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matrix theory

Let AA be a Lawvere theory with generic object TT. For nn \in \mathbb{N}, the full subcategory of AA generated by the cartesian powers of T nT^n is also a Lawvere theory, which we denote by M n(A)M_n(A). In the case of an annular theory (the theory of modules over a ring that we also call AA), this is the construction of n×nn\times n matrices over AA. If we denote by M nM_n the application of this construction to the initial theory (the theory of sets), then we may identify M n(A)M_n(A) with the tensor product theory M nAM_n\otimes A.

It is an amusing exercise to present M nM_n in terms of generating operations and relations between them.

Revised on March 16, 2015 22:58:47 by Maciej Pirog? (129.67.119.218)