Embedding of closed categories into monoidal closed categories

Laplaza, Miguel L.

doi:10.1090/S0002-9947-1977-0480686-8

Embedding of closed categories into monoidal closed categories
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by Miguel L. Laplaza

Trans. Amer. Math. Soc. 233 (1977), 85-91

DOI: https://doi.org/10.1090/S0002-9947-1977-0480686-8

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Abstract:

S. Eilenberg and G. M. Kelly have defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms. A monoidal closed category is a closed category with an associative tensor product which is adjoint to the int-hom. This paper proves that a closed category can be embedded in a monoidal closed category: the embedding preserves any associative tensor product which may exist. Besides the usual tools of the theory of closed categories the proof uses the results of B. Day on promonoidal structures.

References

Jean Bénabou, Catégories avec multiplication, C. R. Acad. Sci. Paris 256 (1963), 1887–1890 (French). MR 148719

Construction of biclosed categories

Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38. MR 0272852
Samuel Eilenberg and G. Max Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 421–562. MR 0225841
Samuel Eilenberg and G. M. Kelly, A generalization of the functorial calculus, J. Algebra 3 (1966), 366–375. MR 190204, DOI 10.1016/0021-8693(66)90006-8
G. M. Kelly, On MacLane’s conditions for coherence of natural associativities, commutativities, etc, J. Algebra 1 (1964), 397–402. MR 182649, DOI 10.1016/0021-8693(64)90018-3
G. M. Kelly and S. Mac Lane, Coherence in closed categories, J. Pure Appl. Algebra 1 (1971), no. 1, 97–140. MR 283045, DOI 10.1016/0022-4049(71)90013-2
Saunders Mac Lane, Natural associativity and commutativity, Rice Univ. Stud. 49 (1963), no. 4, 28–46. MR 170925
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798