bio | website | dpmms.cam.ac.uk/~zll22 |
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age | ||
visits | member for | 4 years |
seen | 25 mins ago | |
stats | profile views | 2,924 |
Dec 17 |
awarded | Yearling |
Dec 13 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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unbounded derived category of a $\infty$-topos
Surely you mean "a chain complex of abelian group objects in sheaves of sets"? |
Dec 4 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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Infinite Fubini rule for co/limits
No, you can't just do transfinite composition here. That would land in some other category. |