Leave $ X $ be a closed symmetric collector equipped with a smooth fibration of the Lagrangian bull $ pi: X rightarrow Q $. Assume that $ pi $ Supports a Lagrangian section. By the work of Kontsevich-Soibelman, one can associate with $ (X, pi) $ A rigid analytical space (in the sense of Tate). $ Y $. In recent work, Abouzaid showed that one can associate with $ (X, pi) $ a gerbe $ beta $ in the rigid analytical space $ Y $, classified by a class $ ( alpha_X) in H ^ 2 (Y, O *) $, and define a $ A _ {infty} $-Functor of the Fukaya category of $ X $ to the category derived from $ beta $– coherent sheaves twisted in $ Y $. He shows that this is full and faithful inlay. I have heard that there are ideas floating around how to extend a result like this in the presence of singular fibers. Are there explicit proposals on how you could try to do this?

# Tag: fibrations

## ct.category theory – Co-Cartesian vs. Coherent Fibrations local coartesians

Tell $ pi: C a J $ it is an internal fibration of $ infty $-categories Then "morally", $ pi $ corresponds to a diagram indexed by $ J $ in the "category of categories with correspondences", and if $ pi $ is *cocartesian* then corresponds to a diagram indexed by $ J $ in the "category of categories with functors".

From this, it would be natural to guess that the CoCartesian condition is the requirement that the correspondence on each arrow in $ J $ It is representable by a functor. This is almost but not quite right. These fibrations are denominated "locally coCartesianas" by Lurie and in chapter 2.4.2 of Higher Topos Theory, offers several ways to quantify the "almost" in question, that is to say, the conditions of a local coCartesian fibration that do it coCartesiano.

A condition that I could not find in HTT (maybe because it's obvious) but that seems enough is the following:

**Guess.** An internal fiber $ C a J $ is coCartesian if the fibrations $ C ([n]) a J ([n]$ They are locally co-Cartesians for all. $ n $, where $ C ([n]$ the category of functors of the $ n $-simply to $ C $.

In fact, it seems that it should be enough to impose this condition only for $ n = 0, 1 $.

My question: is this conjecture correct or solvable? And, can a similar criterion be given for a locally sustainable arrow? $ f $ of $ J $ to be a co-supporter of society (maybe something that involves the regression of $ C $ to the over category of $ f $ in $ J $)?

## Are biquotient fibrations?

I am interested in the following:

Leave $ G $ be a group of lies and $ H $ a closed subgroup. Leave $ K subset H $ be a closed subgroup of $ H $. Then, one can define an action of $ H times K $ in $ G $ as follows:

$$ (h, k) cdot g: = hgk ^ {- 1}. $$

If this action is free, then there is a dive:

$$ pi: M a H setminus M / K $$ and we call the quotient a *biciciente* of $ G $ by $ H, K $.

My question is, does $ pi $ defines a fibration, in the sense that each fiber is diffeomorphic with each other?

## dg.differential geometry – How does the instantaneous Floer homology of the Seifert fibrations relate to that of a trivial fibration?

My question focuses on the relationship of the Chern-Simons theories about a Seifert fibration and the trivial product space. $ Sigma_g times S ^ 1 $, and its implication for the instantaneous homology of Floer. Here, $ Sigma_g $ It is a Riemann gender surface. $ g $.

Leave $ M_ {g, p} $ be the Seifert variety that presents itself as a degree $ -P $, $ U (1) $

package on a Riemann surface of the genus $ g $.

In this Thompson article, it is explained that the Chern-Simons theory for $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $ They are closely related. This is essentially due to the fact that each 3-manifold has a contact structure, and for $ M_ {g, p} $ the natural contact structure, $ kappa $, is the $ U (1) $ connection in the $ U (1) $ Package used in its definition. As explained earlier in equation 2.2 of Thompson's paper, this corresponds to the obvious structure in $ Sigma_g times S ^ 1 $.

This leads to similar forms for the partition functions of the two theories, as shown on page 2 of Thompson. That is, for $ Sigma_g times S ^ 1 $, one finds the Hirzebruch-Riemann-Roch theorem for fundamental beam power in the space of flat connection modules in $ Sigma_g times S ^ 1 $, While for $ M_ {g, p} $ There is a simple generalization.

This brings me to my question. Since the Instanton Floer homology is defined using the Chern-Simons function, **Is there a simple relationship between the Floer instant homology of $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $?**

## Algebraic topology – 2 out of 3 property of fibrations?

Leave $ X xrightarrow {f} Y xrightarrow {g} Z $ Be a diagram in the category of topological spaces. Yes $ g circ $ and one of $ f, g $ Are the fibrations, we can conclude that this is the other?

In this question it is stated that if $ f $ it is surjective and $ g circ $ Y $ f $ they are fibrations then $ g $ It is a fibration. However, no test is given there. Is this true? How can I show it? I understand that the main difficulty is that an arbitrary map $ W to Y $ it can not be lifted along $ f $ to a map $ W to X $. I'm not sure if this elevation property is true …

The motivation for this problem is that I want to show that $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $ Y $ mathbb {CP} ^ {2n + 1} to mathbb {HP} ^ n $ they are the fibrations. I know there are fibrations (in fact fiber bundles) $ mathbb {S} ^ {2n + 1} a mathbb {RP} ^ {2n + 1} $ Y $ mathbb {S} ^ {2n + 1} to mathbb {CP} ^ n $ and the factorization of these two maps gives the map. $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $. I wonder if we could use the previous property to directly deduce that $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $ It is a fibration.

## Etale-analytic comparison without elementary fibrations.

A theorem due to Artin states that for a smooth scheme $ X $ of finite type about $ mathbb {C} $ and a constant locally constructible sheaf $ F $ we have an isomorphism

$$

H ^ * et (X, F) approx H ^ * (X ( mathbb {C}), F)

$$

where the LHS is ethnic cohomology and the RHS is cohomology functor derived from $ F $ In the analytical topology. One of the tests that I know comes from establishing that there is an open cover of $ X $ by $ K ( pi, 1) $ However, the appearance of $ K ( pi, 1) $I feel a little weird. Is there any alternative test that is not based on $ K ( pi, 1) $ neighborhoods