Abstract
We equip the category of properads with a symmetric monoidal closed structure. For topological operads, a symmetric monoidal product was already defined by Boardman and Vogt (Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347, Springer, Berlin, 1973). One main result of this chapter gives a simple description of the tensor product of two free properads in terms of the two generating sets. In particular, when the free properads are finitely generated, their tensor product is finitely presented. This is not immediately obvious from the definition because free properads are often infinite sets.
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References
J.M. Boardman, R.M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces. Lecture Notes in Mathematics, vol. 347 (Springer, Berlin, 1973)
P. Hackney, M. Robertson, On the category of props. Appl. Categ. Struct. 23(4), 1–31 (2014)
I. Weiss, From Operads to Dendroidal Sets. Proceedings of Symposia in Pure Mathematics, vol. 83 (American Mathematical Society, Providence, 2011), pp. 31–70
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Hackney, P., Robertson, M., Yau, D. (2015). Symmetric Monoidal Closed Structure on Properads. In: Infinity Properads and Infinity Wheeled Properads. Lecture Notes in Mathematics, vol 2147. Springer, Cham. https://doi.org/10.1007/978-3-319-20547-2_4
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DOI: https://doi.org/10.1007/978-3-319-20547-2_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20546-5
Online ISBN: 978-3-319-20547-2
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