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Nov
28
comment What exactly is Standard Coordinates?
"Standard" is being used as an ordinary English word here.
Nov
28
comment Can every basic concept of fundamental group be generalized to homotopy group?
I don't agree. For one thing, the higher homotopy groups are all abelian, but $\pi_1$ is not necesarily abelian (and $\pi_0$ is not even a group). For another, there's the beautiful connection between $\pi_1$ and covering space theory...
Nov
27
comment Uniqueness of the long exact sequence in homology
The connecting morphism is not quite unique. You could replace $\delta_n$ with $-\delta_n$, for example. So there is a genuine ambiguity. Regardless, the long exact sequence (including the connecting morphism) can be defined in a very natural way one you have the notion of mapping cone.
Nov
27
comment Functorial cofibrant replacement does not have to be fibration?
Well, if you are only interested in showing that morphisms in $\operatorname{Ho} \mathcal{M}$ (defined by its universal property) can be factored in the usual way, there is no need to bring in homotopy classes at all.
Nov
27
comment Does the five lemma hold true for Lie algebras?
However, the category of Lie algebras is a semi-abelian category.
Nov
26
comment Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$
Strictly speaking, this is the free strict monoidal category, of course. The free monoidal category has non-trivial unitors and associators.
Nov
25
comment Are there some reference books or handbooks on homology and homotopy groups of every manifold which has been calculated?
Crossposted on MO.
Nov
25
comment tensor product and commutation, category theoretical argument
You could probably use the fact that filtered categories are sifted.
Nov
24
comment non-abelian Galois cohomology
In [Cohomologie non-abélienne, Ch. III §4], Giraud describes an extended exact sequence of the desired form for short exact sequences satisfying a special condition. But I do not know any of the details.
Nov
24
comment What does $C_{-1}(X)$ mean?
If $X$ is a simplicial complex (resp. topological space) and $C_{\bullet} (X)$ is the simplicial (resp. singular) complex, then one typically defines $C_n (X) = 0$ for $n < 0$.
Nov
23
comment what is the name of this operation?
It's called currying.
Nov
22
comment Homotopy equivalence of pushouts of topological spaces
Sorry, I mean this one instead.
Nov
22
comment Homotopy equivalence of pushouts of topological spaces
possible duplicate of Homotopy equivalence of two different gluings of $B^n$ and an arbitrary space $X$
Nov
20
comment Equivalent definitions of regular categories?
It's very simple: under (1) or (2), the pullback of a regular/extremal epimorphism is regular/extremal, so (regular/extremal epi, mono)-factorisations are preserved by pullback, hence images are preserved by pullback.
Nov
20
comment Equivalent definitions of regular categories?
Strange, I think that the obvious direction is that (2) implies (4). At least if you go via (1).
Nov
20
comment Equivalent definitions of regular categories?
Well, isn't (2) just a way of making (4) + cartesian precise?
Nov
20
comment Is $\mathbb{Z}$ the initial rook?
Oh, actually I was thinking of the other absorption axiom, $0 x = 0$, because then there is no restriction on the endomaps.
Nov
20
comment Equivalent definitions of regular categories?
(1) and (2) are equivalent. Every extremal epi is regular: take its (regular epi, mono)-factorisation and observe that the monomorphism part must be an isomorphism.
Nov
20
comment Is $\mathbb{Z}$ the initial rook?
Isn't this the structure you get when you look at the set of endomaps of a group, with $+$ being the pointwise group operation and juxtaposition being composition?
Nov
20
comment Equivalent definitions of regular categories?
(3) is stronger than the others because coequalisers don't have to exist in general. (4) is weaker than the others because finite limits don't have to exist in general.