# Techniques: Tensor-restriction categories

## What the paper contributes (one paragraph)
This is a pure category-theory paper bridging two notions of partiality in monoidal categories. Restriction categories record where a morphism is defined with respect to composition; tensor topology records where objects and maps are defined with respect to the spatial structure given by subunits of the tensor unit. The authors give the S-construction, a functorial way of turning a firm monoidal category into a restriction category whose restriction structure comes from its subunits, and characterise exactly which restriction categories arise this way, axiomatising them as tensor-restriction categories. A reverse construction takes the restriction-total maps of a tensor-restriction category back to a firm monoidal category, and the two are mutually inverse up to equivalence. The work is foundational mathematics: it supplies a structural equivalence and an axiomatisation, not a computational method, protocol, or bound that an external quantum result could consume.

## Techniques offered
- **S-construction (firm monoidal to restriction category equivalence)** - a functorial construction turning a firm monoidal category into a restriction category, with an inverse on restriction-total maps, axiomatising the image as tensor-restriction categories.
  - guarantee: a static categorical equivalence (the two constructions are mutually inverse up to equivalence, with a sound axiomatisation of the resulting categories)
  - quote: "We introduce a construction that turns a firm monoidal category $\cat{C}$ into a restriction category $\S[\cat{C}]$ in a functorial way."

## Where it could apply
- Target primitive(s): none
- Target application group(s): none

## Caveats
Pure category-theory foundations with nothing suppliable to an external vendor or application result. There is no quantum content, no algorithm, protocol, decoder, bound, or complexity claim that a quantum primitive or application group could consume; the "techniques" the paper discusses are abstract methods transferred between restriction-category theory and tensor topology, and the graphical (string-diagram) calculus is borrowed from prior work rather than contributed, with its adaptation to subunits explicitly left to future work. The single best-fit bullet above records the structural equivalence only for completeness.