Puzzles 107 and 108: ooh, these are fun. OK, so there seems to be a connection between finitely presented categories and symmetry groups. For 107, we can interpret s as a 90 degree rotation. For 108, we can do the same for s, and have t be either an x-axis reflection or a y-axis reflection.

RCG2: for 108, I gave two interpretations of t. Are there more? What about different interpretations of s? Are all interpretations unique up to isomorphism?

Questions: is the notion of "interpretation" I'm informally using formalizable as a functor? For instance, let F be the map from the category in 107 to the 90 rotation group of a square (maybe let's say that the square has a marked orientation, like a red dot on the top right corner). If F maps from the one object of the category to the one object of the group (viewed as a category), then F is a functor. (Digression: ooh, this can't be quite the right notion of interpretation, because I want to distinguish the interpretation of t as an x-axis flip from the interpretation of t as a y-axis flip, but presumably the symmetry groups are isomorphic? I don't know any group theory, so this is all a bit speculative).

If so, then I could ask RCG3: is it the case that for any finitely presentable one object category C, there exist a functor F from C to a symmetry group? And if there are two such functors F and G, is F(C) isomorphic to G(C)?

I might be getting a bit ahead of myself - not yet sure this is a well-formed question.

RCG2: for 108, I gave two interpretations of t. Are there more? What about different interpretations of s? Are all interpretations unique up to isomorphism?

Questions: is the notion of "interpretation" I'm informally using formalizable as a functor? For instance, let F be the map from the category in 107 to the 90 rotation group of a square (maybe let's say that the square has a marked orientation, like a red dot on the top right corner). If F maps from the one object of the category to the one object of the group (viewed as a category), then F is a functor. (Digression: ooh, this can't be quite the right notion of interpretation, because I want to distinguish the interpretation of t as an x-axis flip from the interpretation of t as a y-axis flip, but presumably the symmetry groups are isomorphic? I don't know any group theory, so this is all a bit speculative).

If so, then I could ask RCG3: is it the case that for any finitely presentable one object category C, there exist a functor F from C to a symmetry group? And if there are two such functors F and G, is F(C) isomorphic to G(C)?

I might be getting a bit ahead of myself - not yet sure this is a well-formed question.