FinVect
{{Short description|Category whose objects are finite-dimensional vector spaces and whose morphisms are linear maps}}
In the mathematical field of category theory, FinVect (or FdVect) is the category whose objects are all finite-dimensional vector spaces and whose morphisms are all linear maps between them.{{Citation |last1=Hasegawa |first1=Masahito |title=Finite dimensional vector spaces are complete for traced symmetric monoidal categories |date=2008 |work=Pillars of computer science |pages=367–385 |publisher=Springer |last2=Hofmann |first2=Martin |last3=Plotkin |first3=Gordon}}
Properties
FinVect has two monoidal products:
- the direct sum of vector spaces, which is both a categorical product and a coproduct,
- the tensor product, which makes FinVect a compact closed category.
Examples
Tensor networks are string diagrams interpreted in FinVect.{{Cite thesis |last=Kissinger |first=Aleks |date=2012 |title=Pictures of processes: automated graph rewriting for monoidal categories and applications to quantum computing |arxiv=1203.0202 |bibcode=2012PhDT........17K }}
Group representations are functors from groups, seen as one-object categories, into FinVect.{{Cite arXiv|last=Wiltshire-Gordon |first=John D. |date=2014-06-03 |title=Uniformly Presented Vector Spaces |class=math.RT |eprint=1406.0786 }}
DisCoCat models are monoidal functors from a pregroup grammar to FinVect.{{Cite journal |last1=de Felice |first1=Giovanni |last2=Meichanetzidis |first2=Konstantinos |last3=Toumi |first3=Alexis |title=Functorial question answering |journal=Electronic Proceedings in Theoretical Computer Science |url=https://scholar.googleusercontent.com/scholar.bib?q=info:hHiyaU_p6-AJ:scholar.google.com/&output=citation&scisdr=CgXXiV6NEJT6uebecrw:AAGBfm0AAAAAY3DYarzhswQRNF4gQmnLJCBdh2EDTL0a&scisig=AAGBfm0AAAAAY3DYah8teHAVmBWJPhJ8JmlTXXJ5fXfI&scisf=4&ct=citation&cd=-1&hl=en |year=2020 |volume=323 |pages=84–94 |doi=10.4204/EPTCS.323.6 |arxiv=1905.07408 |s2cid=195874109 }}