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Jun
30
comment Developing category theory inside ETCS
It should be a universe, but I have no checked. Not all universes are of this form, I think.
Jun
30
comment Is there a logic of sufficiency? Or goals?
Neither: it refers to substitutability.
Jun
29
answered Is there a logic of sufficiency? Or goals?
Jun
29
comment Developing category theory inside ETCS
It essentially means to interpret internal structures as instances of those structures in the meta-logic. It is not meant to be precise.
Jun
29
answered Developing category theory inside ETCS
Jun
29
comment Why Et(X) is not small?
I find it convenient to have all affine open subschemes of $X$ to be in the site – locally noetherian ensures this happens. I suppose it's unnecessary because the basic affine open subschemes are always of finite presentation.
Jun
28
answered Why Et(X) is not small?
Jun
28
comment Why Et(X) is not small?
Well, the canonical map from a disjoint union of copies of $X$ to $X$ is étale, and there are a proper class of isomorphism classes of these things.
Jun
28
comment Finite fiber of scheme morphism is zero-dimensional?
The question is local on the domain and the codomain, and both are locally noetherian, so there's no practical difference between "locally of finite presentation" and "of finite type".
Jun
28
answered Finite fiber of scheme morphism is zero-dimensional?
Jun
28
comment Finite fiber of scheme morphism is zero-dimensional?
What's your definition of étale morphism? In any case, one can show that the fibre $X_y$ is discrete as a topological space, and so must be zero-dimensional (because there are no interesting irreducible sets!).
Jun
24
revised Does this right adjoint of a geometric morphism preserve directed colimits?
added 770 characters in body
Jun
24
answered Does this right adjoint of a geometric morphism preserve directed colimits?
Jun
24
comment Does the nerve of a category preserve directed colimits?
(However, not every compact category is finite.)
Jun
23
comment Is a filtered category necessarily (essentially) small?
Yes, that is a small subtlety. It is similar to the pitfall of saying that $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is the free colimit-completion of $\mathcal{C}$ – this is true when $\mathcal{C}$ is essentially small but can fail when $\mathcal{C}$ is not essentially small.
Jun
23
comment Is a filtered category necessarily (essentially) small?
That is the standard proof as far as I know. Bear in mind that many results we have about filtered colimits are, strictly speaking, for colimits over small filtered diagrams – so your generalisation may be less useful than at first sight.
Jun
23
answered Is a filtered category necessarily (essentially) small?
Jun
23
comment A Topology such that the continuous functions are exactly the polynomials
If every polynomial function $f : K \to K$ is continuous and $\{ 0 \}$ is closed, then the topology on $K$ must be finer than the Zariski topology. However, the Zariski topology does not have the desired property: for example, the absolute value function $\mathbb{Q} \to \mathbb{Q}$ is Zariski-continuous.
Jun
22
comment Construction(s) of new integral domains from “old ones”
The tensor product of two integral domains over an algebraically closed field is again an integral domain: see here.
Jun
21
comment Why is the category of coherent sheaves not grothendieck?
You can check it by applying $\mathrm{Hom}_A (-, A)$.