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Jan
9
comment Why are adjoints usually defined in terms of hom-sets?
Would you rather define adjoints in terms of triangle identities?
Jan
8
comment Prove that theory is not Henkin one
What about, say, the formula $x = y$?
Jan
8
comment Locally presentable sheaves and the associated module functor
Surely we can take $U = X$?
Jan
7
comment How to show that $\mathbb Q/\mathbb Z$ is not a $\mathbb Q-$module?
Can a $\mathbb{Q}$-module have torsion? Does $\mathbb{Q} / \mathbb{Z}$ have torsion?
Jan
7
comment Are there formal criteria for morphisms of stacks?
Well, the point is that there can be more than one isomorphism between two objects, and you should specify which isomorphisms are being used.
Jan
7
comment Are there formal criteria for morphisms of stacks?
Is there a pre-existing definition? I would guess that your phrasing is not correct on the basis that it treats isomorphism as a property rather than a structure. Surely the correct definition would imply that $X (A) \to Y (A) \times_{Y (B)} X (B)$ is, say, an equivalence of groupoids when $X \to Y$ is formally étale.
Jan
6
comment Differential topology and/or geometry from the viewpoint of Cauchy completions
Someone else asked essentially the same question.
Jan
6
comment What is this categorical notion called?
Actually, you don't even get a preordered set – there can be more than monomorphism between any two objects...
Jan
5
comment Adjunction of two functors. If the right adjoint functor is linear, then the left adjoint functor is also linear.
Actually, any left or right adjoint functor between additive categories is additive.
Jan
5
comment $\mathcal{V}$-naturality in enriched category theory
Kelly uses "natural" for a more general notion, sometimes called extranatural or dinatural.
Jan
4
comment Is it ever convenient to mod out reparameterizations in a category of spaces?
If $f \circ \phi = g$ and $f' \circ \psi = g'$ then $g' \circ g = f' \circ \psi \circ f \circ \phi$, but what you want is $g' \circ g = f' \circ f \circ \theta$ for some $\theta$.
Jan
4
comment Is it ever convenient to mod out reparameterizations in a category of spaces?
Is this a congruence? That is to say, is composition of equivalence classes well defined?
Jan
2
comment Functor of section over U is left-exact
The sections functor doesn't preserve images.
Jan
2
comment What does it mean for a subfunctor to be “defined by an open subscheme?”
That is exactly it.
Dec
31
comment Proving that $(Tor_n(\_\ ,N))_n$ is a universal homological $\delta$ functor
You could use Grothendieck's theorem on effaceable functors instead of trying to directly show that $\mathrm{Tor}$ is universal.
Dec
30
comment Applications of small object argument outside model categories.
Not especially. It's quite easy once you understand the problem.
Dec
30
comment On the join of simplicial sets as a dependent product
Use the Yoneda lemma and compute. It's exactly what Joyal says.
Dec
30
comment On the join of simplicial sets as a dependent product
Your intuition about $i_*$ is wrong. The first part of the proof tells you how to compute $i_*$ – it's a lot more complicated than you think.
Dec
29
comment Quillen equivalence vs $\infty$-categorical equivalence
That's more difficult. If you have a locally presentable $(\infty, 1)$-category then there are some things that can be said.
Dec
28
comment Quillen equivalence vs $\infty$-categorical equivalence
Two model structures on the same category yield the same $(\infty, 1)$-category if and only if they have the same weak equivalences.