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 Yearling
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Dec
18
comment Are dagger categories truly evil?
For plain categories, one can also express the compostionality in terms of properties of the nerve, which is a particular kind of simplicial set. Is there an analogous description of the compositionality in a dagger category? What is the nerve of a dagger category, if not just a simplicial set? Maybe we should discuss that in person soon ;)
Dec
18
comment Are dagger categories truly evil?
That rings very much true. One could even go a little bit further in that 'plain' categories are composition structures in which one can compose a bunch of morphisms aligned like $\bullet\to\bullet\to\bullet\to\bullet$, while in a dagger category one can compose a bunch of morphisms aligned also as e.g. in $\bullet\to\bullet\leftarrow\bullet\leftarrow\bullet\rightarrow\bullet$. In both cases, the axioms form a presentation of a limit sketch whose models are plain categories or dagger categories, respectively.
Oct
22
comment Form of binary function over poset that is monotone over first and antitone over second argument
See also ncatlab.org/nlab/show/profunctor
Oct
22
comment Form of binary function over poset that is monotone over first and antitone over second argument
You can consider $f$ also as a monotone function $f:P\times P^{\mathrm{op}}\to\mathbb{R}$. A nice example of such an $f$ is the hom-functor $(a,b)\mapsto 1$ if $a\geq b$ and $(a,b)\mapsto 0$ otherwise.
Oct
16
revised Examples of common false beliefs in mathematics
deleted 52 characters in body
Oct
15
comment Existence of polytope
yes, that's exactly it!
Oct
14
comment Existence of polytope
This is indeed not possible, since the polar of such a polytope would be $d$-neighbourly: en.wikipedia.org/wiki/Neighborly_polytope
Oct
6
comment What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?
In line with Todd's suggestion, computer scientists also use the term "cotupling" for the dual notion involving a coproduct: startpage.com/do/search?q=cotupling
Oct
4
awarded  Yearling
Sep
29
comment cross sections of semialgebric sets
@Patrik: you probably mean that any plane contains at most three of the points? (Every three points are contained in some plane, so your statement can't be quite right.)
Sep
29
comment cross sections of semialgebric sets
Actually already for $n=1$ there are plenty of counterexamples, since any subset of the real line has semialgebraic sections ;)
Sep
29
comment cross sections of semialgebric sets
In the plane, any convex set that is not semialgebraic will be a counterexample. For example, consider the set which is an intersection of infinitely many hyperplanes, such that no finite subset of hyperplanes defines the set. This is convex, but probably not semialgebraic; I'm not sure how to formally prove the latter, though. Maybe you know?
Aug
30
comment Can we drop commutativity assumption?
By construction, the multiplication map $A^{\otimes n}\otimes A^{\otimes n}\to A^{\otimes n}$ is an intertwiner for the permutation representation of the symmetric group $S_n$. Since $S^nA$ is the invariant subspace of this representation, this intertwiner restricts to $S^n A\otimes S^n A\to S^n A$, independently of whether the original multiplication is commutative or not. On the other hand, the resulting map $\bigwedge^n A\otimes\bigwedge^n A\to A^{\otimes n}$ also lands in $S^n A$, so that $\bigwedge^n A$ is closed under multiplication only in the degenerate cases $char(k)=2$ or $dim(A)<n$.
Aug
26
awarded  Nice Answer
Aug
21
revised Algebras for probability monad
added 43 characters in body
Aug
21
answered Algebras for probability monad
Aug
21
comment Vector Fields in a Riemannian Manifold
Actually I'm not sure that my statement on "isospectralities" makes sense: the principal symbol of the Laplace operator is the metric itself. So doesn't this imply that every vector field that commutes with the Laplacian must be Killing?
Aug
21
comment Vector Fields in a Riemannian Manifold
Clearly Killing vector fields have this property. But in general, I guess that one can think of such vector fields as generators of "isospectralities" (instead of isometries). What would be a nice example of a manifold with such a vector field that is not Killing?
Aug
17
answered General additive function of probability
Aug
17
awarded  Informed