String diagrams for traced and compact categories are oriented 1-cobordisms

Abstract

We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob /O , where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully encoded by Cob /O in the sense that there is an equivalence between Cob /O -algebras, for varying O, and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O. As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.

String Diagrams for Regular Logic (Extended Abstract)

“…Presenting regular categories using monoidal maps FRg(T) → Poset fits into an emerging pattern. In [21] it was shown that lax monoidal functors 1-Cob T → Set present traced monoidal categories, and in [12] it was shown that lax monoidal functors Cospan T → Set present hypergraph categories. But now in all three cases, the domain of the functor represents a particular language of string diagrams, and the codomain represents a choice of enriching category.…”

Hypergraph categories

“…The above operadic viewpoint on wiring diagrams was put forth by Spivak and collaborators [Spi13; RS13; VSL15]. In particular it was shown in [SSR16] that the operad governing traced monoidal categories is Cob, the operad of oriented 1-dimensional cobordisms.…”

Dynamical Systems and Sheaves

“…Operads (or monoidal categories) of various 'wiring diagram shapes' have been considered in [Spi13,RS13]. More recently, [SSR16] showed a strong relationship between the category of algebras on a certain operad (namely Cob, the operad of oriented 1-dimensional cobordisms) and the category of traced monoidal categories; results along similar lines were proven in [Fon16]. The operads W we use here are designed to model what could be called 'cartesian traced categories without identities'.…”

“…The diagrams in a wheeled prop look very similar to diagrams in this paper like (11), so it is worth elaborating on their differences. First of all, it was shown in [SSR16] that the category of wheeled props is equivalent to that of algebras on the operad 1-CatCob of oriented 1-cobordisms. The operad of wiring diagrams does indeed compare to that of cobordisms, but the two are not equivalent.…”

“…In Section 2.2 we restrict our interest to a certain symmetric monoidal category W (and its associated operad) studied in previous works like [VSL15]. The objects and morphisms in W classify a certain sort of string diagrams, which for reasons explained elsewhere [SSR16] we call wiring diagrams. The category of wiring diagram algebras, i.e.…”