Sridhar Ramesh
Reputation
2,490
Top tag
Next privilege 2,500 Rep.
Create tag synonyms
 Feb 11 revised Hat Problem/Hamming Codes Nitpick about 1-based indexing Feb 11 comment Can we recognize when a category is equivalent to the category of models of a first order theory? And composition is straightforward: the binary composition of F(X, Y) and G(Y, Z) is the predicate H(X, Z) := There exist Y such that F(X, Y) and G(Y, Z). And the identity morphisms are given by equality predicates [or, technically, given the description as I've stated it so far, the identity morphism on A(X) is the predicate F(X_1, X_2) := A(X_1) & A(X_2) & (X_1 = X_2)]. Feb 11 comment Can we recognize when a category is equivalent to the category of models of a first order theory? As for how to construct the classifying logos of an arbitrary theory T, take as objects the definable predicates (of any arity) in the theory, with morphisms being definable, provably functional relations on the extensions of those predicates up to provable equivalence. I.e., a morphism from D(X) to R(Y) is a predicate F(X, Y) such that the theory proves "For all X and Y, F(X, Y) implies (D(X) and R(Y)), and for all X such that D(X), there exist unique Y such that F(X, Y).", where X is a tuple of variables of any length and similarly for Y. Feb 11 comment Can we recognize when a category is equivalent to the category of models of a first order theory? There are precisely two of these, and neither has a natural transformation into the other; thus, the category lacks a terminal object. Feb 11 comment Can we recognize when a category is equivalent to the category of models of a first order theory? The category I describe won't necessarily have finite limits: for example, consider the four-element Boolean algebra, viewed as a preorder category. This is a Boolean logos, and the functors from it to Set preserving Boolean logos structure are those which send its top element to 1 (by virtue of being a terminal object), its bottom element to 0 (by virtue of its arrow into the top element being the bottom subobject of the top element), one of the middle elements to 1 and the other to 0 (by virtue of their arrows into the top element being complementary subobjects). Feb 11 comment Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments Oh, that makes sense. Thanks. Feb 11 revised Hat Problem/Hamming Codes added 21 characters in body; added 25 characters in body; added 1 characters in body Feb 11 answered Hat Problem/Hamming Codes Feb 11 comment Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments Is it something like a higher-dimensional analogue of that same phenomenon? (Man, these character limits are brutal... this and the last two posts were meant to be one single comment) Feb 11 comment Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments (e.g., if A, B, and C happened to all be identical, in addition to composing the left and right morphisms, there would also be simply projection the left morphism, projecting the right morphism, producing the square of the left morphism, etc.)? That is, the theory of categories doesn't concern itself with such cases as where the three objects in binary composition all happen to accidentally line up; it can't see all these functions from Hom(A, A) x Hom(A, A) to Hom(A, A) for objects A, and so it doesn't impose coherence equalities between them. Feb 11 comment Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments Is the idea here analogous to the fact that, in some sense, the theory of ordinary categories (which I want to describe using a multicategory of some sort, but can't quite, but suppose I could if I had the multicategory live in some fancy category other than Set) sees one and only one function from Hom(A, B) x Hom(B, C) to Hom(A, C) for objects A, B, and C [binary composition], even though, for any particular actual objects in an actual category, there may be many non-equivalent functions of this type which can be built out of the structure of a category Feb 11 comment Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments I'm not sure how to read the diagram you linked to. It looks like a diagram of a plain-vanilla Eckmann-Hilton argument, except for the $l$s and $l^{-1}$ pieces, and I'm not sure what those are. That having been said, I think I understand anyway. At least, the last paragraph above seems like what I was thinking the answer was anyway (as for why there's no demand for a coherence isomorphism between the path all the way around and the identity). But let me make sure I understand: (Question coming in a followup comment with more characters left...) Feb 11 revised Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments Retitled Feb 11 awarded Student Feb 11 revised Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments added 88 characters in body Feb 11 asked Effects of “weak” vs. “strict” categories in Eckmann-Hilton arguments Feb 11 answered What is information-theoretic lower bound? Feb 11 awarded Critic Feb 11 revised Can we recognize when a category is equivalent to the category of models of a first order theory? Rewording Feb 11 comment Compact Hausdorff and C^*-algebra “objects” in a category. Actually, it occurs to me that there's a much easier way to describe this than the detour through infinitary Lawvere theories I've been using. Given a monad $M$ on $Set$, an algebra for this in the category $C$ should be an algebra (in the standard sense) for the monad $M^{C^{op}}$ whose carrier is in the range of the Yoneda embedding, where $M^{C^{op}}$ is the monad on presheaves on $C$ induced by postcomposition with $M$. This should be equivalent to what we've both been saying, but seems to me now much clearer (though perhaps others will differ).