Note On Analytic Functors As Fourier Transforms

Note On Analyt ic F unctors As F o urier T ransfor ms. Brian J. D A Y April 19, 20 09. Ab str act: Sever al notions of “analytic” functor intr o duc e d r e c ently in the li ter atur e fit into the gr aphic fourier tr ansform c ontext pr esente d in [D]. V ar ious concepts of “ analytic” functor a re well c ha racterize d in differen t places in the literature. Here we want to men tion explicitly ho w t w o ideas, in- tro duced in [A V] and [F GHW], ca n also be viewed from [D]. Howev e r, we don’t offer further characterization results here . Example [FGHW]: Let N : B A → [ A op , Set ] b e the canonical “inclusion” f unctor from the (monoidal) group oid B A constructed in [FGHW] fro m the small Set -ca tegory A ; i.e. B A is the g roup oid of all iso mo rphisms in the free finite-copro duct completion of A in [ A op , Set ]. Then, assuming (here) that C is small, the functor ∃ N : [ BA , [ C op , Set ]] → [[ A op , Set ] , [ C op , Set ]] which is precisely the proce s s of left Kan extension along N, is conse rv ative (be- cause B A is a gro up o id) and tensor pro duct pr eserving (b ecaus e N pr eserves finite copr o ducts). This is then consistent with [D] for V = [ C op , Set ] (car te- sian monoida l). Example [A V]: In [A V] Remark 4.5, the authors complain (justly) that many of their “an- alytic” functors for V = V ect k are not k -linear . But this is not to o serious a matter b ecause the set-up in [D] p ermits a k - linearizatio n (this is no t a tautol- ogy , but merely a n adjunction). Th us the “ordinar y” k ernel discus sed in [A V] § 4, namely K : B × V 0 − → V 0 , K ( n, X ) = ⊗ n X , yields the corresp onding (multiplicativ e) V -kernel E : k ∗ B ⊗ k ∗ V 0 ∼ = / / k ∗ ( B × V 0 ) k ∗ K / / k ∗ V 0 can. / / V where k ∗ denotes the free- V ect k -structure functor, and V 0 is the or dina ry ca t- egory underlying V (here k ∗ V 0 has the comono idal structure dir e ctly induced by that on V 0 ). Then the V -functor E : [ k ∗ B , V ] − → [ k ∗ V 0 , V ] , 1 is conserv a tive (since B is a gr oup oid) and tensor pro duct preser v ing (since E is multiplicative). The F ourier transforms E ( f ) can th us be view ed as either k -linear “ E -analytic” functor s k ∗ V 0 − → V , or just ordinary “[A V]-ana lytic” functors V 0 − → V 0 , in the sense of [A V] Definition 4 .1. Then the co ns iderations of [D] Section 1 .3 apply . Remark 1 The term “analytic” functor se ems quite appr opriate in such c ases. Example [D]: A t yp e of “q ua ntum category” example ev olves from an y V -pro mo noidal category ( A , p, j ). Namely , the left “ Cayley” functor K : [ A , V ] − → [ A op ⊗ A , V ] given by K ( f )( A, B ) = X Z p ( X , A, B ) ⊗ f ( X ) , the V -kernel functor K : A op ⊗ A op ⊗ A − → V here b eing just the pr omultiplication p . This K is b oth conserv ative and tensor preserving , where [ A , V ] has the conv o lution s tr ucture and [ A op ⊗ A , V ] has the tensor pro duct defined b y bimo dule comp o sition. Th us K qua lifies as a “F ourier transforma tio n” [D]. 2 References. [A V] J. Ada mek and J. V elebil, “Analytic functors and weak pullbacks”, Theory Appl. Ca tegories, 2 1 (11), (2008 ) 191- 209. [D] B. J . Day , “Mo noidal functor categories and graphic F o urier transforms”, arXiv:mathQA/0 6124 9 6v1, 18 Dec. 20 06. [F GHW] M. Fior e, N. Gam bino, M. Hyland and G. Winskel, “ The car tesian clos ed category o f genera lized sp ecies of structures”, London Math. So c. (20 07), 1-18 . Mathematics Dept., F aculty of Science, Macquarie Un iversi ty , N SW 2109, A ustralia. Any replies are welc ome th rough T om Booker (th omas.bo oker@students. mq.edu.au), who k indly typ ed the manuscript. 3