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By Pillay A.

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Compatibility between units and commutation: unr ◦ com = unl . A symmetric monoidal category is a braided monoidal category that further fulfils v. the idempotency of the commutation isomorphism: comA,B ◦ comB,A = idA . Proof. 2. We now introduce an important finiteness condition on an object of a monoidal category: that of having a monoidal dual. 5. Let (C, ⊗) be a monoidal category. A dual pair in (C, ⊗) is given by a pair (A, A∨ ) of objects of C and two morphisms i : ✶ → A∨ ⊗ A and e : A ⊗ A∨ → ✶ fulfilling the triangle equalities meaning that the composite maps A A∨ A⊗i G i⊗A∨G A∨ ⊗ A ⊗ A∨ A ⊗ A∨ ⊗ A GA e⊗A A∨ ⊗e G A∨ are the identities.

This approach is equivalent to the other ones (in particular, to Maclane and Segal’s approaches), but looks much more natural. , finitely presented or finite limit theories). • The idea of parametrized and functional geometry (see Chapter 2), was already present in the literature, for example in the work of Grothendieck’s and Souriau’s schools, and in the synthetic geometry community, grounded by Lawvere’s categorical approach to dynamics. However, it’s systematic use for the formalization of the known physical approaches to quantum field theory is new.

Any category with finite products is a monoidal category with monoidal structure given by the product. 2. The category Mod(K) of modules over a commutative unital ring K with its ordinary tensor product is a closed monoidal category. If one works with the standard construction of the tensor product as a quotient M ⊗K N := K (M ×N ) / ∼bil , of the free module on the product by the bilinearity relations, the associativity isomorphisms are not equalities. Instead, they are given by the corresponding canonical isomorphisms, uniquely determined by the universal property of tensor products.

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Differential Galois theory 1 by Pillay A.

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