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Oct
14
comment Classification of transitive G-sets for a given group of small order
It is true though: every transitive $G$-set is isomorphic to the coset space $G / H$ for some subgroup $H$.
Oct
14
comment What is required for a monoidal category to have products/coproducts?
There are no reasonable conditions, because any category with products or coproducts is a monoidal category.
Oct
13
comment Repeating the Yoneda embedding
False. There are various issues here that prevent this operation from being idempotent.
Oct
12
comment Infinite coproduct of abelian groups
It exists. It is the infinite direct sum, of course.
Oct
11
comment Where to get help with Homotopy type theory?
You will not learn about higher groupoids by studying algebraic topology or category theory, at least not at the introductory level. Rather, I advise you to just ignore the phrase entirely and replace it with ‘space’ – that, after all, is the content of the homotopy hypothesis.
Oct
10
comment One-point-functors and the Yoneda Lemma
What do you mean by "recovered"? The Yoneda embedding is fully faithful and so conservative in particular. Therefore $X$ is unique up to isomorphism, and even up to equality if you know how to recognise $\mathrm{id}_X$.
Oct
10
comment The morphism of rings inducing the diagonal morphism.
Well, the ring morphism $R \otimes R \to R$ you describe is defined by the dual universal property.
Oct
9
comment Right exact functors are tensor products
$F (A)$ is an $A$-module because $A$ acts on itself and $F$ is an additive functor. (Consider the maps $A \to A$ defined by $x \mapsto a x$ for a fixed element $a$.) I already explained how $F$ is determined by $F (A)$.
Oct
8
comment Difference between colon and membership symbol?
@TobiasKildetoft Depends on the school of thought. $\exists x \in \mathbb{N}$ is strictly speaking an abbreviation in set theory, but $\exists x$ on its own is malformed in type theory.
Oct
8
comment Pullback of maximal ideal in $k[y]$ is not maximal in $k[x]$.
Jacobson rings and the generalised Nullstellensatz are also discussed in Eisenbud's textbook.
Oct
8
comment Zariski tangent space.
The Zariski tangent space is (isomorphic to) the dual space of the Zariski cotangent space.
Oct
7
comment What is the product and coproduct of Morphism category(Arrow category)?
If $\mathcal{C}$ has an initial object and a terminal object, then the arrow category of $\mathcal{C}$ has exactly the same limits and colimits as $\mathcal{C}$, and they are calculated componentwise.
Oct
7
comment Compact Riemann surfaces are projective varieties.
The first statement is a tautology, at least compared to the second statement, which is a remarkable result of Riemann!
Oct
6
comment Factoring of a Pro-$\mathcal{C}$ morphism
Well, take a morphism $A \to B$. Then it has components $A \to B j$ (this is the ${\varprojlim}_j$ part) and each of those factors through some projection (this is the ${\varinjlim}_i$ part).
Oct
6
comment Factoring of a Pro-$\mathcal{C}$ morphism
What is there to explain? I am just using the definition of morphsim in $\mathbf{Pro}(\mathcal{C})$.
Oct
5
comment Meaning of Fundamental group of a graph
Think of a graph as being a collection of points and lines glued together with the induced topology. For instance, you can topologise a planar graph as a subset of the plane. Then it is a topological space and the usual definitions apply.
Oct
5
comment $\textbf{C}$-Monoids and products
I imagine $\mathcal{C}$ has finite products. Show that finite products (or, indeed, any limits that exist) in $\mathbf{Mon}(\mathcal{C})$ are created by the forgetful functor.
Oct
3
comment Is there such a thing as “combinatorial category theory”?
Category theorists tend not to study small categories like that... to the extent that, lately, some have taken to calling the locally presentable categories of Gabriel and Ulmer simply as "presentable categories" – despite the fact that they are "presentable" only in a sense that is very far from the sense in which groups are "presentable"!
Oct
3
comment How are expressions like $(Df)(x)$ treated in categorial set-theories (in a foundational context)?
The trouble with pulling back to a point is that the result is the germ of the function/differential at the point, rather than the value.
Oct
2
comment Prove that this infinite planar map can still be colored with four colors.
@NasuSama What you say is correct. Where is the problem?