FinVect

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.[1]

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.[2]

Group representations are functors from groups, seen as one-object categories, into FinVect.[3]

DisCoCat models are monoidal functors from a pregroup grammar to FinVect.[4]

See also

• FinSet
• ZX-calculus
• category of modules

References

1. Hasegawa, Masahito; Hofmann, Martin; Plotkin, Gordon (2008), "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", Pillars of computer science, Springer, pp. 367–385
2. Kissinger, Aleks (2012). Pictures of processes: automated graph rewriting for monoidal categories and applications to quantum computing (Thesis). arXiv:1203.0202. Bibcode:2012PhDT........17K.
3. Wiltshire-Gordon, John D. (2014-06-03). "Uniformly Presented Vector Spaces". arXiv:1406.0786 [math.RT].
4. de Felice, Giovanni; Meichanetzidis, Konstantinos; Toumi, Alexis (2020). "Functorial question answering". Electronic Proceedings in Theoretical Computer Science. 323: 84–94. arXiv:1905.07408. doi:10.4204/EPTCS.323.6. S2CID 195874109.