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1d
comment Basic localizers contain adjoint functors
As I said, you can deduce it from the 2-out-of-6 property; but then you have to prove the 2-out-of-6 property. If you want to know why Cisinski called it a corollary, perhaps you should ask him directly.
2d
answered Basic localizers contain adjoint functors
2d
comment Show that Q, as a Z module, is a direct summand in a direct product of copies of Q/Z.
Who says that a homomorphism $\prod A \to B$ has to factor through a finite number of projections?
2d
comment $f_*(O_X)=O_Y$ and connectedness of fibers
Well, then take $X = Y \times_k \mathbb{A}^1_k$, say.
2d
comment $f_*(O_X)=O_Y$ and connectedness of fibers
Last question: No chance. Take $Y$ to be the point and $X$ any connected scheme/variety.
2d
answered Canonically isomorphic but not equal
Apr
17
comment Arrow between endofunctors over a symmetric monoidal category.
There seems to be some confusion here. An arrow between two categories is usually a functor. A natural transformation is an arrow between functors.
Apr
17
comment First and Second Fundamental Form Intuition
The coefficients depend on the choice of coordinate system. I am inclined to say that means they have no geometric meaning.
Apr
17
comment exercise involving exactness
The definition of exactness says the kernel of $A \to B$ vanishes. Show this implies that $A \to B$ is a monomorphism.
Apr
16
comment Axiomatizing topology through continuous maps
What you propose is, essentially, to study the category of topological spaces as an abstract category. What do you hope to achieve by doing this?
Apr
16
comment Topological groups, why need them?
I think one needs some assumption on $X$ in order to make $\mathrm{Homeo} (X)$ a topological group – the compact–open topology doesn't always have all the expected properties!
Apr
14
comment Finding a formula for a $C^{\infty}$ 1-form $\omega$.
Use the chain rule for differentials: $d x^j = \sum \frac{\partial x^j}{\partial y^i} d y^i$.
Apr
14
comment Topological construct
It's not that simple. The left adjoint has a universal property with respect to all $A$, here we only want a universal property with respect to the given $A$ (or rather, the given source).
Apr
12
comment A map $f: X\rightarrow Y$ is a homotopy equivalence if and only if $h\circ f,f\circ k$ are homotopy equivalences of $X,Y$ respectively.
It's probably easiest to introduce the homotopy category, so as to avoid having to keep track of individual homotopies.
Apr
12
comment Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$
Linearly independent over $\mathbb{C}$, or some other ring?
Apr
11
comment Derivatives on Functors
First of all you should tell us what you expect a derivative to behave like!
Apr
11
answered If the functor on presheaf categories given by precomposition by F is ff, is F full? faithful?
Apr
10
comment Combinatorial definition of the homotopy groups of a quasi category?
The fundamental group of a the nerve of a monoid is, if I'm not mistaken, precisely the group obtained by freely adjoining inverses.
Apr
10
comment Combinatorial definition of the homotopy groups of a quasi category?
Why not refer to "weak Kan complexes" as quasicategories? In which case it becomes clear that defining the fundamental group will at least include the process of making a group out of a monoid.
Apr
9
comment What does 'real-valued' function mean in topology?
Yes, it means exactly that. What else could it mean?