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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.
Nov
20
comment Colimits in full subcategory (of all monics) of arrow category
That's one way of seeing it, yes.
Nov
19
comment Why are Truth, Conjunction, and Implication called “Negative” fragments of IPL (intuitionisti logic Proposition Logic)?
At a guess, I'd say because $\top$, $\land$, and $\to$ are all coinductive (whereas $\bot$ and $\lor$ are inductive).
Nov
19
comment name this “hybrid” categorical construction
The lax pullback is the lax limit of the obvious diagram. It is not equivalent to the comma category. On the other hand, the pseudopullback (= pseudolimit of the obvious diagram) is equivalent to the iso-comma category. There is no substantial difference between the pseudopullback (also called "homotopy pullback", but I prefer to avoid that phrase in this context) and the ordinary pullback if one of the functors is an isofibration, which is the case in the OP's specific example.
Nov
19
comment Coordinate-free definition of integration of differential forms?
Ah, right. You only want $\int \omega = \int f^* \omega$, which is the case.
Nov
18
comment Coordinate-free definition of integration of differential forms?
Well, if by "orientation" you mean a chosen $n$-form $\omega$ and if by "orientation-preserving" you mean $f^* \omega = \omega$, then yes, we have naturality. But this is a rather boring kind of naturality, in some sense...
Nov
18
comment name this “hybrid” categorical construction
In your specific example, $A^*$ is the same as the pullback in the usual sense.
Nov
18
comment Coordinate-free definition of integration of differential forms?
I'm not convinced. Actually, the failure of diffeomorphism-invariance implies there must be some cheating going on – it certainly cannot be natural in the sense of category theory.
Nov
17
comment Dedekind(?) representation lemma on posets?
Perhaps it is named for Dedekind in recognition of his work on Dedekind cuts?
Nov
17
comment Examples of algebro-geometric moduli problems without a “natural” choice of pullback?
The fibre product of schemes is well defined up to unique isomorphism. If you are so inclined, you might say that there is a contractible groupoid of choices of fibre product. But that is not the reason why stacks show up; that has to do with non-trivial automorphisms.
Nov
17
answered Colimits in full subcategory (of all monics) of arrow category
Nov
16
comment Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?
Let me be clear: you can compute it as a limit for a strictly commutative diagram of the same shape, but the vertices and arrows are different.
Nov
16
comment Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?
Yes, but that involves many technical details. (In the first place, what we have is not a commutative diagram but rather a pseudocommutative diagram.)
Nov
16
answered underlying set of direct limit not the direct limit of underlying sets
Nov
16
revised underlying set of direct limit not the direct limit of underlying sets
edited tags
Nov
15
comment Coordinate-free definition of integration of differential forms?
And we can also not have equality of the obvious volume-forms. For example, let $x'^k = 2 x^k$, then $\mathrm{d} x'^1 \wedge \cdots \wedge \mathrm{d} x'^n = 2^n \, \mathrm{d} x^1 \wedge \cdots \wedge \mathrm{d} x^n$. In particular, this shows that it is not diffeomorphism-invariant. (It is isometry-invariant, of course.)
Nov
15
comment Coordinate-free definition of integration of differential forms?
But how would you distinguish the $n$-form $\mathrm{d} x^1 \wedge \cdots \wedge \mathrm{d} x^n$ from $2 \, \mathrm{d} x^1 \wedge \cdots \wedge \mathrm{d} x^n$?
Nov
15
comment Coordinate-free definition of integration of differential forms?
Why do you believe it should be definable without coordinates? For convenience, suppose $M$ is compact. Consider linear operators from the space of $n$-forms to $\mathbb{R}$. Of course, $\int_M$ is one such operator; but so is $2 \int_M$. How would you distinguish between the two?