strict symmetric monoidal

strict symmetric monoidal

[Thorsten Altenkirch]

Hi everybody,

I haven't been reading this list for a while but now I have a
question people on categories may have thought about. May be
trivial... apologies in advance.

A strict monoidal category is a monoidal one where all the isos are
indentities, e.g. we have that A(x)(B(x)C) = (A(x)B)(x)C and also
f(x)(g(x)h) = (f(x)g)(x)H because the natural iso is the identity. A
nice example is the skeletal cat of finite sets, i.e. objects are
natural numbers and Hom m n = {i<m} -> {i<n}. This cat is even
strictly bimonoidal with + and *. However, even though both operators
are symmetric and A(x)B = B(x)A we don't have f(x)g = g(x) f - hence
I wouldn't call this one strictly symmetric monoidal.

Are there any (constructive) examples of non-trivial strict symmetric
monoidal cats?

I am thinking about reversible computations and quantum, btw.

Cheers,
Thorsten

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