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- W2964238351 abstract "It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call “Frobenius monoidal”, is sufficient. Further properties of Frobenius monoidal functors are developed. The idea of this note became apparent from Proposition 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. Walters [5]. Throughout suppose that A and B are strict1 monoidal categories. Definition 1. A Frobenius monoidal functor is a functor F : A B which is monoidal (F, r, r0) and comonoidal (F, i, i0), and satisfies the compatibility conditions ir = (1⊗ r)(i⊗ 1) : F (A⊗B)⊗ FC FA⊗ F (B ⊗ C) ir = (r ⊗ 1)(1 ⊗ i) : FA⊗ F (B ⊗ C) F (A⊗B)⊗ FC, for all A,B,C ∈ A . The compact case (⊗ = ⊕) of Cockett and Seely’s linearly distributive functors [2] are precisely Frobenius monoidal functors, and Frobenius monoidal functors with ri = 1 have been called split monoidal by Szlachanyi in [6]. A dual situation in A is a tuple (A,B, e, n), where A and B are objects of A and e : A⊗B I n : I B ⊗A Received May 16, 2008. Mathematics Subject Classification. 18A22, 18D10, 18D25." @default.
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- W2964238351 date "2008-12-01" @default.
- W2964238351 modified "2023-09-26" @default.
- W2964238351 title "Note on Frobenius monoidal functors" @default.
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