Combinatorics And Algebra Of Tensor Calculus

In this thesis,motivated by the theory of operads and PROPs we reveal the combi-natorial nature of tensor calculus for strict tensor categories and show that there exists a monad which is described by the coarse-graining of graphs and characterizes the al-gebraic nature of tensor calculus.More concretely,what we have done are listed in the following:1.We give a combinatorial formulation of a progressive plane graph introduced by Joyal and Street and some of their properties are investigated.2.We introduce the category T.Sch of tensor schemes to make the construct of a free strict tensor category a functor F:T.Sch ? Str.T from the category T.Sch of tensor schemes to the category Str.T of strict tensor categories.We also construct a right adjunction U:Str.T ? T.sch of F.3.We analysis the associated monad of the adjunction which is named as the monad of tensor calculus and show that it is described by the coarse-graining of graphs.A algebra of this monad is named as a tensor manifold.4.Identity morphisms in a tensor manifold and several natural algebraic operations on a tensor manifold such as tensor product,composition and fusion are introduced.We show that they satisfy some natural compatible conditions.We also show that under these compatible conditions the appointment of identity morphisms and these operations can totally characterize the algebraic structure of a tensor manifold.5.We construct a functor ?:T.SchT?Str.T from the category T.SchT of tensor manifolds to the category of strict tensor categories,and we show that ? is a left inverse of the natural comparison functor ?:Str.T ? T.SchT which means that ? is an embedding.6.We show that the adjunction F(?)U is not monadic,hence we can interpret a strict tensor category as a special kind of tensor manifold.7.We prove that ? is also a left adjunction of the comparison functor?.