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May 18 |
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Uniformly solvable families of polynomials Very interesting! Perhaps it would be better to consider the question in the setting of commutative algebra: assume $f$ is monic, irreducible, and separable; let $A$ be the subring of the coefficient field generated by $\mathbb{Q}[a_0, \ldots, a_k]$ and the coefficients of $f$, and consider the $A$-algebra $A [x] / (f)$. Then the automorphism group of $A [x] / (f)$ (as an $A$-algebra) will also act on each of the specialisations $A [x] / (f) \otimes_A A / \mathfrak{m}$, for each maximal ideal $\mathfrak{m}$ of $A$, and the "generic" situation is obtained by considering $\mathfrak{p} = (0)$. |
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May 17 |
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Graph of a morphism of affine schemes In what sense is the image of a scheme morphism a sheaf? |
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May 16 |
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Borceux. Handbook of Categorical Algebra I. Proposition 3.4.2. Presumably $\eta$ and $\epsilon$ are adjunction unit and counit and $\alpha$ and $\beta$ are mutually inverse, or something like that? Then just push $\epsilon$ past $\beta$ using naturality. |
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May 16 |
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Is $\operatorname{Aut}(\mathbb{I})$ isomorphic to $\operatorname{Aut}(\mathbb{I}^2)$? Actually, isn't every order automorphism (or anti-automorphism) of the interval automatically continuous? |
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May 16 |
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When is $T$-Alg monoidal closed? Every monad on sets has a unique strength and being a commutative monad is a property of that strength. The tensor product of abelian groups doesn't come into the question. |
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May 16 |
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When is $T$-Alg monoidal closed? (Also, the abelian group monad is commutative.) |
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May 16 |
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When is $T$-Alg monoidal closed? I don't see any reason why the algebras would be closed if the base isn't closed. After all, the hom module between two modules is a subset of the hom set... Not to mention, the identity monad is the nicest possible monad there is, so if its algebras are closed, then the base is already closed! |
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May 16 |
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When is $T$-Alg monoidal closed? Yes, that is the theorem of Kock. Why are you seeking different conditions? |
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May 16 |
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When is $T$-Alg monoidal closed? It is not necessary to assume that the coequalisers commute with $T$, but some condition about existence of coequalisers and preservation of epimorphisms seems to be required. |
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May 16 |
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Functoriality of the Fundamental group @Drew As is clear from the construction of $B G$ I give, I am only discussing discrete groups. |
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May 15 |
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Elementarily equivalent You should at least indicate the textbook you have copied this from, so that we can figure out what "2.7.4(b)" means... |
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May 15 |
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How to directly show that Figure 8 injective immersion is not a monomorphism I prove that no such maps exist in my second paragraph. I believe your understanding is incorrect: if I recall correctly, the regular monomorphisms are embeddings. |
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May 15 |
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Functoriality of the Fundamental group Aren't you basically asking whether there is a choice of Eilenberg–MacLane spaces $K(G, 1)$ that is functorial in $G$? The answer is yes. |
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May 15 |
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Plus construction of a presheave factors every sheaf-valued morphism. Something like that. Doing things internally in $\mathcal{S}$ is a bit too restrictive, however; it would be like working only with small categories. |
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May 15 |
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Plus construction of a presheave factors every sheaf-valued morphism. Yes, but that's not what I mean. There is a notion of $\mathcal{S}$-topos, where $\mathcal{S}$ is another topos, which one thinks of as being a topos "based on" $\mathcal{S}$, and there is a surprising amount of topos theory that carries over to this relative context including, yes, the machinery of Grothendieck topologies. The whole of Part B of Sketches of an elephant is devoted to relative topos theory. |
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May 15 |
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Plus construction of a presheave factors every sheaf-valued morphism. General advice: if you're doing topos theory, you can almost always do everything intuitionistically; in particular, you should not need to appeal to the law of excluded middle. So don't do case analysis! |
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May 14 |
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Do Boolean rings always have a unit element? My definition of ring includes a $1$, and my definition of boolean algebra also includes a $1$. |
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May 12 |
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Why does every countable limit ordinal have cofinality $\omega$? If there is a cofinal $\alpha$-sequence in $\beta$ and a cofinal $\beta$-sequence in $\gamma$, then there is a cofinal $\alpha$-sequence in $\gamma$. Therefore cofinalities are initial ordinals. |
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May 11 |
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Space modelled on ring Functors are not the best way to think about schemes if you're looking for spaces modelled on rings. Rather, a scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring. There is a functorial way of looking at schemes but I find it much more artificial. |
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May 11 |
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Space modelled on ring Do you mean a scheme? |
