DEFORMATION OF MONOIDAL CATEGORY AND YETTER MULTI PRE-COMPLEX
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Abstract
We introduce the topic of algebraic deformation theory developed by Murray Gerstenhaber in 1960’s to the Vietnamese audiences in this paper. Currently, this topic is studied very extensively in the field of algebraic geometry. On the other hand, we applied this theory to study the first order deformations of linear monoidal categories and found a new result in completing components in low degrees (degree 1, 2 and 3) of the differential map in the Yetter multi pre-complex. Shrestha (2010) introduced a formula for components in low degrees (degree 1, 2 and 3) of the differential map in the Yetter multi pre-complex. His formula was not fully completed. In this paper, we offer a completed formula for these components of low degrees with nice explanations.
Keywords
Homological algebra, algebraic deformation
Article Details
References
Dinh, V. H., & Lowen, W. (2018). On the gerstenhaber-schack complex for prestacks. Advances in Mathematics, 330, 173-228.
Gerstenhaber, M. (1964). On the deformation of rings and algebras. Ann. of Math. (2), 79, 59-103.
Gerstenhaber, M., & Schack, S. D. (1983). On the deformation of algebra morphisms and diagrams. Trans. Amer. Math. Soc., 279, 1-50.
Gerstenhaber, M., & Schack, S. D. (1988). The cohomology of presheaves of algebras.
I. Presheaves over a partially ordered set. Trans. Amer. Math. Soc., 310, 135-165.
Kontsevich, M. (2003). Deformation quantazation of poisson manifolds. Letter of Mathematical Physics, 66, 157-216.
Kodaira, K., & Spencer, D. (1958) On deformations of complex analytic structures I & II. Ann. of Math., 67, 328-466.
Lowen, W. (2008). Algebroid prestacks and deformations of ringed spaces. Trans. Amer. Math. Soc., 360, 1631-1660.
Lowen, W., & Van den Bergh, M. (2006). Deformation theory of abelian categories. Trans. Amer. Math. Soc., 358, 5441-5483.
Nijenhuis, A., & Richardson, R. (1967). Deformations of homomorphisms of lie groups and lie al¬gebras. Bull. Amer. Math. Soc., 73, 175-179.
Shoikhet, B. (2010). Koszul duality in deformation quantization and tamarkin's approach to Kontsevich formality. Advances in Mathematics, 224, 731-771.
Shrestha, T. (2010). Algebraic deformation of a monoidal category. PhD Thesis, Kansas University. Retrieved from http://krex.k-state.edu/dspace/handle/2097/6393
Shrestha, T., & Yetter, D. (2014). On deformations of pasting diagram (2). Theory Appl. Categ., 29, 569-608.
Yetter, D. (2009). On deformations of pasting diagrams. Theory Appl. Categ., 22, 24-53.
Gerstenhaber, M. (1964). On the deformation of rings and algebras. Ann. of Math. (2), 79, 59-103.
Gerstenhaber, M., & Schack, S. D. (1983). On the deformation of algebra morphisms and diagrams. Trans. Amer. Math. Soc., 279, 1-50.
Gerstenhaber, M., & Schack, S. D. (1988). The cohomology of presheaves of algebras.
I. Presheaves over a partially ordered set. Trans. Amer. Math. Soc., 310, 135-165.
Kontsevich, M. (2003). Deformation quantazation of poisson manifolds. Letter of Mathematical Physics, 66, 157-216.
Kodaira, K., & Spencer, D. (1958) On deformations of complex analytic structures I & II. Ann. of Math., 67, 328-466.
Lowen, W. (2008). Algebroid prestacks and deformations of ringed spaces. Trans. Amer. Math. Soc., 360, 1631-1660.
Lowen, W., & Van den Bergh, M. (2006). Deformation theory of abelian categories. Trans. Amer. Math. Soc., 358, 5441-5483.
Nijenhuis, A., & Richardson, R. (1967). Deformations of homomorphisms of lie groups and lie al¬gebras. Bull. Amer. Math. Soc., 73, 175-179.
Shoikhet, B. (2010). Koszul duality in deformation quantization and tamarkin's approach to Kontsevich formality. Advances in Mathematics, 224, 731-771.
Shrestha, T. (2010). Algebraic deformation of a monoidal category. PhD Thesis, Kansas University. Retrieved from http://krex.k-state.edu/dspace/handle/2097/6393
Shrestha, T., & Yetter, D. (2014). On deformations of pasting diagram (2). Theory Appl. Categ., 29, 569-608.
Yetter, D. (2009). On deformations of pasting diagrams. Theory Appl. Categ., 22, 24-53.