4,412 reputation
11126
bio website dpmms.cam.ac.uk/~zll22
location
age
visits member for 3 years, 7 months
seen 4 hours ago

Jul
20
comment Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure
Alternatively, one notes that every object is "canonically cofibrant", so the model structure (if it exists) must be left proper and so pushouts of cofibrations "must" be homotopy pushouts.
Jul
19
comment Is a finite-dimensional simplicial set the homotopy colimit over a truncated simplicial category?
Taking homotopy colimits is also a left adjoint (at least, if you work at the level of $(\infty, 1)$-categories). Let $p$ be the unique functor $\Delta \to \mathbb{1}$. Then $\operatorname{colim} \cong p_!$, so $\operatorname{colim} \operatorname{sk}_n \cong p_! i_! i^* \cong \operatorname{colim} i^*$.
Jul
19
comment Is a finite-dimensional simplicial set the homotopy colimit over a truncated simplicial category?
Yes. The point is that left adjoints compose, and $\mathrm{sk}_n \cong i_! i^*$.
Jul
17
comment Do non-subcanonical Grothendieck topologies always induce a category of fractions?
If the underlying category of the site is a groupoid then this always fails. (For example, take a non-trivial group considered as a groupoid; then the Grothendieck topology generated by the empty sieve is not subcanonical.)
Jul
17
comment Proof correctness problem
It might be amusing to look at these slides from a more recent talk of Voevodsky's – he gives some examples.
Jul
16
comment $i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism
This question already has an answer on math.SE.
Jul
16
answered How should I be thinking about object classifiers / universal fibrations / universes?
Jul
15
comment Pushouts of equivalences of categories
In view of the fact that pushouts along "h-cofibrations" or "flat morphisms" are already homotopy pushouts, why not call them "homotopically coquadrable"?
Jul
14
comment Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?
In other words, if you have a simplicial model category then you will be fine. But in general it would seem that more work is needed.
Jul
13
comment Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
I suppose you mean spaces with trivial $\pi_n$ for $n > 1$? Then yes, because 1-groupoids are a reflective $(\infty, 1)$-subcategory of $\infty$-groupoids.
Jul
13
comment Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
@ColinTan Yes, but not all pushouts are homotopy pushouts.
Jul
12
answered Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
Jul
12
comment Comonads from monoids
We can regard $\mathcal{C}^\mathrm{op}$ as a $\mathcal{C}$-enriched category with tensors (or category with a left $\mathcal{C}$-action); then this is a special case of the monad associated with a monoid. (A monad on $\mathcal{C}^\mathrm{op}$ is the same thing as a comonad on $\mathcal{C}$.)
Jul
12
comment What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
What's wrong with writing $x$ for a projection? I suppose you might prefer $x, y \vdash x$, but contexts are supposed to be, well, contextual.
Jul
8
comment Does simplicial localization with a 3-arrow calculus commute with functor categories?
If the answer is yes, it would appear to say that homotopy-coherent diagrams can be rectified to strict diagrams. (I previously asked a question along those lines.) In particular, if this is not possible for the given $(C, W)$ and $D$, then one cannot have $L (C^D, W^D) \simeq L(C, W)^D$.
Jul
6
comment Retracts of 2-categories
You probably don't need the whole 2-category to be freely generated; it's probably enough that the subcategory to be "collapsed" is obtained by a nice pushout. You might want to look at [Dwyer and Kan, Simplicial localizations of categories] for inspiration.
Jul
3
comment When does a sheaf of categories represent a homotopy sheaf?
Ah, I see. I thought you were talking about the other kind of descent condition. So the problem is to compute some homotopy limits... I don't know of any theorems in that direction.
Jul
3
comment When do localizations of presentable (infinity) categories commute?
One sufficient condition is to have $S \subseteq R$. No?
Jul
3
comment When does a sheaf of categories represent a homotopy sheaf?
Have you looked at the old Joyal–Tierney paper on "strong stacks"? There they describe strict sheaves of groupoids but I am informed it also extends to strict sheaves of categories.
Jul
2
awarded  Curious