bio | website | dpmms.cam.ac.uk/~zll22 |
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visits | member for | 3 years, 7 months |
seen | 4 hours ago | |
stats | profile views | 2,553 |
Jul 20 |
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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 |
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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 |
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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 |
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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 |
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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 |
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$i^{-1} F$ 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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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When do localizations of presentable (infinity) categories commute?
One sufficient condition is to have $S \subseteq R$. No? |
Jul 3 |
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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 |