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Languages for monoidal categories. (English) Zbl 0693.18003
As the title suggests, the author introduces typed languages for monoidal categories, in analogy with what has been done for toposes and cartesian closed categories (for example). As more and more examples of non-strict monoidal categories arise - quantum groups come to mind - such languages should become more widely used and useful. To quote the author, “The power of this language is that it manipulates not only the formal monoidal structure, which the coherence theorem also does, but the data specific to [the monoidal category].” The language depends both on the theory of a monoidal category and the specific model with which one wants to work.
Reviewer: D.H.Van Osdol

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
##### Keywords:
typed languages for monoidal categories
Full Text:
##### References:
 [1] Barr, M.; Beck, J., Homology and standard constructions, (), 245-335 [2] Barr, M.; Wells, C., Toposes, triples and theories, (1985), Springer Berlin · Zbl 0567.18001 [3] Boileau, A.; Joyal, A., La logique des topos, J. symbolic logic, 46, 6-16, (1981) · Zbl 0544.03035 [4] Borceux, F.; Day, B., Universal algebra in a closed category, J. pure appl. algebra, 16, 133-147, (1980) · Zbl 0426.18004 [5] Bruno, O.P., Internal mathematics in toposes, () [6] Bunge, M., Relative functor categories and categories of algebras, J. algebra, 11, 64-101, (1969) · Zbl 0165.32902 [7] Day, B., On closed categories of functors, (), 1-38 [8] Day, B., On adjoint-functor factorisation, (), 1-19 · Zbl 0367.18004 [9] Eilenberg, S.; Kelly, G.M., Closed categories, (), 421-562 · Zbl 0192.10604 [10] Fröhlich, A.; Wall, C.T.C., Graded monoidal categories, Comp. math., 28, 3, 229-285, (1974) · Zbl 0327.18007 [11] C. Barry Jay, Languages for triples, bicategories and braided monoidal categories, to appear. · Zbl 0702.18005 [12] Johnstone, P.T., Topos theory, () · Zbl 0368.18001 [13] Keigher, William F., Symmetric monoidal closed categories generated by commutative adjoint monads, Cashiers topologie Géom. différentielle, 19, 3, 269-293, (1978) · Zbl 0381.18015 [14] Kelly, G.M., On Maclane’s conditions for coherence of natural associativities, commutativities, etc., J. algebra, 4, 397-402, (1964) · Zbl 0246.18008 [15] Kelly, G.M., Adjunction for enriched categories, (), 166-177 · Zbl 0214.03201 [16] Kelly, G.M., An abstract approach to coherence, (), 106-147 · Zbl 0243.18016 [17] Kelly, G.M., Basic concepts of enriched category theory, () · Zbl 0709.18501 [18] Kelly, G.M.; MacLane, S., Coherence in closed categories, J. algebra, 1, 97-140, (1971) · Zbl 0212.35001 [19] Kock, A., Monads on symmetric monoidal closed categories, Arch. math., 21, 1-10, (1970) · Zbl 0196.03403 [20] Lambek, J., Deductive systems and categories I. syntatic calculus and residuated categories, Math. systems theory, 2, 287-318, (1968) · Zbl 0176.28901 [21] Lambek, J., Deductive systems and categories II. standard constructions and closed categories, (), 76-122 · Zbl 0198.33701 [22] Lambek, J.; Scott, P., Introduction to higher order categorical logic, () · Zbl 0596.03002 [23] Lewis, G., Coherence for a closed functor, (), 148-195 [24] Lindner, H., Adjunctions in monoidal categories, Manuscripta math., 26, 123-139, (1978) · Zbl 0389.18004 [25] Lindner, H., Center and trace, Arch. math., 35, 476-496, (1980) · Zbl 0453.18007 [26] Lindner, H., Enriched categories and enriched modules, Cahiers topologie geom. différentielle, 22, 2, 161-173, (1981) · Zbl 0463.18004 [27] Linton, F.E.J., Relative functional semantics: adjointness results, (), 384-418 · Zbl 0227.18004 [28] MacLane, S., (), 28-46 [29] MacLane, S., Categorial algebra, Bull. amer. math. soc., 71, 40-106, (1965) · Zbl 0161.01601 [30] MacLane, S., Categories for the working Mathematician, (1971), Springer Berlin [31] May, J.P., Infinite loop space theory, Bull. amer. math. soc., 83, 4, 456-494, (1977) · Zbl 0357.55016 [32] Street, R., Fibrations in bicategories, Cahiers topologie Géom. différentielle, 21, 111-160, (1980) · Zbl 0436.18005 [33] Sweedler, M.E., Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901 [34] Voreadou, R., Coherence and non-commutative diagrams in closed categories, Mem. amer. math. soc., 9, 1, (1977) · Zbl 0347.18008
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