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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 $\textbf{Set}$.
Nov
17
comment Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
In the above (counter)example $X$ is representable and $Y$ is a subobject of $X$.
Nov
17
comment Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
Yes, that's right.
Nov
17
revised Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
edited body
Nov
17
comment Why do number rings have no endomorphisms
$\operatorname{Spec} \mathbb{Z}$ is affine, so perhaps the affine line $\mathbb{A}^1$ would make a better analogy...
Nov
17
answered Does the Yoneda extension of a functor preserving monomorphisms also preserve monomorphisms?
Nov
17
comment How to show $V(f_{1},f_{2})$ is a finite set?
The claim as stated is false: let $f_1 = f^2$ and $f_2 = f^3$; then $V(f_1, f_2) = V(f)$. You should probably assume $f_1$ and $f_2$ are radical.
Nov
17
comment Exactness of short exact sequences
A slightly less trivial example is $P_1 \oplus 0$ over the ring $A_1 \times A_2$, where $P_1$ is a projective $A_1$-module...
Nov
17
comment What good are free groups?
Have you ever stopped to think what it means to give a presentation of a group by generators and relations? You'll find that you will need the concept of free group in order to make sense of this rigorously.
Nov
16
comment Why can I choose to work in a strict monoidal category without loss of generality?
The coherence theorem can also be interpreted as, roughly speaking, "everything commutes". This is how you justify assuming strictness.
Nov
15
comment Complete but not cocomplete category
I don't see any obvious reason for it to be true, but at the same time it is difficult to come up with a counterexample because the cocomplete abelian categories I know of are Grothendieck categories...
Nov
15
comment How to express inclusion with arrows?
Please formulate exactly what you mean: as it stands your claim does not generalise.
Nov
15
comment How to express inclusion with arrows?
In the category of sets, yes, provided you label morphisms $1 \to X$ by their values. Otherwise false in general.
Nov
14
comment Objects and its morphisms
Please be aware there is a limit on the number of questions you can ask within a period of time. You should spend some time thinking about your own questions before asking them.
Nov
14
comment Example of non-associative composition of morphisms
Of course, we have to take homotopy classes of homotopies in order to have an associative composition for those...
Nov
14
comment Is the formal semantics of first-order logic ambiguous?
Once you agree on what the semantics of the basic logical connectives and quantifiers, you can show that the purely syntactic inference rules only allows you to deduce true statements from other true statements.