# Tom Leinster

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 Name Tom Leinster Member for 3 years Seen yesterday Website Location Age
 May10 awarded ● Nice Answer May9 comment A question in category theoryAnother apposite phrase: If it walks like a duck and quacks like a duck, it probably is a duck. What the proof of the Yoneda lemma reveals is that the best judge of whether something's a duck is a duck. May7 comment An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference requestPS to Butch: I'll acknowledge you when I update the paper for which I needed this, arxiv.org/abs/1209.3606. Forgive the impertinence, but is Butch Malahide your real name? Feel free to contact me by email. (I'd contact you myself, but there's no address on your profile.) May7 comment An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference requestExcellent. Thanks very much. For the sake of precision, let me add that they don't quite do the case $n=3$ in the sense described in my question. They do show that if a set $\mathcal{U}$ of subsets of $X$ satisfies the partition condition for all $n\leq 3$, then $\mathcal{U}$ is an ultrafilter. But they seem not to observe that it suffices to require it for $n=3$ (which implies it for $n=0,1,2$). Quite possibly they knew it but just didn't think it was worth mentioning. May4 awarded ● Notable Question Apr20 accepted Surface area of a convex set Apr20 answered Surface area of a convex set Mar29 awarded ● Popular Question Mar12 awarded ● Good Answer Feb25 comment Understanding Gibbs’s inequalityNice! Thanks. Nevertheless, the "trick" aspect of it means that it doesn't entirely satisfy me: what I want is to get an intuitive picture in my head which makes the result seem obvious (just as for the isoperimetric inequality). @Noah: I suspect the uniqueness part can't be extended to arbitrary real weights too easily. Indeed, in the introduction to their book Inequalities, Hardy, Littlewood and Pólya comment on this limitation of such rational-approximation arguments. Feb25 asked Understanding Gibbs’s inequality Feb25 comment “Philosophical” meaning of the Yoneda LemmaThanks. Fixed. Feb25 revised “Philosophical” meaning of the Yoneda LemmaUpdated link Jan29 comment Any recommendations on good papers I should read?This article comes from the excellent Princeton Companion to Mathematics. If you (profiles117...) can get hold of a copy, you might enjoy browsing it. Certainly it's pitched at a level significantly higher than you're likely to be used to. Nevertheless, you might find that it gives you an impression of what the wide world of mathematics looks like. Jan22 comment Category and the axiom of choice Ah, maybe I can guess: an equivalence relation, viewed as a groupoid? Jan22 comment Category and the axiom of choice Nice answer, but what is a "setoid"? Jan21 comment How would set theory research be affected by using ETCS instead of ZFC?@François: re whether mathematicians act as if functions come equipped with codomains, I think this varies from subject to subject. One subject where codomains are crucial is algebraic topology. For example, suppose we construe the circle $S^1$ as a subset of the plane $\mathbb{R}^2$. The identity $S^1 \to S^1$ definitely has to be distinguished from the inclusion $S^1 \to \mathbb{R}^2$, since when you pass to first homology, the former gives an isomorphism but the latter gives $\mathbb{Z} \to 0$, which isn't even injective. Jan18 awarded ● Popular Question Jan17 comment What methods have been used to study AW*-algebras up to now?Thanks for the explanation, Yemon. I agree, it does seem to be a very open-ended question, maybe too much so. Jan16 revised What methods have been used to study AW*-algebras up to now?Further English fixes; added 1 characters in body Jan16 comment What methods have been used to study AW*-algebras up to now?English fixed (correctly, I hope: I hadn't even heard of AW*-algebra). Jan16 revised What methods have been used to study AW*-algebras up to now?Corrected English Jan15 comment Does this poset have a unique minimal element?Kind of off topic, but: why do people so often say "unique minimal element" when "least element" is shorter and (I would say) more vivid? They're the same, at least under the axiom of choice. But undeniably, the definition of local ring is phrased in terms of a "unique maximal (proper) ideal" waaay more often than "largest (proper) ideal". Why? Jan15 comment Elementary applications of linear algebra over finite fieldsAnyone who's ever been single knows the phenomenon whereby everyone else in the room is a couple. But I've never heard anyone say they liked that. Jan14 comment real symmetric matrix has real eigenvalues - elementary proofGreat, thanks. Jan13 awarded ● Nice Answer Jan12 comment real symmetric matrix has real eigenvalues - elementary proof@marjeta: the point is that we shouldn't have to spend time guessing exactly what your question means, which is what many of these comments are trying to do. You should make it clear what your question means. Jan12 comment real symmetric matrix has real eigenvalues - elementary proof@marjeta: we probably all know this... More importantly: better than clarifying your question in the comment thread is to clarify your question by editing it. You should be able to see an "edit" button. Jan11 comment real symmetric matrix has real eigenvalues - elementary proofI'm with Gerald in not being sure exactly what the question's asking. By definition, the eigenvalues of a matrix over a field $k$ are elements of $k$. So strictly speaking, the question is trivial; looking for a nontrivial interpretation, I guess it must be one of the two possibilities that Gerald mentions. @Z254R: yes, I think Gerald is helping to formulate the problem. Jan10 comment Usage of set theory in undergraduate studiesJustin: I'll just observe that set theory and category theory are not disjoint subjects. For example, ETCS is a set theory with a categorical flavour. Jan10 comment Usage of set theory in undergraduate studies"Set theory is not in competition with category theory, in spite of what category theory thinks." -- Fully agreed that set theory is not in competition with category theory, but "in spite of what category theory thinks"?? I don't think I've ever heard a category theorist say that set theory is in competition with category theory. On the contrary, I have heard category theorists say explicitly that there's no such competition. Jan9 comment Locales and Topology.@Qiaochu: thanks for the plug. As I've now moved university, a safer link is arxiv.org/abs/1012.5647. Jan8 comment Bifunctor: Vector space of linear transformations betw vector spaces as bifunctorA better place for this question would be math.stackexchange.com, but I'd suggest thinking about Michael Murray's comment and rewriting your question before posting it there. Jan7 comment How has “what every mathematician should know” changed?Dieter, what's a Christmas Eve question? Jan3 answered The symmetric monoidal category of finite sets Jan2 revised Not-so-symmetric monoidal categoriesadded 14 characters in body; edited body Jan2 revised Not-so-symmetric monoidal categoriesadded 260 characters in body; added 87 characters in body Jan1 answered Not-so-symmetric monoidal categories Dec30 comment trigonometric non-identityIf I stick $a=b=c=\pi/6$ into this apparent identity (with cos instead of C, etc), I get 15/8 on the LHS and 13/8 on the RHS. Dec30 comment trigonometric non-identityAlso, when Michael says "$z = a + by$", these aren't the same a and b as earlier in the question. Dec30 comment trigonometric non-identityIgor, I think Michael is using "<-" to mean "is defined to be equal to". Thus, the second displayed block gives the definitions of y and z. Dec26 comment Fixed objects of the M endofunctor on category MeasReference: Michèle Giry, A categorical approach to probability theory. In B. Banaschewski, editor, Categorical Aspects of Topology and Analysis, Springer LNM 915, 1982. It's almost inconceivable that there could be two different reasonable monad structures on this endofunctor, so I bet it's the same. Dec26 comment Topologizing the category of measure spacesDmitri, it seems to me that there are interesting non-discrete possibilities for the topology on the objects. For example, take a skeleton of the category of measure spaces with finite underlying set. The set of objects is in natural bijection with the disjoint union of the topological simplices $\Delta^n$ ($n \geq 0$). This set carries a natural, non-discrete topology, namely the disjoint union of the Euclidean topologies. Dec26 comment Topologizing the category of measure spacesIt's an interesting answer, but not exactly an answer to the question asked. I'd still like to know about ways of making the category of measure spaces (not measurable spaces) internal to (not enriched in) the category of topological spaces. As both Dmitri and I have pointed out, there's a set-theoretic niggle to deal with, but that shouldn't be a serious obstacle. Dec26 comment Topologizing the category of measure spacesDmitri, you're technically right of course (and I mentioned this in my first comment). But the question about internalization (not enrichment!) is still a good one if you interpret it with a bit of generosity: e.g. take the category of measure spaces whose underlying sets have cardinality less than some fixed cardinal. Dec25 comment Topologizing the category of measure spacesRonnie, you may be right; all the same, the question does seem reasonable to me. For example, in my own work I've used the fact that the category of measures spaces with finite underlying set can naturally be regarded as an internal category in Top. This question removes the restriction of finiteness, which makes it harder. Dec24 awarded ● Nice Answer Dec22 answered The origin of sets? Dec21 awarded ● Populist Dec21 revised Subadditivity for Renyi entropiesdeleted 78 characters in body