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2h
comment Maps between direct limits and functoriality of $f^{-1}:Shv(Y) \rightarrow Shv(X)$ and $f_{*}:Shv(X) \rightarrow Shv(Y)$
"Natural" here is meant in the sense of category theory. You should look up adjoint functors.
2h
comment How much set theory does the category of sets remember?
Well, another way to phrase the question to ask which properties of $M$ can be detected in $\mathbf{Set}_M$. One of them is the axiom of choice, so we can't have $M$ and $L^M$ related in that way in general. The axiom of global choice is more subtle though, I think, and it would be interesting to know if global choice for $M$ can be detected in $\mathbf{Set}_M$.
8h
comment Being contractible in homotopy theory vs. homotopy type theory
The machinery needed to convert topological spaces into actual homotopy types does not yet exist in homotopy type theory (due to some technical difficulties).
9h
comment Being contractible in homotopy theory vs. homotopy type theory
The *****set***** $[0, 1]$ is not contractible, of course.
2d
comment Resemblance between product and homotopy
The only way $X \times X$ can be the path object is if the diagonal $\Delta : X \to X \times X$ is a trivial fibration, which would imply (if $X$ is fibrant) that the unique morphism $X \to 1$ is a homotopy monomorphism. I am inclined to conclude that there are no interesting examples.
2d
comment Direct limit with non-injective maps
Well, you can always look at the boring direct system $A \to A \to A \to A \to \cdots$ where $A$ is the direct limit of your original direct system.
2d
comment Strong (trivial) cofibration in Lurie's HTT
The word "strong" does not appear in the version I have. Can you quote the relevant part in full?
2d
comment Gluing sheaves together
It can be done without the axiom of choice, but then the construction becomes much more elaborate.
2d
comment Gluing sheaves together
Choose one or the other arbitrarily and remember your choice.
Oct
28
comment basic question regarding the definition of sheaf of rings
The real point to make, though, is that the data of a sheaf also includes the restriction maps, so the object part really has to be defined on the nose.
Oct
28
comment Functoriality of fundamental group
I'm not sure why you say that $G$ is supposed to land in abelian groups. I think it is reasonable to rule out $G (g) = 0$ axiomatically: after all, for any choice of basepoint $c$ in $C$, $\pi_1 (g) : \pi_1 (C, c) \to \pi_1 (K, c)$ is non-zero.
Oct
28
comment Resemblance between product and homotopy
There is nothing interesting to say here, except perhaps to notice that $X \times X \cong X^2$.
Oct
26
comment presentable vs inductive categories
You have not understood the definition of $\kappa$-filtered (quasi)categories. If you are not familiar with the ordinary notion, look at that first before looking at the $(\infty, 1)$-version.
Oct
26
comment presentable vs inductive categories
You seem to be confused. $\kappa$-filtered diagrams are not necessarily $\kappa$-small. Even in the case where $\mathcal{D}$ has $\kappa$-filtered colimits $\mathcal{D} \to \mathrm{Ind}^\kappa (\mathcal{D})$ is rarely an equivalence.
Oct
26
comment Is a subobject classifier logically equivalent to set-inclusion?
The choice of an injection carries more information than merely knowing there is an injection. There is no need to bring subobject classifiers into this discussion.
Oct
26
comment What categorical limits and colimits does $\pi_1$ preserve?
However, the relation between (homotopy) colimits in $\mathbf{Grpd}$ and colimits in $\mathbf{Grp}$ is not so clear. (For instance, think about the groupoid version of the pushout of $\partial \Delta^1 \hookrightarrow \Delta^1$ along itself.)
Oct
26
comment Are adjoint functors between additive categories additive?
Moreover the adjunction itself is additive, in the sense that one has an enriched natural isomorphism of hom-groups $\mathrm{Hom} (F X, Y) \cong \mathrm{Hom} (X, G Y)$.
Oct
25
comment Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?
This is a standard transport-of-structure argument. I'm sure it has been asked before.
Oct
24
comment Words in the Category of Sets
But then you may as well work with a Lawvere theory?
Oct
24
comment Words in the Category of Sets
As I understand it, there is no operad whose algebras are groups.