For the (very) general case, see Theorem 4.83 in [Kelly, _Basic concepts of enriched category theory_]. It’s somewhat confusing that Kelly refers to these functors as “accessible”.

]]>Actually, I think it stems from the fact that the natural numbers are inadequately defined. For example, multiplication is commutative on the natural numbers, and each natural has a unique prime factorization. But, there are no infinitely long natural numbers, for if I had a box containing all the prime numbers, and I chose one at random handing it to you to handle the commutativity part and form the prime factorization of some “number”, in theory, I could hand you a 2, then an infinite number of other primes before I handed you another 2, so how would you invoke commutativity on an infinite string of primes? Moreover, in some book on number theory, I think it was either Hardy or Apostol, it showed that the termination or periodicity in the decimal expansions of rational numbers with respect to some base is determined by the exponents of the remaining primes in the denominator after reducing and factoring out the base. By this reasoning, we should find that irrational numbers are really just ratios of infinite products of primes, so that the decimal expansion is actually cyclic, just infinitely so. This is actually a fact since for each real number we can create an infinite sum converging to said real number, each of whose terms in the rationals. Yet, in transforming such a sum into an infinite product, these infinite ratios are not in the rationals, since the numerator and denominator are not in the integers since they are not in the naturals since they are infinitely long. And also, in every single proof of irrationality I have ever seen, there is a deduction assuming “reducibility” or something logically equivalent to it. Now, “reducibility” in terms of the current definition of the rationals, therefore definition of the integers, therefore definition of the naturals, is well-defined, but on the other hand, how could one “reduce” an infinitely long product? You can only “reduce” if the infinite sequence of powers of the primes in both of these prime factorizations terminates, thus leaving you with the current definitions of the rationals as “reducible” and naturals as “countable”. There doesn’t seem to be any intuitive way to circumvent these inconveniences and expand these concepts either, so, like you, until there is, I don’t trust any uncountable set.

In other words, I believe that the inadequacy in the current definition of the naturals and everything that comes from them to be the fact that we expect them to be countable. I think this is just a property that some person came up with and then some other assumed to hold for the entire set of naturals. Likewise, with reducibility in the rationals.

Above all else, I think that in the current epsilon-delta approximations of analysis, the only thing guaranteed about the real numbers is an approximation to them by some countable group.

]]>Different possibilities arise from using goroupoids, since group objects in groupoids, or, equivalently, , groupoid objects groups, are equivalent to crossed modules, which are in some sense “more nonabelian” than groups, and should be thought of as 2-dimensional groups, since they model weak, pointed, homotopy 2-types. There is much more on this in my expository article

R. Brown `Groupoids and crossed objects in algebraic topology’,

Homology, homotopy and applications, 1 (1999) 1-78.(electronic)

which also contains a matrix version of the Eckmann-Hilton argument,

This also explains about the use of certain kinds of double groupoids, which have advantages over the commonly used 2-groupoids. A homotopy strict double groupoid is defined for a pointed pair of spaces, $(X,A,x)$ and is “equivalent” to the second relative homotopy group of the pair, considered as a crossed module. One takes homotopy classes of maps of a square to $X$ which take the edges of the square to $A$ and the vertices to the base point. I find these much easier to use than the globular 2-groupoids, and

I get confused about the use of the term “fundamental 2-groupoid” of a space, since it is not a strict structure, even in dimension 1!