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Nov
21
comment Does the functor of points commutes with inverse limits?
As indicated in my answer, if $\operatorname{Spec} \mathbb{C}[[t]]$ is the filtered colimit of $\cdots \to \operatorname{Spec} \mathbb{C}[t]/(t^n) \to \cdots$ in $\textbf{Sch}$ (and I have not checked this, but it appears to be true) then $X(\mathbb{C}[[t]])$ will indeed be the inverse limit of $\cdots \to X(\mathbb{C}[t]/(t^n)) \to \cdots$, by the first paragraph of my answer.
Nov
21
comment Does $E$ in a model $\langle M, E\rangle$ of ZFC have to be wellfounded?
What's the definition of "strictly wellfounded"? It is easy to produce non-standard models of ZFC in which there are externally infinite descending chains.
Nov
21
comment Learning path to the proof of the Weil Conjectures and étale topology
Crossposted to MO.
Nov
21
comment What does the maximal basis for a Grothendieck topology look like?
It is when the category $\mathcal{C}$ has pullbacks.
Nov
21
revised Learning path to the proof of the Weil Conjectures and étale topology
deleted 9 characters in body
Nov
21
comment Does the functor of points commutes with inverse limits?
As far as I can tell, $\textrm{Spec}$ transforms that into a filtered colimit, so that's not a counterexample.
Nov
20
answered What does the maximal basis for a Grothendieck topology look like?
Nov
20
comment What is the significance of multiplication (as distinct from addition) in algebra & ring theory?
Since you're mentioning composition and rings, I simply cannot resist mentioning composition rings, which can be thought of as an algebraic structure with three binary operations.
Nov
20
answered Does the functor of points commutes with inverse limits?
Nov
20
comment Overview and introduction to strong logics
Compactness is generally sacrificed when passing to infinitary logics: it's a fun exercise to find an (infinite) list of propositions (in, say, infinitary propositional logic) whose finite fragments are consistent while the list as a whole is inconsistent. (Hint: pigeonhole principle.)
Nov
20
answered $F(M)$ as a $\text{Hom}_{R_1}(M,M)$-module for some functor $F:R_1-\text{mod}\rightarrow R_2-\text{mod}$
Nov
19
comment Category theory without codomains?
There is no category theory that supports it, because this is a far too concrete notion. A category is fundamentally a system of arrows that can be joined head-to-tail in an associative way: the fact that the first and foremost examples of such things are functions is purely an accident of history.
Nov
19
comment Category theory without codomains?
@IttayWeiss While what you say is all correct, I suspect the heart of the contention is that the OP believes two functions are equal when their graphs are equal. In order to use the categories-without-objects formulation, it is essential that we do not identify functions with their graphs. (The fact that the domain of a function is recoverable from its graph is an accident of our demand that functions be total; if we worked with partial functions instead then the situation is more symmetrical and neither domain nor codomain are recoverable from the graph.)
Nov
19
comment Category theory without codomains?
Also let me quote Mac Lane (1988) on the topic: "At that time, a homomorphism in algebra always meant a surjective homomorphism (a mapping onto). Now homomorphisms also arise for homology groups of spaces; in such cases they are not necessarily onto – the familiar map $x \mapsto e^{2 \pi i x}$ of the real line to the circle is onto the circle, but the induced homomorphism in homology is not onto. [...] The problems forced on us the consideration of homomorphisms (and other maps) which are not necessarily surjective or injective."
Nov
19
comment Category theory without codomains?
This is a terrible idea, as it makes composition very difficult, not to mention it renders useless the notion of cokernel. To be entirely frank: if you think maps should always be surjective, then you have not done enough mathematics.
Nov
19
comment When and why do products preserve pushouts?
Any non-distributive lattice (e.g. the lattice of vector subspaces of a non-trivial vector space) will give a counterexample.
Nov
19
comment When and why do products preserve pushouts?
You are asking when the functor $(-) \times D$ preserves pushouts. One situation where this happens is when $(-) \times D$ has a right adjoint: this happens, for example, in $\textbf{Set}$, or more generally in any cartesian closed category. Unfortunately, $\textbf{Top}$ is not such a category.
Nov
18
comment “Good” closure conditions
If your theory has any existential axioms then it becomes much harder for the intersection of submodels to still be a submodel. If your axioms are all of the form "for all [...] there exists unique [...] such that [...]" (more precisely, if you have a "cartesian" theory in the sense of Johnstone, [Sketches of an elephant, Part D]) then the intersection of submodels should still be a submodel.
Nov
17
comment Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
You should ask a new question and state all the hypotheses clearly there – or even the concrete example you are interested in.
Nov
17
comment Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
That's a very different question, then, because it boils down to a question of what colimits preserve monomorphisms in $\textbf{Set}$.