Tobias Fritz
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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)