The smash product of monoidal theories
Amar Hadzihasanovic (Tallinn University of Technology)
Abstract: The smash product of pointed spaces is a classical construction of topology. The tensor product of props, which extends both the Boardman-Vogt product of symmetric operads and the tensor product of Lawvere theories, seems firmly like a piece of universal algebra.
In this talk, we will see that the two are facets of the same construction: a “smash product of pointed directed spaces”. Here, “directed spaces” are modelled by combinatorial structures called diagrammatic sets, developed as a homotopically sound foundation for diagrammatic rewriting in higher dimensions, while the cartesian product of spaces is replaced by a form of Gray product. Most interestingly, the smash product applies to presentations of higher-dimensional theories and systematically produces oriented equations and higher-dimensional coherence data (oriented syzygies). This introduces a synthetic, compositional method in rewriting on higher structures.
This talk is based on my preprint arXiv:2101.10361 with the same title.
category theory
Audience: researchers in the topic
( paper )
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