The (Tetra) Category of Pseudocategories in an Additive 2-category with Kernels

Martins-Ferreira, N.

http://hdl.handle.net/10400.8/15141

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The (Tetra) category of pseudocategories in an additive 2-category with kernels.pdf	We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47-78, 2006), internal to an additive 2-categorywith kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387-410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudomorphism and prove the equivalence of categories: PsCat(A) ̃PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels- extending thus the well known equivalence Cat(Ab)̃Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229-249, 1990).	510.28 KB	Adobe PDF	Download

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Authors

Martins-Ferreira, N.

Abstract(s)

We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47-78, 2006), internal to an additive 2-categorywith kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387-410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudomorphism and prove the equivalence of categories: PsCat(A) ̃PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels- extending thus the well known equivalence Cat(Ab)̃Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229-249, 1990).

Keywords

Internal bicategory Internal pseudo-double category Pseudo-category Pseudo-functor Pseudo-natural transformation Modification 2-Ab-category Additive 2-category Pseudo-morphism

URI

http://hdl.handle.net/10400.8/15141

Citation

Martins-Ferreira, N. The (Tetra) Category of Pseudocategories in an Additive 2-category with Kernels. Appl Categor Struct 18, 309–342 (2010). https://doi.org/10.1007/s10485-008-9158-z.