Abstract
This paper proves coherence results for categories with a natural transformation called intermutation made of arrows from (A ∧ B) ∨ (C ∧ D) to (A ∨ C) ∧ (B ∨ D), for ∧ and ∨ being two biendofunctors. Intermutation occurs in iterated, or n-fold, monoidal categories, which were introduced in connection with n-fold loop spaces, and for which a related, but different, coherence result was obtained previously by Balteanu, Fiedorowicz, Schwänzl and Vogt. The results of the present paper strengthen up to a point this previous result, and show that two-fold loop spaces arise in the manner envisaged by these authors out of categories of a more general kind, which are not two-fold monoidal in their sense. In particular, some categories with finite products and coproducts are such. Coherence in Mac Lane’s “all diagrams commute” sense is proved here first for categories where for ∧ and ∨ one assumes only intermutation, and next for categories where one also assumes natural associativity isomorphisms. Coherence in the sense of coherence for symmetric monoidal categories is proved when one assumes moreover natural commutativity isomorphisms for ∧ and ∨. A restricted coherence result, involving a proviso of the kind found in coherence for symmetric monoidal closed categories, is proved in the presence of two nonisomorphic unit objects. The coherence conditions for intermutation and for the unit objects are derived from a unifying principle, which roughly speaking is about preservation of structures involving one endofunctor by another endofunctor, up to a natural transformation that is not an isomorphism. This is related to weakening the notion of monoidal functor. A similar, but less symmetric, justification for intermutation was envisaged in connection with iterated monoidal categories. Unlike the assumptions previously introduced for two-fold monoidal categories, the assumptions for the unit objects of the categories of this paper, which are more general, allow an interpretation in logic.
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References
Baez, J.C., Dolan, J.: Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36, 6073–6105 (1995)
Baez, J.C., Neuchl, M.: Higher-dimensional algebra I. Braided monoidal 2-categories. Adv. Math. 121, 196–244 (1996)
Balteanu, C., Fiedorowicz, Z., Schwänzl, R., Vogt, R.: Iterated monoidal categories. Adv. Math. 176, 277–349 (2003)
Bénabou, J.: Introduction to bicategories, reports of the midwest category seminar. In: Bénabou, J., et al. (eds.) Lecture Notes in Mathematics, vol. 47, pp. 1–77. Springer, Berlin (1967)
Brünnler, K.: Atomic cut elimination for classical logic, computer science logic. In: Baaz, M., Makowsky, J.A. (eds.) Lecture Notes in Computer Science, vol. 2803, pp. 86–97. Springer, Berlin (2003)
Brünnler, K.: Cut elimination inside a deep inference system for classical predicate logic. Stud. Log. 82, 51–71 (2006)
Brünnler, K., Tiu, A.F.: A local system for classical logic, logic for programming, artificial intelligence and reasoning. In: Nieuwenhuis, R., Voronkov, A. (eds.) Lecture Notes in Computer Science, vol. 2250, pp. 347–361. Springer, Berlin (2001)
Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22, 465–476 (1979)
Došen, K., Petrić, Z.: Cartesian isomorphisms are symmetric monoidal: a justification of linear logic. J. Symb. Log. 64, 227–242 (1999)
Došen, K., Petrić, Z.: Proof-Theoretical Coherence. KCL (College Publications), London (2004). Revised version available at: http://www.mi.sanu.ac.rs/~kosta/coh.pdf
Došen, K., Petrić, Z.: Proof-Net Categories. Polimetrica, Monza (2007). Available at: http://www.mi.sanu.ac.rs/~kosta/pn.pdf
Došen, K., Petrić, Z.: Coherence for star-autonomous categories. Ann. Pure Appl. Logic 141, 225–242 (2006). Available at: http://arXiv.org/math.CT/0503306
Došen, K., Petrić, Z.: Medial commutativity. Ann. Pure Appl. Logic 146, 237–255 (2007). Available at: http://arXiv.org/math.CT/0610934
Eilenberg, S., Kelly, G.M.: Closed categories. In: Eilenberg, S., et al. (eds.) Proceedings of the Conference on Categorical Algebra, La Jolla 1965, pp. 421–562. Springer, Berlin (1966)
Forcey, S., Siehler, J., Seth Sowers, E.: Operads in iterated monoidal categories. J. Homotopy Relat. Struct. 2, 1–43 (2007). (electronic, previously entitled Combinatoric n-fold categories and n-fold operads, available at: http://arXiv.org/math.CT/0411561)
Ježek, J., Kepka, T.: Medial Groupoids. Rozpravy Československé Akademie Věd, Řada matematických a přirodních věd, Ročnik 93, Sešit 2, p. 93. Prague (1983)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Kelly, G.M., Mac Lane, S.: Coherence in closed categories. J. Pure Appl. Algebra 1, 97–140, 219 (1971)
Lamarche, F.: Exploring the gap between linear and classical logic. Theory Appl. Categ. 18, 473–535 (2007)
Lawvere, F.W., Schanuel, S.H.: Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press, Cambridge (1997)
Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49, 28–46 (1963)
Mac Lane, S.: Categories for the Working Mathematician, expanded 2nd edn. Springer, Berlin (1998)
Street, R.: Two constructions on lax functors. Cah. Topol. Géom. Différ. 13, 217–264 (1972)
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Došen, K., Petrić, Z. Intermutation. Appl Categor Struct 20, 43–95 (2012). https://doi.org/10.1007/s10485-010-9228-x
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DOI: https://doi.org/10.1007/s10485-010-9228-x
Keywords
- Coherence
- Associativity
- Commutativity
- Monoidal categories
- Symmetric monoidal categories
- Iterated monoidal categories
- Loop spaces