TY - JOUR T1 - Internal Categories in Crossed Semimodules and Schreier Internal Categories AU - Temel, Sedat PY - 2020 DA - October Y2 - 2020 DO - 10.36753/mathenot.691956 JF - Mathematical Sciences and Applications E-Notes JO - Math. Sci. Appl. E-Notes PB - Murat TOSUN WT - DergiPark SN - 2147-6268 SP - 86 EP - 95 VL - 8 IS - 2 LA - en AB - 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. . KW - Crossed module KW - Crossed semimodule KW - Schreier internal category KW - Double category. CR - [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). CR - [2] Baez, J.C., Lauda, A.D.: Higher Dimensional Algebra V: 2-Groups. Theory Appl. Categ. 12, 423–491 (2004). CR - [3] Brown, R.: Topology and Groupoids. BookSurge LLC, North Carolina (2006). CR - [4] Brown, R, Spencer, C.B.: Double groupoids and crossed modules. Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (4), 343-362 (1976). CR - [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). CR - [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). CR - [7] Brown, R., Mucuk, O.: Covering groups of non-connected topological groups revisited. Math. Proc. Camb. Phil. Soc. 115, 97–110 (1994). CR - [8] Ehresmann, C.: Catégories doubles et catégories structurées. C. R. Acad. Sci. Paris 256, 1198-1201 (1963). CR - [9] Ehresmann, C.: Catégories structurées. Ann. Sci. Ec. Norm. Super. 80, 349-425 (1963b). CR - [10] Huebschmann, J.: Crossed n-fold extensions of groups and cohomology. Comment. Math. Helvetici. 55: 302-314 (1980). CR - [11] Kerler, T. and Lyubashenko, V.V.: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Springer-Verlag. Berlin, Heidelberg, (2001). CR - [12] Loday, J.-L.: Cohomologie et groupes de Steinberg relatifs. J. Algebra. 54 178-202 (1978). CR - [13] Maclane, S.: Categories for the Working Mathematician, Graduate Text in Mathematics. 5, Springer-Verlag. New York (1971). CR - [14] Mucuk, O., Demir S.: Normality and quotient in crossed modules over groupoids and double groupoids. Turk J Math, 42, 2336 – 2347 (2018). CR - [15] Patchkoria, A.: Crossed Semimodules and Schreier Internal Categories In The Category of Monoids. Georgian Math. J. 5(6), 575-581 (1998). CR - [16] Porter, T.: Crossed Modules in Cat and a Brown-Spencer Theorem for 2-Categories. Cah. Topol. Géom. Différ. Catég. XXVI-4 (1985). CR - [17] ¸Sahan, T., Mohammed, J.J.: Categories internal to crossed modules Sakarya University Journal of Science. 23 (4), 519-531, (2019). CR - [18] Temel, S., ¸Sahan, T. and Mucuk, O.: Crossed modules, double group-groupoids and crossed squares. Preprint arxiv:1802.03978v2 (2018). CR - [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). CR - [20] Temel, S.: Crossed semimodules of categories and Schreier 2-categories. Tbilisi Math. J. 11 (2), 47-57 (2018). CR - [21] Temel, S.: Normality and quotient in crossed modules over groupoids and 2-groupoids. Korean J. Math. 27 (1), 151-163 (2018). CR - [22] Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, 453-496 (1949). CR - [23] Whitehead, J.H.C.: Note on a previous paper entitled "On adding relations to homotopy group". Ann. Math. 47,806-810 (1946). UR - https://doi.org/10.36753/mathenot.691956 L1 - https://dergipark.org.tr/en/download/article-file/976686 ER -