# Number of ways to construct mathematical objects

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.

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That's only 6 methods, and your indentation makes it look like algebraic geometry encompasses all of them. –  Nate Eldredge Sep 23 '10 at 18:51
That's because I was quoting from the original question, which had only 6 of the 7. –  Thierry Zell Sep 23 '10 at 18:57
Cohomology of complex algebraic manifolds: smooth deRham, algebraic deRham, Cech, singular, derived functor, Alexander-Spanier, canonical flasque resolution, etale topology, infinitesimal site,... They're all "the same", yet each viewpoint useful for various reasons. –  BCnrd Sep 23 '10 at 19:11
The word 'truly' in the question is problematic. –  HJRW Sep 23 '10 at 19:34
somewhat related, though different: mathoverflow.net/questions/15731/cryptomorphisms –  j.c. Sep 25 '10 at 0:43

• Turing machines
• Register machines
• Cellular automata
• $\mu$-recursive functions
• The untyped $\lambda$-calculus
• Unrestricted grammars
• Term-rewriting systems
• Recursively-enumerable subsets of $\mathbb{N}$
• $\ldots$

To this list we can add finite presentations of semigroups and groups, diophantine sets, and many other mathematical structures that realize universal computation.

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I imagine I would be tempted to lump at least some of them together. –  Thierry Zell Sep 23 '10 at 19:04
To this list we can add finite presentations of semigroups and groups, diophantine sets, and many other mathematical structures that realize universal computation. –  John Stillwell Sep 23 '10 at 19:11
I'll note: "can add" is literally true here, since the answer is community wiki. –  Ben Webster Sep 23 '10 at 20:10

There are lots of different ways to construct a finitely generated (discrete) group. For example (in no particular order):

1. Fundamental groups of topological spaces;

2. Groups of symmetries of mathematical objects (including fields, manifolds, simplicial, real and $\Lambda$-trees, etc.);

3. Groups given by presentations satisfying various small cancelation conditions (from Tartakovsky to Olshanskii to Gromov);

4. Lattices in Lie groups;

5. Wreath products of various kinds;

6. Direct limits of sequences of groups and their homomorphisms (including various "monsters", etc.);

7. Free constructions (HNN extensions and amalgamated products);

8. Groups simulating various computing devices (there are several different constructions here);

9. Groups acting on locally finite rooted trees (including Grigorchuk groups and iterated monodromy groups of Nekrashevych);

10 Automatic groups,

..................

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