4,851 reputation
11326
bio website dpmms.cam.ac.uk/~zll22
location
age
visits member for 4 years
seen 25 mins ago

Dec
17
awarded  Yearling
Dec
13
comment Can one make a category concrete by “enlarging the universe”?
In fact, $\mathbf{Set}^\mathrm{op}$ and $\mathbf{Rel}$ are concrete. First, note that the former embeds in the latter in an obvious way. Then note that $\mathbf{Rel}$ can be embedded in $\mathbf{Set}$ by sending each set $X$ to $2^X$.
Dec
10
comment Relation between $BG$ in topology and in algebraic geometry
It's not obvious to me that the model structure on simplicial presheaves restricts to a model structure on simplicial schemes. How does one verify the factorisation axiom?
Dec
9
answered Cartesian product of small objects
Dec
9
comment Cartesian product of small objects
I don't think it's true, but I don't have a counterexample to hand. However, it is not true that the terminal object must be $\lambda$-presentable. For example, take $\mathbf{Set}^I$ where $I$ is any set; this is locally finitely presentable, but the terminal object is finitely presentable if and only if $I$ is a finite set.
Dec
9
comment Morphism on schemes induced by continuous morphism of sites
It is somewhat puzzling that the author does not comment on the choice of morphisms here. As far as I know it is conventional to ask for the relevant triangles of geometric morphisms to commute up to isomorphism. But I think one really has to go with the lax version here.
Dec
9
comment Morphism on schemes induced by continuous morphism of sites
@DavidCarchedi I am not familiar with this paper. I suppose you refer to Theorem 47? It appears to me that the structure sheaf is being carried around, albeit disguised as a geometric morphism to the base topos, which is the big étale topos (i.e. the classifying topos for strictly henselian rings).
Dec
7
comment Morphism on schemes induced by continuous morphism of sites
You will have to be more specific. For instance, let $X$ be $\operatorname{Spec} k$ and let $Y$ be $\operatorname{Spec} K$ for some algebraically closed transcendental extension of $k$. Then the corresponding small étale toposes are equivalent, but there is no morphism $X \to Y$.
Dec
6
comment Clarification about Joyal's notation
I would guess, based on the statement of Theorem C, that $A_0$ is simply the set of vertices of $A$. If you prefer, you can think of it as a discrete simplicial set.
Dec
6
comment unbounded derived category of a $\infty$-topos
Surely you mean "a chain complex of abelian group objects in sheaves of sets"?
Dec
4
comment Small objects vs Compact objects
I would think so. But that claim is even harder to disprove because it is vacuous in a locally presentable category.
Dec
3
comment Small objects vs Compact objects
It would be good to know if $\kappa$-smallness and $\kappa$-compactness really are different. But given that $\aleph_1 \not\triangleleft \aleph_{\omega + 1}$ is basically the smallest non-example of $\kappa \triangleleft \lambda$, it's probably going to be quite intricate...
Dec
3
answered Small objects vs Compact objects
Nov
27
comment Localizations or quotients of categories?
In your specific example, $\mathcal{K}$ is actually isomorphic to the localisation of $\mathbf{Ch}(\mathcal{A})$ with respect to chain homotopy equivalences. This is true more generally when you have suitable cylinder objects or path objects.
Nov
26
comment Functoriality of the adjoint functor construction?
What does "2-functorial in $\mathcal{A}$" mean? Do you mean a 2-functor $\mathfrak{Cat} \to \mathfrak{Cat}_{/ \mathcal{A}}$, where $\mathfrak{Cat}_{/ \mathcal{A}}$ is the strict slice 2-category? Or some other variation?
Nov
11
comment Infinity category of functors from a relative category to a model category
The special case $\mathcal{M} = \mathbf{sSet}$ is addressed implicitly in [Dwyer and Kan, Equivalences between homotopy theories of diagrams].
Nov
4
comment The real numbers object in Sh(Top)
Don't we need it to deduce that $\mathbf{Geom}(\mathbf{Sh}(Y), \mathbf{Sh}(\mathbb{R}))$ is equivalent to the set $\mathbf{T}(Y, \mathbb{R})$?
Nov
4
answered The real numbers object in Sh(Top)
Nov
3
comment Infinite Fubini rule for co/limits
It's really easy: if you have $f_n : X_n \to X_{n+1}$ then the transfinite composition $\cdots \circ f_2 \circ f_1 \circ f_0$ lands in $\varinjlim_n X_n$.
Nov
3
comment Infinite Fubini rule for co/limits
No, you can't just do transfinite composition here. That would land in some other category.