bio | website | math.berkeley.edu/~sramesh |
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location | Berkeley, CA | |
age | 29 | |
visits | member for | 5 years, 2 months |
seen | Apr 13 at 17:32 | |
stats | profile views | 3,766 |
I was a graduate student in the Logic program at Berkeley, broadly interested in categorical logic and foundations of mathematics, as well as in applications of category theory to the semantics of programming languages. I work for Google now.
Feb 11 |
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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 |
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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 |
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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 <i>think</i> 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 |
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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). |
Feb 10 |
revised |
Can we recognize when a category is equivalent to the category of models of a first order theory?
deleted 160 characters in body |
Feb 10 |
answered | Can we recognize when a category is equivalent to the category of models of a first order theory? |
Feb 10 |
awarded | Commentator |
Feb 10 |
comment |
Compact Hausdorff and C^*-algebra “objects” in a category.
Such morphisms will automatically satisfy the appropriate commutative diagrams (by virtue of the appropriate equations holding in each algebra Hom(c^k, c)). Thus, they can be combined into what I thought of as an M object. I haven't sat down and checked all the details, but I am quite confident now that it works and I was wrong.) |
Feb 10 |
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Compact Hausdorff and C^*-algebra “objects” in a category.
(Why? The latter amounts to just putting the structure of a Set-algebra for M on Hom(x, c) for each x, such that precomposition is a homomorphism of such algebras. For every element in M(k), thought of as a k-ary operation, we obtain a morphism from c^k to c by applying that operation to the k many projections in Hom(c^k, c), from which the result of that operation on arbitrary Hom(x, c) is determined... [continued in next comment] |
Feb 10 |
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Compact Hausdorff and C^*-algebra “objects” in a category.
Egads, no, you're right. The correspondence does go both ways. I failed to see it before, being so used to viewing things one way, but if M is a monad on Set, then product-preserving functors from the dual of M's Kleisli category to C (what I was thinking of as an M object) are in correspondence with tuples of the form <contravariant functor F from C to the category of Set-algebras of M, object c in C such that the product of any set of copies of c exists, and natural isomorphism between Hom_C(-, c) and UnderlyingSet(F(-))> (what you were thinking of as an M object). So, I retract my "No". |
Feb 10 |
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Compact Hausdorff and C^*-algebra “objects” in a category.
I agree that what you (Tom) said is presumably equivalent to what I said. However, I have a question about the part where you say "(suitable kind of) monad on Set". What do you mean by "suitable kind of"? It seems to me this general idea should work for every monad on Set. [That is, pulling it through the correspondence between monads on Set and categories with set-sized products generated by a single object [the (Set-)algebras of the former also corresponding to the Set-models of the latter], and then using the latter to give an account of such algebras in other categories with products] |
Feb 10 |
revised |
Compact Hausdorff and C^*-algebra “objects” in a category.
Minor rewording |
Feb 10 |
revised |
Compact Hausdorff and C^*-algebra “objects” in a category.
Hesitation, considering deletion |