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Dec
6
comment Catsters Video Question
Of course adjunctions have a direction (but it is a matter of convention as to what that direction is). Anyway, $1$ means identity. (I avoid this notation.)
Dec
6
comment About mapping cone complex
Yes. The mapping cone induces a long exact sequence and you can use the five lemma. There are some details in the introduction of Weibel's book on homological algebra.
Dec
5
comment About mapping cone complex
The mapping cone of chain complexes is something that can be defined purely algebraically. It is analogous to the mapping cone of spaces but there is no reason to define one in terms of the other.
Dec
4
comment 2-natural equivalences
Well, consider the case where $\mathfrak{D}$ is trivial. Then you are asking whether it is possible to put a 2-categorical structure on $\mathcal{X}$ such that the equivalences (in the sense of 2-category theory) are the weak equivalences. This is not possible in general, because weak equivalences can fail to have quasi-inverses.
Dec
3
comment Does the Coend have a right adjoint?
$H (c', c) (c'', c''') = \mathcal{C} (c', c'') \times \mathcal{C} (c''', c)$, so $\int^{c'' : \mathcal{C}} H (c', c) (c'', c'') \cong \mathcal{C} (c', c)$.
Dec
3
comment A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?
Have a look here.
Dec
3
comment Equivalence of Modules
I really don't think the analogy works like that.
Dec
2
comment Associativity of arrow composition counter example?
Well, isn't that why it's an axiom? Or as you asking why it needs to be required? I'm sure you can easily concoct artificial examples yourself.
Dec
2
comment Unwinding descent via Barr-Beck
If I could be bothered to write down all the details in the answer box... the link won't stick around forever, after all. Also, why do algebraic geometers keep saying "Barr–Beck"? Barr himself attributes the theorem to Beck (e.g. "Beck's precise tripleability theorem" here).
Dec
2
comment Unwinding descent via Barr-Beck
It's not completely trivial to go from one description to another. See here.
Dec
2
comment Pullbacks and pushouts in the category of graphs
Consider $\{ \bullet - \bullet \} \leftarrow \{ \bullet \quad \bullet \} \rightarrow \{ \bullet \}$.
Dec
2
comment Representable open immersion of functors is a monomorphism
By hypothesis, $h_T \times_G F$ is (the representable functor associated with) an open subscheme of $T$, but $\mathrm{id}_{h_T} : h_T \to h_T$ factors through the projection $h_T \times_G F \to h_T$, so in fact this open subscheme is $T$ itself.
Dec
1
comment Pullbacks and pushouts in the category of graphs
Hmmm. So your graphs are irreflexive. Then pushouts don't always exist.
Dec
1
comment Representable open immersion of functors is a monomorphism
Your first question is answered by the second: I use the hypothesis on $p$ to show that the diagram is a pullback square.
Dec
1
comment Pullbacks and pushouts in the category of graphs
It's more like sets than modules.
Dec
1
comment Understanding the stack $B\mathbb{Z}$
I would guess that quasicoherent sheaves on $B G$ for any discrete group $G$ is just $G$-modules.
Dec
1
comment Pullbacks and pushouts in the category of graphs
Yes. Just construct them the only way you can.
Dec
1
comment Inaccessible Cardinals and Grothendieck Universes
It suffices to show that $\left| U \right| = U \cap \mathbf{On}$ is an inaccessible cardinal for every Grothendieck universe $U$.
Dec
1
comment Basic misunderstanding of the theorema egregium
To the extent that it is possible to measure infinitesimal distances, sure.
Nov
30
comment Something I don't understand about Hilbert's grand hotel
You don't have to enumerate the whole set. It suffices to choose a countable infinite subset and an enumeration thereof.