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Nov
11
comment Do pushouts exist in a cartesian closed category?
@MaliceVidrine Of course completeness is not automatic. Look at $\mathbf{FinSet}$.
Nov
11
comment Help in a proof in basic Algebraic Geometry
Well, it's true for all $g$. So it's true for $s_i$ as well.
Nov
10
comment Formulas in a Field and in a Field Extension.
Well, (1) involves a non-linear equation but (2) does not. This heuristic is not foolproof but is a good starting point.
Nov
10
comment Is there anything to be learned from the spectrum of a cohomology ring?
$H^* (X)$ is not always a commutative ring, so how do you propose to apply $\operatorname{Spec}$ or $\operatorname{Proj}$ to it?
Nov
10
comment Can we define the homology of the homology chain complex
Of course you can make any sequence of modules into a chain complex: set $d = 0$.
Nov
9
comment Is the restriction map of structure sheaf on an irreducible scheme injective?
Except if $U = \emptyset$...
Nov
9
comment Is the subobject classifier of the sheaves the sheaffication of the one from the presheaves?
It doesn't have any particular significance.
Nov
8
comment Meaning of “same underlying topological space”
Can you give the quotation in full?
Nov
8
comment Is the category of topological spaces coregular?
There's no chance of $\mathbf{Top}$ being cocartesian coclosed: if it were, then we would have $X + (Y \times Z) \cong (X + Y) \times (X + Z)$, which is not even true for numbers.
Nov
8
comment Question about Lemma D1.4.4(iii) in the Elephant - possible typo?
Adding $\bar{y}$ to the context works.
Nov
6
comment Easy characterization of Cohomology in an Abelian Category
I suppose you are asking about whether the two possible definitions of (co)homology object coincide. They do, even in semi-abelian categories: see Proposition 2.3 in [Everaert and van der Linden, Baer invariants in semi-abelian categories II].
Nov
6
comment W-types and inverse image functor
Initial algebras for the simplest polynomial endofunctors (i.e. those of the form $X \mapsto A_0 + A_1 \times X + A_2 \times X^2 + \cdots + A_n \times X^n$) can be constructed using only colimits and finite products, so they are preserved by any inverse image functor. But I imagine there are difficulties for the general case, because inverse image functors don't usually preserve exponential objects – so it's not even clear whether algebras get mapped to algebras.
Nov
6
comment Pullbacks and pushouts
These have very little to do with pullbacks in the sense of category theory and nothing at all to do with pushouts. For one thing, $\phi_*$ is called the pushforward.
Nov
6
comment 2 out of 3 axiom and simplicial sets
There's no reason to expect anything like that. Model structures are surprisingly difficult to construct.
Nov
6
comment A question about pre-additive category
The formal dual of the proof given here works.
Nov
6
comment Natural transformation defined by one element
But every endofunctor on $\mathbf{Set}$ is strong, and every natural transformation between endofunctors on $\mathbf{Set}$ is strong.
Nov
5
comment Hilbert Basis Theorem applied to integral domains
You really don't need the Hilbert basis theorem for this. For one thing, $A$ may not be noetherian.
Nov
5
comment Functors that are the homology of a chain complex
Hmmm. You could do it that way. But then it's not obvious that the set of homotopy classes of maps $X \to K (G, n)$ has a group structure, let alone an abelian group structure when $G$ is abelian. And it's not obvious that any $K (G, n)$ spaces exist at all: after all, we know it's impossible to have a $K (G, 2)$ space if $G$ is non-abelian. All in all, it seems like a bad definition!
Nov
5
comment Functors that are the homology of a chain complex
How do you define singular homology/cohomology if not as the homology/cohomology of the singular complex?
Nov
5
comment Natural transformation defined by one element
You should ask the natural transformation to be a strong natural transformation.