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Stream: deprecated: id my structure

Topic: quiver with tensors?

Samuel Gélineau (Nov 16 2022 at 18:49):

Is there a special name for the following structure? It's useful as a generator for a monoidal category, but it is not itself a category.

a set T of objects {A, B, ...}
a set of morphisms, each of which has an input type and an output type. A type is not an object, but a list of objects, written as a tensor A ⊗ B ⊗ C or as 1 for the empty list.

This is not a quiver, because T does not include 1 nor A ⊗ B.

For example, the following instance of that structure represents what sigfpe calls the "four essential components" of a tangle:

a singleton set of objects, where the single object is called "Wire"
the set of morphisms {over, under, cup, cap}, whose input and output types are as follows:

over : Wire ⊗ Wire -> Wire ⊗ Wire
under : Wire ⊗ Wire -> Wire ⊗ Wire
cup : Wire ⊗ Wire -> 1
cap : 1 -> Wire ⊗ Wire

A tangle, as illustrated later in the linked blog post, can then be defined as the free monoidal category generated by those essential components.

Mike Shulman (Nov 16 2022 at 18:54):

One thing it's called is a [[pre-net]].

Mike Shulman (Nov 16 2022 at 18:55):

As noted there, it's also been called a "tensor scheme". I think I have occasionally called it a "polygraph", because it's related to polycategories in the same way that (directed) graphs are related to categories, but that probably clashes with too many other things.

Spencer Breiner (Nov 16 2022 at 20:24):

Sometimes called a "monoidal signature" (e.g., here)

Amar Hadzihasanovic (Nov 16 2022 at 21:29):

Mike Shulman said:

As noted there, it's also been called a "tensor scheme". I think I have occasionally called it a "polygraph", because it's related to polycategories in the same way that (directed) graphs are related to categories, but that probably clashes with too many other things.

You probably already know this, but the clash is not so bad here, since the other use of “polygraph” (that I know of) is as a synonym for “computad”, and this may be identified with the data of a 2-computad/2-polygraph with a single 0-cell.

Matt Earnshaw (Nov 17 2022 at 13:52):

yet another: Walters and coauthors called these monoidal graphs

Samuel Gélineau said:

A tangle, as illustrated later in the linked blog post, can then be defined as the free monoidal category generated by those essential components.

A tangle is a morphism in this category :-)