topological group. Mathematical Sciences and Applications E-Notes. Indagat. Math. 79 (4), 296-302 (1976).
  • [6] Brown, R., Higgins, P. J. and Sivera, R.: Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids. European Mathematical Society Tracts in Mathematics 15 (2011).
  • [7] Brown, R., Mucuk, O.: Covering groups of non-connected topological groups revisited. Math. Proc. Camb. Phil. Soc. 115, 97–110 (1994).
  • [8] Ehresmann, C.: Catégories doubles et catégories structurées. C. R. Acad. Sci. Paris 256, 1198-1201 (1963).
  • [9] Ehresmann, C.: Catégories structurées. Ann. Sci. Ec. Norm. Super. 80, 349-425 (1963b).
  • [10] Huebschmann, J.: Crossed n-fold extensions of groups and cohomology. Comment. Math. Helvetici. 55: 302-314 (1980).
  • [11] Kerler, T. and Lyubashenko, V.V.: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Springer-Verlag. Berlin, Heidelberg, (2001).
  • [12] Loday, J.-L.: Cohomologie et groupes de Steinberg relatifs. J. Algebra. 54 178-202 (1978).
  • [13] Maclane, S.: Categories for the Working Mathematician, Graduate Text in Mathematics. 5, Springer-Verlag. New York (1971).
  • [14] Mucuk, O., Demir S.: Normality and quotient in crossed modules over groupoids and double groupoids. Turk J Math, 42, 2336 – 2347 (2018).
  • [15] Patchkoria, A.: Crossed Semimodules and Schreier Internal Categories In The Category of Monoids. Georgian Math. J. 5(6), 575-581 (1998).
  • [16] Porter, T.: Crossed Modules in Cat and a Brown-Spencer Theorem for 2-Categories. Cah. Topol. Géom. Différ. Catég. XXVI-4 (1985).
  • [17] ¸Sahan, T., Mohammed, J.J.: Categories internal to crossed modules Sakarya University Journal of Science. 23 (4), 519-531, (2019).
  • [18] Temel, S., ¸Sahan, T. and Mucuk, O.: Crossed modules, double group-groupoids and crossed squares. Preprint arxiv:1802.03978v2 (2018).
  • [19] Temel, S.: Topological Crossed Semimodules and Schreier Internal Categories in the Category of Topological Monoids. Gazi University Journal of Science. 29 (4), 915-921 (2016).
  • [20] Temel, S.: Crossed semimodules of categories and Schreier 2-categories. Tbilisi Math. J. 11 (2), 47-57 (2018).
  • [21] Temel, S.: Normality and quotient in crossed modules over groupoids and 2-groupoids. Korean J. Math. 27 (1), 151-163 (2018).
  • [22] Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, 453-496 (1949).
  • [23] Whitehead, J.H.C.: Note on a previous paper entitled "On adding relations to homotopy group". Ann. Math. 47,806-810 (1946).
Primary Language en
Subjects Mathematics
Journal Section Articles
Authors

Orcid: 0000-0001-6553-8758
Author: Sedat TEMEL (Primary Author)
Institution: Recep Tayyip Erdogan University
Country: Turkey


Dates

Publication Date : October 15, 2020

Bibtex @research article { mathenot691956, journal = {Mathematical Sciences and Applications E-Notes}, issn = {}, eissn = {2147-6268}, address = {}, publisher = {Murat TOSUN}, year = {2020}, volume = {8}, pages = {86 - 95}, doi = {10.36753/mathenot.691956}, title = {Internal Categories in Crossed Semimodules and Schreier Internal Categories}, key = {cite}, author = {Temel, Sedat} }
APA Temel, S . (2020). Internal Categories in Crossed Semimodules and Schreier Internal Categories . Mathematical Sciences and Applications E-Notes , 8 (2) , 86-95 . DOI: 10.36753/mathenot.691956
MLA Temel, S . "Internal Categories in Crossed Semimodules and Schreier Internal Categories" . Mathematical Sciences and Applications E-Notes 8 (2020 ): 86-95 <https://dergipark.org.tr/en/pub/mathenot/issue/57179/691956>
Chicago Temel, S . "Internal Categories in Crossed Semimodules and Schreier Internal Categories". Mathematical Sciences and Applications E-Notes 8 (2020 ): 86-95
RIS TY - JOUR T1 - Internal Categories in Crossed Semimodules and Schreier Internal Categories AU - Sedat Temel Y1 - 2020 PY - 2020 N1 - doi: 10.36753/mathenot.691956 DO - 10.36753/mathenot.691956 T2 - Mathematical Sciences and Applications E-Notes JF - Journal JO - JOR SP - 86 EP - 95 VL - 8 IS - 2 SN - -2147-6268 M3 - doi: 10.36753/mathenot.691956 UR - https://doi.org/10.36753/mathenot.691956 Y2 - 2020 ER -
EndNote %0 Mathematical Sciences and Applications E-Notes Internal Categories in Crossed Semimodules and Schreier Internal Categories %A Sedat Temel %T Internal Categories in Crossed Semimodules and Schreier Internal Categories %D 2020 %J Mathematical Sciences and Applications E-Notes %P -2147-6268 %V 8 %N 2 %R doi: 10.36753/mathenot.691956 %U 10.36753/mathenot.691956
ISNAD Temel, Sedat . "Internal Categories in Crossed Semimodules and Schreier Internal Categories". Mathematical Sciences and Applications E-Notes 8 / 2 (October 2020): 86-95 . https://doi.org/10.36753/mathenot.691956
AMA Temel S . Internal Categories in Crossed Semimodules and Schreier Internal Categories. Math. Sci. Appl. E-Notes. 2020; 8(2): 86-95.
Vancouver Temel S . Internal Categories in Crossed Semimodules and Schreier Internal Categories. Mathematical Sciences and Applications E-Notes. 2020; 8(2): 86-95.
IEEE S. Temel , "Internal Categories in Crossed Semimodules and Schreier Internal Categories", Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, pp. 86-95, Oct. 2020, doi:10.36753/mathenot.691956