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Jun
18
comment Sub(P) is complete?
That's automatic if you define subfunctors the way I said.
Jun
17
comment Sub(P) is complete?
The relevant definition of subfunctor here is the one where $S (C)$ is literally a subset of $P (C)$ and $s : S (C) \to P (C)$ is the inclusion.
Jun
16
comment short exact sequences of complexes and triangles in the homotopy category
A short exact sequence of complexes goes to a distinguished triangle in the derived category, however.
Jun
13
comment $S$ subring of $R$. Is a projective objects in $R$-$\bmod$ still projective in $S$-$\bmod$?
Well, what about some easy examples, like $\mathbb{Z} \subset \mathbb{Q}$?
Jun
12
comment properties of pullback diagrams
This is called the pullback (pasting) lemma.
Jun
11
comment What is the correct analogue of $\mathbb N$ in a ring of integers?
The obvious map $n \mapsto (n)$ is a bijection between $\mathbb{N}$ and the set of ideals of $\mathbb{Z}$. So perhaps rather than elements one should consider the set of ideals.
Jun
11
comment Points of Weil restriction
The bijection is part of the data of the Weil restriction.
Jun
10
comment Intersection of affine open subschemes
The intersections are always separated: any open subscheme of a separated scheme is separated, after all.
Jun
9
comment Definition of a Cartesian Closed Category
You've written down a bunch of symbols without indicating what they mean. This is hardly a definition.
Jun
9
comment Flatness over tensor product
What if $A = B = M$ is a non-trivial field extension of $k$?
Jun
9
comment Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?
@tcamps The homomorphisms of models of your sketch will have to preserve the $\{ 0 \} \amalg Y$ decomposition, so in particular they will be injective. So you are talking about two different categories of torsionfree abelian groups here.
Jun
8
comment What is a Monad in the two category $Rel$?
Of course, that only says what happens on the level of objects. The next question one should ask is, what is a morphism of monads?
Jun
8
comment Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$
It would be a bit harder to get a counterexample for the étale case: given morphisms $f : X \to Y$ and $g : Y \to Z$ (of schemes), assuming $g : Y \to Z$ is étale, $g \circ f : X \to Z$ is étale if and only if $f : X \to Y$ is étale. So any counterexample would need to have $g : Y \to Z$ not étale.
Jun
6
comment Indices at the left of a tensor in mathematical physics/differential geometry?
Incidentally, the parentheses have a meaning here: they indicate symmetrisation.
Jun
5
comment canonical map of a monoid to its classifying space
It is essentially what Martin Brandenburg mentions in his answer. Have you understood that?
Jun
5
comment canonical map of a monoid to its classifying space
Giving a continuous map $f : M \to \Omega B M$ is the same thing as giving a continuous map $g : M \times S^1 \to B M$ that is basepoint-preserving in the second variable. If you build $B M$ using the bar construction, there is an obvious such map.
Jun
5
comment Properties preserved by fppf morphisms
Did you look at the descent chapter in the Stacks project? You are basically asking about whether something or other is a property that is fppf-local on the source.
Jun
4
comment Sheafification as a Kan Extension of the Identity?
Any left (resp. right) adjoint is a right (resp. left) Kan extension of the identity.
Jun
4
comment Unit and co-unit of Exponential Product and Sets
Describe them in terms of what? $\mathcal{C}$ could be a completely abstract category here.
Jun
4
comment Homotopy colimits preserve weak equivalences
It really depends on what you mean by homotopy colimit.