bio | website | math.berkeley.edu/~sramesh |
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location | Berkeley, CA | |
age | 29 | |
visits | member for | 4 years, 7 months |
seen | 21 hours ago | |
stats | profile views | 3,535 |
I am 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.
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). |
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 |
comment |
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 |
comment |
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 |
comment |
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 |
Feb 10 |
answered | Compact Hausdorff and C^*-algebra “objects” in a category. |
Feb 10 |
answered | Geometric Interpretation of Trace |
Feb 10 |
comment |
Can we recognize when a category is equivalent to the category of models of a first order theory?
Just for clarification/making the question more concrete, what exactly do you mean by "In other words, is being Mod(T) a category-theoretic concept?". That is, what is it about something like "There exists a first-order theory (equivalently, a Boolean logos) T such that C is equivalent to Mod(T)?" that would make it not immediately a category-theoretic concept? [It is, after all, a property of categories which is preserved by categorical equivalence] |
Feb 10 |
comment |
When a formal power series is a rational function in disguise
(That having been said, this latter problem, despite being a priori easier, is also unsolvable, for the reasons I outline in my answer, which, although they don't invoke Rice's theorem, are similarly general and require little knowledge of the properties of rational power series specifically [they amount to a simple variant of the more powerful Rice-Shapiro theorem]) |
Feb 10 |
revised |
When a formal power series is a rational function in disguise
Minor wording change |