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bio website dpmms.cam.ac.uk/~zll22
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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.
Nov
3
comment Infinite Fubini rule for co/limits
I think there's more than just a notational problem here. How do you define infinitely iterated colimits?
Nov
2
comment Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects
@QiaochuYuan The trivial monoid with respect to a cartesian monoidal structure is a zero object, yes. But consider for instance $\mathbb{Z}$ as a monoid in $\mathbf{Ab}$...
Oct
28
comment Why do filtered colimits commute with finite limits?
It is true. The usual proof uses the fact that sheafification preserves finite limits as well as filtered colimits.
Oct
23
revised Characterize the category of rings
added 4 characters in body
Oct
23
answered Characterize the category of rings
Oct
21
comment Higher vector spaces
Well, there is a trivial observation: an idempotent-complete category $\mathcal{C}$ is (equivalent to) the category of finitely presented $R$-modules if and only if $\mathbf{Ind}(\mathcal{C})$ is (equivalent to) the category of $R$-modules.
Oct
15
revised $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
added 1 character in body
Oct
15
answered $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
Oct
15
comment $(LLP(Epi), Epi)$ is a WFS on any variety of algebras
$U$ should be only faithful, not fully faithful.
Oct
15
comment Cocomplete but not complete abelian category
This looks like the direct limit of an $\mathbf{On}$-indexed sequence of cocomplete abelian categories along colimit-preserving exact functors, so it should be just abstract nonsense that it is a cocomplete abelian category.
Oct
13
comment Homotopy uniqueness of some coends
Essential uniqueness here is referring to the fact that some classifying space is contractible. I wouldn't read too much into it.
Oct
13
comment Homotopy uniqueness of some coends
But the diagrams in question are indeed cofibrant?
Oct
11
comment Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?
This seems to be a folklore result. It is alluded to in [Higher topos theory, §6.5.3].
Oct
9
comment The Karoubi model structure on Cat
Yes, it is a left Bousfield localisation of the canonical model structure, almost by definition. The generating cofibrations are therefore the same as for the canonical model structure.
Oct
8
comment The Karoubi model structure on Cat
I describe the model structure here. There is just one extra generating trivial cofibration, namely the inclusion of the free idempotent into the free split idempotent.