# Generic Freeness and Chevalley’s Theorem II (Applications)

This post together with the last one is my work on the exercises 7.4 A-7.4.O of section 7.4 in Vakil’s note. This post is to be continued.

Following the last post, we now discuss some consequences of Chevalley’s Theorems.

View the full article as pdf here: Generic Freeness and Chevalley’s Theorem.

# Topos as A Model for Set Theory and Independence Proof

### Introduction

Part 1:  For any elementary topos, its internal logic, considered as propositional calculus, is a Heyting algebra. Some constructions are required to make a topos a suitable model of set theory, where the propositional calculus needs to be Boolean and two-valued.  We discuss some properties of topos desired for a suitable model: the existence of a natural number object (NNO), being Boolean, satisfying the axiom of choice (AC) and well-pointedness. I gave some proofs for some exercises in [1] and organise them into a thread.

Part 2: We sketch the construction of a Cohen topos, which is Boolean and satisfies the axiom of choice, where the continuum hypothesis (CH) fails [1, pp. 277-290]. After that, we improve the result by a filter-quotient construction for obtaining a well-pointed topos satisfying AC with an NNO, where CH fails. Such a filter-quotient construction indeed provides a stronger model than the Cohen topos, with the four properties mentioned above satisfied.

### Main

Topos as A Model for Set Theory and Independence Proof is an excerpt from my previous work (2018) [2]. Here is the full work of my project: Topos Foundation for Set Theory. It discusses some desired properties of topos as a suitable model of set theory and ideas on the independence proof of the axiom of choice. I gave detailed proofs for some exercises in [1] and organise them into a thread. I have also cited necessary definitions and propositions for completeness (I omitted the proofs for propositions that can be directly found in [1]). Assuming basic familiarity with category theory,  this article should be self-contained.  Here is an appendix for quick reference: Appendix.

Note: there are some missing cross-references which should not affect reading.

### Summary

To show the foundation aspects of topos, we have clarified the differences and connections between external and internal concepts and discussed the properties required for a topos as a suitable model of set theory, that is, the existence of a natural number object (NNO), being Boolean, satisfying the axiom of choice (AC) and well-pointedness.

The universality of the NNO was discussed and we showed how the recursion, addition, multiplication and the partial order can be defined from the NNO and how they correspond to that for sets. The propositional calculus, the Heyting algebra and some equivalent conditions for a topos being Boolean were discussed. The presheaf topos and sheaf topos, as special examples, were illustrated for the Heyting algebras and the equivalent conditions for being Boolean. Some propositions about external and internal projective objects were proved and related to the external and internal axiom of choice (AC and IAC). We proved the equivalent conditions for a presheaf topos satisfying AC or IAC, thus had an example of a presheaf topos satisfying IAC but not AC.
We also proved the equivalence of AC and IAC for a well-pointed topos and some equivalent conditions for well-pointedness.

To show an application of the foundation aspect of topos, we sketched the process of construction of the Cohen topos, which is Boolean and satisfies AC, where the continuum hypothesis (CH) fails, with some details examined. We then proved that the filter-quotient construction on the Cohen topos preserves the cardinal inequality and the NNO. We also proved that the filter-quotient topos constructed from a Boolean topos satisfying AC is well-pointed and again satisfies AC. Together with the fact that a well-pointed topos is both Boolean and two-valued, we concluded that the filter-quotient topos constructed from the Cohen topos satisfies all the properties for sets and violates CH, thus is an improved model in the independence proof.

### Further Remark

Later in the book [1], the Mitchell-Benabou language, as a first-order language for topos, is constructed, and then the relation to set theory will become clear. The internal properties we have discussed can be expressed by some formulas that correspond to the formulas expressing related properties in set theory. For example, a topos is Boolean iff the formula  $\forall p( p\vee \neg p)$  holds; a topos satisfies IAC iff the formula for AC holds. Also, the geometric construction for the independence proof can be translated to the language of forcing. On the other hand, we can say that the symbols in logic can be interpreted diagrammatically through categories. In this way, we see the symbolic and diagrammatic aspects of logic. Topos theory provides a diagrammatic view of logic, which is particularly good when dealing with structures. Though the idea in the independence proof is actually equivalent to that in set theory, the diagrammatic view and by topos theory gives us a different viewpoint.

### References

[1] MacLane, S., & Moerdijk, I. (2012). Sheaves in geometry and logic: A first introduction to topos theory. Springer Science & Business Media.

[2] Likun Xie. (2018). Topos Foundation for Set Theory. Unpublished Bachelor Thesis. University of Manchester, UK.

# Categorical Construction of Fibre Product of Schemes

Following our discussion of glueing schemes: Categorical descriptions for glueing sheaves and schemes. We now discuss the construction of the fibre product of schemes by glueing.

Given arbitrary schemes $X,Y,S$, let $q:X\to S$ and $r: Y\to S$ be the given morphisms. Let $\{S_i\}$ be an open affine cover of $S$. Let $X_i=q^{-1}(S_i)$, $Y_i=r^{-1}(S_i)$, choose an affine open cover $X_{ij}$ for $X_i$ and an affine open cover $Y_{jk}$ for $Y_k$. The fibre product is constructed by glueing various $X_i\times_{S_i} Y_i$  together.

We rewrite the glueing construction of fibre product in a more categorical way as follows. Note that the colimit here is glueing construction and the consequences of the two pullback squares should be clear thinking in terms of schemes.

View as pdf: Construction of fibre product

Theorem[Thm 3.3, [1]/ Thm 9.1.1, [2]] For any two schemes $X$ and $Y$ over a scheme $S$, the fibre product $X\times _S Y$ exists and is unique up to unique isomorphism.

# Variations of Yoneda Lemma; Monos, Epis and Isomorphisms of (Pre)sheaves

The first part is my work on a variation of Yoneda Lemma. The second part is my work on Exercises 2.4A, 2.4.C-2.4.D of section 2.4 in Vakil’s notes.

### 1.  Variations of Yoneda Lemmas (Monos, Epis and Isomorphisms of Presheaves)

Here are a few variations of Yoneda Lemma I played around a few years ago, which bear similar ideas of Yoneda Lemma. Recently I have been dealing with sheaves again, so I just reviewed some old stuff here. I used these variations to show that a morphism of presheaves is monic resp. epic if and only if it’s injective resp. surjective on the level of sections (For another proof see here https://stacks.math.columbia.edu/tag/00V5).

Here are the variations and proofs for presheaves: (pdf version: Variations of Yoneda lemma)

### 2.  Monos, Epis and Isomorphisms of Sheaves

Here we give a detailed discussion for sheaves, following exercises in Section 2.4 of Vakil’s notes: (pdf version: Monos, epis and isomorphisms of sheaves)

# Categorical descriptions for glueing sheaves and schemes

In this post, we give a categorical proof of Lemma 33.2 (Tag 00AK) by showing that a glued sheaf is defined as an equaliser. This equaliser provides a tool for calculating global sections of glued schemes. We later present some examples for calculating global sections and glueing constructions using (co)limits descriptions. (Sometimes people say they may lose insights about the details and the real maths behind abstraction. But it really depends on one’s approach and ways of thinking. ) The goal of this post is to give a structured summary of glueing constructions of schemes after meditating on explicit constructions, using categorical language. It is not meant to replace explicit arguments for schemes, but to give some ideas on how general a construction is and whether it can be transferred to a different setting. For example, we will know what to do if we are working on a site with a different Grothendieck topology instead of the Zariski topology. Note that some consequences of the categorical facts we use are indeed straightforward for schemes, for example, the two pullback squares in Example 4. This post is open-ended and more examples of glueing will be added.

Remark. Note that for a sheaf $\mathcal {F}$ on a topological space and an open cover $U=\bigcup U_i,$

$\displaystyle \mathcal {F}(U) =\mathrm{lim}_J \mathcal{F}(U_i),$

where $J$ is the covering sieve generated by the covering $\{U_i\}$, namely, $J$ is the collection of all those $V\subset U$ with $V\subset U_i$ for some $i.$ This limit is equivalent to the following equaliser diagram in the usual sheaf definition:

where for $t\in FU$, $e(t)=\{t|_{U_i} \mid i\in I\}$ and for a family $t_i\in FU_i, p\{t_i\}=\{t_i| _{U_i\cap U_j}\}, q\{t_i\}=\{t_j| _{U_i\cap U_j}\}.$

For a scheme $X=\bigcup X_i$ with an open cover $\{X_i\}$ in the Zariski topology, $X$ is the colimit indexed over the covering sieve generated by the covering $\{X_i\}$. This colimit can also be simplified to be a coequaliser diagram.

First, we include the explicit constructions for glueing sheaves and some sources for details check of these constructions. The categorical proof we are going to show is a categorical rephrasing by meditating on these constructions, which gives a more structured presentation.

### Glueing Morphisms

Proposition A.1  [Tag 00AK] Let $X$ be a topological space. Let $X=\cup U_i$ be an open covering. Let $\mathcal{F},\mathcal{G}$ be sheaves of sets on $X$. Given a collection

$\phi_i:\mathcal{F}|_{U_i}\to \mathcal {G}|_{U_i}$

of maps of sheaves such that for all $i,j\in I$ the maps $\phi_i,\phi_j$ restrict to the same map $\mathcal {F}_{U_i\cap U_j}\to \mathcal{G}_{U_i\cap U_j}$, then there exists a unique map of sheaves

$\phi: \mathcal {F}\to \mathcal{G}$

whose restriction to  each $U_i$ agrees with $\phi_i$.

Proof. Take any $s\in \mathcal{F}(V)$, where $V\subset X$ is open, and let $V_i=U_i\cap V$. Then we have an element $\phi_i (s| _{V_i})\in \mathcal{F}(V_i)$ and $\phi_i (s| _{V_{ij}})=\phi_j (s| _{V_{ij}})$ by the glueing condition. Thus by the sheaf condition for $G$, the sections $\phi_i(s|_{V_i})\in G(V_i)$ patch together to give a section in $G(V)$, define this section to be $\phi(s)$. (We omitted the checking details. )                                                   $\square$

### Glueing Sheaves

Explicit construction of glueing sheaves is given in Lemma 6.33.2, Tag 00AK, but the details of checking have been omitted.  For some details of a reality check,  see this post.

### Glueing Schemes

To glue schemes, one needs to define the glued topological spaces which will be a quotient space of the disjoint union of the glued spaces and then verify the structure sheaves of the glued schemes satisfy the condition of Lemma 6.33.2 [see Tag 01JA].

Example 1. (The affine line with doubled origin is not affine).   Let $k$ be a field. Let $X= \text{Spec} (k[t])$, $Y= \text{Spec}(k[u])$. Let $U=D(t) =\text{Spec}k[t,1/t]\subset X \text{ and } V= D(u)=\text{Spec}k[u,1/u]\subset Y$.  Consider the ismorphism $U\cong V$ given by $t\leftrightarrow u$.  Let $Z$ be the glued scheme, from the equaliser diagram in the proof of Lemma 6.33.2,  we see that the structure sheaf $\mathcal{O}_Z$ is given by

$\mathcal{O}_{Z}(W) = \mathcal{O}_X(W \cap X) \times_{\mathcal{O}_{X(W \cap X \cap Y)} \cong \mathcal{O}_{Y}(W \cap X \cap Y)} \mathcal{O}_Y(W \cap Y)$.

Thus the global section $\mathcal{O}_Z(Z) \text{ is } k[t] \times_{k[t,t^{-1}] \cong k[u,u^{-1}]} k[u] \cong k[t]$. From this we see that $Z$ is not affine, since  $Z=\text{Spec}(k[t])$ which is not the case: the underlying topological space $Z$ has one more point- the doubled origin.

Example 2. (Quasiseparated scheme is glued from affine schemes). Note that every scheme is a colimit of affine schemes. This is true in general by the fact that every sheaf is a colimit of representables and that the Zariski topology is subcanonical (see here for more details).

For the case that a scheme $X$ is separated (for which the intersection
of any two affine open sets is affine), take an affine open cover $\bigcup U_i=X$ such that each intersection $U_{ij}$ is affine, then $X$ is just the coequaliser of the diagram $\coprod_{i,j} U_{ij} \rightrightarrows \coprod_{i} U_i$. This is the glueing construction as we described above.

If $X$ is not separated, one can still write $X$ as a colimit of affines but not with the same diagram as we used for glueing, see here for a description of the diagram.

Remark. One implication of viewing $X$ as a colimit is: let Sch and Rings be the categories of schemes and rings respectively, given that $\text{Hom}_{\textbf{Rings}}(A,B)\cong \text{Hom}_{\textbf{Sch}}(\text{spec}(B),\text{spec}(A))$, one can deduce that for any scheme $X$$\text{Hom}_{\textbf{Rings}}(R,\Gamma(X,\mathcal{O}_x))\cong \text{Hom}_{\textbf{Sch}}(X,\text{spec}(R))$ by the fact Hom-functor preserves (co)limits.

Example 3 (Proj construction). [Section 4.5.7, Vakil]

Let $S_\bullet=\oplus _{n\in \mathbb{Z}} S_n$ be a $\mathbb{Z}$-graded ring and $S_+=\oplus_{i>0} S_i$ be the irrevalant ideal. Suppose $f\in S_+$  is homogeneous, there is a bijection between the prime ideals of $((S_\bullet)_f)_0$ and the homogeneous prime ideal of $(S_\bullet)_f$. The projective distinguished open set $D(f)= \mathrm{Proj} S_\bullet \setminus V(f)$ is identified with $\mathrm{Spec}((S_\bullet)_{f})_0$. If $f,g\in S_{+}$ are homogeneous and nonzero, $D(f)\cap D(g)= \mathrm{Spec} ((S_\bullet)_{fg})_0)$ is isomorphic to the distinguished open subset $D(g^{\mathrm{deg} f}/f^{\mathrm{deg}g})$ of $\mathrm{Spec} ((S_\bullet)_f)_0$, similarly for $\mathrm{Spec} ((S_\bullet)_g)_0$. $\mathrm{Proj}S_\bullet$ is glued from various $\mathrm{Spec}((S_\bullet)_{f})_0$ along the pairwise intersections $\mathrm{Spec}((S_\bullet)_{fg})_0$.

Example 4. (Fibre product of schemes)[For detailes of a categorical proof of this construction, see the post: Construction of Fibre Product of Schemes or the pdf here: Construction of fibre product.]

Given arbitrary schemes $X,Y,S$, let $q:X\to S$ and $r: Y\to S$ be the given morphisms. Let $\{S_i\}$ be an open affine cover of $S$. Let $X_i=q^{-1}(S_i)$, $Y_i=r^{-1}(S_i)$, choose an affine open cover $X_{ij}$ for $X_i$ and an affine open cover $Y_{jk}$ for $Y_k$. The fibre product is constructed by glueing various $X_i\times_{S_i} Y_i$  together.

# Notes and Remarks on Unstable Motivic Homotopy Theory

I have made some notes and remarks about motivic homotopy theory while working on my master dissertation during the summer of 2019.

Affine representability results:

For remarks on section 2.3 and Lemma 2.3.2 of [2] (also quoted as Proposition 5.5 and Proposition 5.6 in [1]), we take the approach of stack and descent.  By relating hyperdescent condition for simplicial presheaves with descent for stacks [3], we show that how this implies classifying space of a stack satisfies hyperdescent.

After that, we give a different proof and some remarks of Lemma 2.3.2 based on my understanding, see Proposition 0.0.2 and Proposition 0.0.3 in Stack and descent.

Exercises on the construction of motivic homotopy theory:

I also have some informal notes about the exercises in [1]. The following are some of my proofs for the exercises, in the remarks, I also pointed out some typos I found which may be helpful for other readers: Exercises on A Primer.

[1] Antieau, B., & Elmanto, E. (2017). A primer for unstable motivic homotopy theory. Surveys on recent developments in algebraic geometry95, 305-370.

[2] Asok, A., Hoyois, M., & Wendt, M. (2018). Affine representability results in 𝔸1–homotopy theory, II: Principal bundles and homogeneous spaces. Geometry & Topology22(2), 1181-1225.

[3] Jardine, J. F. (2015). Local homotopy theory. Springer.

# Descent, Affine Representability and Classifying Space in A1-Homotopy Theory

This is my master thesis in Warwick, supervised by Marco Schlichting.

Abstract
We relate hyperdescent condition for simplicial presheaves, with hyperdescent in $$\infty$$-topos and descent for stacks. One consequence is that the classifying space of a stack satises hyperdescent which simplies existing proofs in some cases. We show an algebraic topological approach to some properties of singular constructions for sites with interval. Using results of affine representability and A1-algebraic topology over a field, we show that the A1-homotopy theory is not a model topos. As an extension of this result, we prove that an A1-local monoid is strongly A1-invariant if and only if its 0-th Nisnevich homotopy sheaf is strongly A1-invariant. This can be used for calculation of A1-loop space and we apply it to BBGm. Finally we present some calculations of A1-homotopy sheaves of BGLn and BSLn and in particular
the first a few A1-homotopy sheaves of BGL2.

# Thoughts on Physics, Maths and Reality (Part I)

Here are some casual thoughts related to my old passion for physics and different mindsets between people doing maths and physics. Most of the things discussed here are open to interpretations.

At the early age of physics, physical laws tended to fit into some sort of common sense or intuitions. As physics developed further, people found the existence of “reality” more elusive , particularly for quantum mechanics which has been thought to be counterintuitive. Physics uses maths to construct a “physical reality” that is in some sense highly subjective. Only a tiny part of the presumed existence of an outside world interacts with us by reflecting itself on our perceptions. The rest of the physics story is filled out with mathematical fabrications. How much is fabricated depends on how much we can perceive, to measure or to build a physical picture for it. Lack of direct observations and perceptions in quantum mechanics makes its story less intuitive so that sometimes people feel the need to find a more sensible interpretation which fit in their philosophical views better, like the alternative Bohmian interpretation for Copenhagen interpretation.

I was ever driven to quest for an “interpretation” of quantum mechanics by studying a few alternative quantum theories including quantum Baysianism, and Bohmian Mechanics (the pilot-wave model). These theories are almost equivalently good at predictions. But the orthodox quantum theory has remained in textbooks as it came first in history. These alternatives are all good candidates of quantum mechanics. There might be some areas in which some of them work slightly better than the others. But none of them stands out to resolve the inconsistency problem in general relativity. People in favour of one of them is in favour of a kind of interpretation or worldview they are happy to accept.

Indeed, all these quantum theories are mathematical theories with their own beauty. That’s how I like pure maths: it stands out as a subject with a lot more freedom creating an abstract reality which does not explicitly depends on the external world.

# [Reading Notes] Mechanics by Landau and Lifshitz

The following is an excerpt from [$\S 43$ The action as a function of coordinate]. I would like to comment on the formal derivation of Hamilton’s equations and the problems of independence of variations.

Note that in the derivation above, the variations $\delta p$ and $\delta q$ are regarded as independent.* Actually, $\delta q$ is arbitrary but $\delta p$ is not, even though p, q are both independent variables. Since $p$ in connected with $\dot{q}$ and  $\delta p$ and $\delta \dot{q}$ are not independent.

Notice that before (43.8) is derived, we have applied Ledrendre  Transformation which requires that

$\dot{q}=\frac{\partial H}{\partial p}$.                                               (1)

So the coefficient of $\delta p$ is  $0$,  and $\delta q$ is arbitrary, so its coefficient must be $0$, Hence we get another Hamilton’s  equation

$\dot{p}=-\frac{\partial H}{\partial q}$.                                              (2)

Notice that we only derive half of Hamilton’s equations from the procedure above.

Since we can not say that we derive Hamilton’s equations by applying Hamilton’s equations. In order to make this induction above complete, we have to give the proof of another half of Hamilton’s equations, that is (1).

(1) is related to the definition of  $p=\frac{\partial L}{\partial \dot{q}}$. From the definition of Hamiltonian

$H=\Sigma \dot{q}p-L$

and $\dot{q}=\dot{q}(p,q,t)$, then

With the definition of p, $p=\frac{\partial L}{\partial \dot{q}}$, we have

$\dot{q}=\frac{\partial H}{\partial p}$.

Hence the half part of Hamilton’sequations is derived.

In this way of deriving Hamilton’s equation, strictly, we first derive (1) from the definition of p, and then by applying (1) in $\delta S$, (2) can be derived.

*We should notice that variations here are simultaneous variations and it’s for a complete system.