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Feb
22
comment Is the Fourier transform a special case of this version of the Yoneda lemma?
Well, for instance, the apparent analogy you bring up.
Feb
22
comment Is the Fourier transform a special case of this version of the Yoneda lemma?
I believe this is nothing more than a (deliberate) coincidence of notation.
Feb
22
comment Motivation for the mapping cone complexes
The mapping cone of a morphism of chain complexes is a homotopy-theoretic generalisation of the cokernel.
Feb
22
comment Precise definition of free group
Well, "word" is just another way of saying "list" or "finite sequence". But there is a small problem with the "definition" quoted because $W (S)$ itself does not become a group – one has to take the quotient by a certain equivalence relation...
Feb
21
comment Eilenberg–Zilber as abstract nonsense - why is it important?
Lax monoidality doesn't quite capture everything about the Eilenberg–Zilber theorem. I think it is much more important to know that the comparison is a quasi-isomorphism.
Feb
21
comment Where are sheaves in the functor of points perspective?
I discuss how to think of $\mathscr{O}_X$-modules as functors here.
Feb
19
comment Will Homotopy Type Theory ever be as accessible as traditional Set Theory?
That's a bit more difficult. There is no single "homotopy type theory" at the moment. One reason is that we don't yet have a sufficiently general mathematical definition of higher inductive type – so instead of postulating that all higher inductive types exist (whatever that might mean), we are forced to postulate each one we want separately. Regardless, there is a basic/core system which you can find described in e.g. the appendix of the HoTT book. But please keep in mind that HoTT is not a first order theory or anything like that.
Feb
19
comment Will Homotopy Type Theory ever be as accessible as traditional Set Theory?
@AlexYoucis Let's distinguish between the model theory of HoTT and HoTT itself. The former uses model categories, the latter does not.
Feb
18
comment A compact Hausdorff space $X$ is finite if and only if $C(X)$ is finite-dimensional
For each $x \in X$, consider the linear map that sends $f$ to $f (x)$.
Feb
18
comment A cartesian closed category is pointed if and only if it has a zero object
Any terminal object in a pointed category is a zero object.
Feb
17
comment Does a subcategory with right adjoint always have limit?
In the case of a full subcategory, yes.
Feb
16
comment What $\bigwedge_{k=1}^{+\infty}$ symbol means?
I would guess that $\bigwedge$ is being used in the sense of infimum here.
Feb
15
comment Tensoring a connective chain complex with a simplicial set
There's a closed monoidal structure on $\mathbf{Ch}_{\ge 0} (R)$ transported from $\mathbf{sMod} (R)$ via Dold–Kan, but it's not the standard one. On the other hand, $\mathbf{sMod} (R)$ is a simplicial model category.
Feb
15
comment Homotopy category of a simplicial category
I suppose it depends on what you mean "homotopy class" but under any reasonable interpretation it should be true. (For example, you might define it as a morphism in the homotopy category.)
Feb
15
comment Converse of realisation lemma for bisimplicial sets
One usually uses degreewise weak equivalences of bisimplicial sets.
Feb
15
comment Is groups with binary operation alone a variety?
A variety in a signature without constants will contain the empty algebra. But the empty set is not a group.
Feb
14
comment equality of two natural transformations and two morphisms
Actually, equality of objects in a category or functors between categories is a contested issue...
Feb
14
comment Is there a geometric meaning associated with the condition “dot product equals $1$?”
There a general trick for embedding affine geometry in linear algebra: replace points $\mathbf{x} = (x_1, \ldots, x_n)$ with vectors $(1, \mathbf{x}) = (1, x_1, \ldots, x_n)$, so that affine transformations become linear transformations. Under this translation, your condition $\mathbf{x} \cdot \mathbf{y} = 1$ translates to $(1, \mathbf{x}) \cdot (1, -\mathbf{y}) = 0$.
Feb
14
comment Is the scheme-theoretic image stable under taking products?
Have you thought about the case where everything is affine? It seems to me that you are asking whether tensoring preserves kernels, or thereabouts.
Feb
14
comment Geometric interpretation of different types of field extensions?
If you want to think about Galois theory in geometric terms then the geometric side is the theory of covering spaces. Although the technical details involve scheme theory, the intuition comes from algebraic topology.