A physical instrument is used to measure physical systems and return the values for those measurements in classical data.

I was wondering if anyone here has been looking at instruments in terms of the categories of mathematical objects that describe them. For instance, a telescope has for main morphisms: rotate the telescope in the plain parallel to the ground, rotate the telescope in a plane perpendicular to the ground, rotate the focus knob, the view through the eyepiece changes.

I have come to believe that the category of instruments is a 2-category of groupoids. Every morphisms of an instrument has an inverse. A lab can be seen as a category of instruments, and every morphism is invertible. If there was a non invertible morphism, you would essentially break the instrument. For instance, grinding the eyepiece into dust has no inverse, and so it is not a +"telescope morphism".

An extremely advanced and powerful notion you get from this is that the category of labs, being a 2-cat of groupoids, supports the particle theory of physics because it admits a polynomial monad that is the free commutative monoid monad, (aka multiset, bag etc). See this post. Scroll down to Jeffery Morton's comments. You can also see Kock's paper here, where he talks about the bag monad and particles.

Is there anyone here thinking in this direction? I know there is no publication in this direction. What is the fallacy?