Finally split two-sided fibration off of Grothendieck fibration. Thanks to Emily Riehl for adding the definitions here.

]]>So, I was just reading the page on twisted cohomology, and I got really excited about the paragraph (sorry, I don’t know how to make it look “quotey”):

Higher-order approximations should involve a notion of higher-order forms of the tangent $(\infty,1)$-topos, in parallel with the relationship between the jet bundles and tangent bundle of a manifold. It is clear that whatever we may say in detail about the $k$th-jet $(\infty,1)$-topos $J^k\mathbf{H}$, its intrinsic cohomology is a version of twisted cohomology which is in between nonabelian cohomology and stable i.e. generalized (Eilenberg-Steenrod) cohomology.

Is this paragraph explained at greater length anywhere?

I seem to recall some recent work of Ching and Arone on classifying second order approximations to the identity functor on spaces? Does this perhaps clarify what the next level of approximation might be? Or, in other words, what might be considered second order Eilenberg-Steenrod axioms?

I feel like this also goes back to a question of perennial interest of mine: if we think of the category of spectra as the tangent space to the category of spaces, then what’s the “quadratic space”? And what does it mean to complete with respect to this tower?

]]>I am reading the article on Hurewicz connection

Theorem. A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.

I have two questions:

(1) How to formally construct the Hurewicz connection for $\pi_!$ ?

(2). It has scratched an idea of proof. May I know if anyone can reference a completed formal proof to me. (a paper or an article will be great !).

]]>I am reading your article on Dold fibration

I have two questions:

How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$. Please reference some good readings to me on Dold fibration.

Thanks,

]]>I am reading your article on Dold fibration

I have two questions:

How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi :E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$. Please reference some good readings to me on Dold fibration.

Thanks,

]]>I am reading your article on Dold fibration

I have two questions:

- How to construct Dold fibration ? Just like Hurewicz fibration, Hurewicz fibration has a theorem: A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.
- Please reference some good readings to me on Dold fibration.

Thanks,

Tom

]]>I am reading the article on Hurewicz connection

Theorem. A map $\pi:E \to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for $\pi_!$.

I have two questions:

(1) How to formally construct the Hurewicz connection for $\pi_!$ ?

(2). It has scratched an idea of proof. May I know if anyone can reference a completed formal proof to me. (a paper or an article will be great !).

]]>Suppose we have the sequence of sets $\mathbb{R}$, $\mathbb{R}^2$, $\mathbb{R}^3$, … Is there a Kan simplicial structure on this sequence of sets, that is not $n$-coskeletal for some $n \in \mathbb{N}$?

To be more precise, is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n+1}$ that is not $n$-coskeletal for some $n \in \mathbb{N}$?

And very closely related: is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n}$ (with $R([0]))=\{0\}$), that is not $n$-coskeletal for some $n \in \mathbb{N}$ ?

]]>Suppose we have a simplicial set X and a m-truncated Kan simplicial set Y. Then how is it possible to construct $Hom_{Simpl}(X,Y)$ as a subset $H \subset Hom_{Set}(X_m,Y_m)$?

Since Severa used this in his work on the n-jet functor (for X the nerve of the pair groupoid over an arbitrary set), it should be possible. Nevertheless I can’t find an explicit construction including a proof that what he constructed is indeed $Hom_{Simpl}(X,Y)$.

By an explicit construction I mean something like: Let $f \in H$ be given, then the appropriate simplicial morphism $F$ is given by $F[n]= X_n \rightarrow Y_n$ as follows : ??? where the commutation with the face and degeneracy maps is seen as follows ??? … On the other side we that any simplicial morphism is given that way, because ???

….

So if someone could give me a proof (I think it will be an induction on something like $Hom_{Simpl}(Sk^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ or $Hom_{Simpl}(Horn_j^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ ) it would be great.

Likely this doesn’t work for arbitrary simplicial sets X, so another topic is to find the appropriate conditions on X .

Moreover this should be put into the nLab, too…

If nobody knows a proof it would be nice, if we could work it out together. At the end I will take the time to put in the nLab. Unfortunately my skills on simplicial sets are not good enough, to do it by myself.

]]>New entry representable fibered category.

]]>This is an excerpt I wrote at logical functor:

As far as cartesian morphism there are two different universal properties in the literature, which are equivalent for Grothendieck fibered categories but not in general. In what Urs calls the “traditional definition” (but is in fact a later one) one has for every $x'$, for every $h$, for every $g$ such that … there exist a unique da da da. This way it is spelled in Vistoli’s lectures. This is in fact the strongly cartesian property, stronger than one in Gabriel-Grothendieck SGA I.6. The usual, Grothendieck, or weak property takes for $g$ the identity, and the unique lift is of the identity at $p(x_1)$. Then a Grothendieck fibered category is the one which has cartesian lifts for all morphisms below and all targets, and cartesian morphisms are closed under composition. With the strong cartesian property one does not need to require the closedness under composition. Now a theorem says that in a Grothendieck fibered category, a morphism is strongly cartesian iff it is cartesian.

Now I have made some changes to cartesian morphism, so that the entry is aware of the two variants of the universal property, which are not equivalent in general but are equivalent for Grothendieck fibered categories.

There was also a statement there

In words: for all commuting triangles in Y and all lifts through p of its 2-horn to X, there is a unique refinement to a lift of the entire commuting triangle.

which is too vague and I am not happy with, as it does not involve the essential parameter: the morphism for which we test cartesianess. I made a hack to it, and still it is not something I happy with (I like the idea of *horn* mentioned, however not the lack of appropriate quantifiers/conditions etc.). It is cumbersome to talk horn. (Maybe we could skip the whole statement in this imprecise form, and just mention *please note the filling of the horn in $X$ with prescribed projection in $Y$* or alike). Here is the temporary hack:

]]>In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-horn to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.

Given all the discussion on the categories list, I decided it would be worth creating Street fibration. While writing it I had occasion to put up a stub at strict 2-equivalence of 2-categories.

]]>Created essential fiber in order to refer to it on the categories list.

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