This question stems from this other one mentioning 7 ways of constructing smooth manifolds. I quote:

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds.[...]

- Algebraic geometry (affine and projective varieties, ...)
- Lie groups (homogeneous spaces, ...)
- General position arguments (Morse theory, Pontryagin-Thom construction, ...)
- Solutions to PDE (Moduli spaces in gauge theory, Floer theory, ...)
- Surgery (Cut and paste techniques, ...)
- Markov processes
- [and also bundles seems to be the consensus in the answers to the cited question]

Note that all of these methods are actually areas of mathematics in their own right (five of the six listed in that question involve trailing dots), so that got me thinking that 7 methods is actually a sign of a rich subject and a fairly ubiquitous concept. I have trouble comparing this to anything else: e.g., would you say that there are fewer ways to build a group? I don't know nearly enough about groups to answer that one. So here's my question:

**Do you know of an abstract math construct that can be built in truly more than seven ways?**

I realize that this is somewhat in the eye of the beholder (hence the soft question tag) since it may not be obvious where to draw the line between methods. But for the comparison to make sense, you need to consider broad categories like Gromov does.