The “Dunce Cap” Space Is Contractible

Here is the exercise 6 on P. 50 in the book Topology and Geometry by Glen Bredon. I put it here because I found the drawing of this cap very lovely. Indeed I like that most of the pictures in this book are lovely sketches.

Question. The “dunce cap” space is the quotient of a triangle (and interior) obtained by identifying all three edges in an inconsistent manner. That is, if the vertices of the triangle are $p, q, r$ then we identify the line segment $(p, q)$ with $(q, r)$ and with $(p, r)$ in the
orientation indicated by the order given of the vertices. (See Figure 1-6.) Show that
the dunce cap is contractible.

Following the development in the book, I will use the following theorem that the homotopy type of a mapping cylinder or cone
depends only on the homotopy class of the map [Theorem 14.18, Topology and Geometry by Glen Bredon]. The idea is to identify the dunce cap as a mapping cone.

Theorem 14.18. If $f_0\simeq f_1:X\to Y \text{ then } M_{f_0 } \simeq M_{f_1}\text { rel } X+Y \text{ and }C_{f_0}\simeq C_{f_1}\text{ rel } Y+\mathrm{vertex}.$

Proof of the Qustion. Suppose $f: S^1\to S^1$ is a map from $S^1$ to itself. The cone $C_f=M_f/S^1 \times\{1\}$ for $f$ is obtained by pinching the top of the mapping cylinder to a point. As $M_f$  is the cylinder $S_1\times [0,1]$ with the bottom pasted to $S_1$ by the map $f$, $C_f$ is $D_2$ with $\partial D_2$ pasted to $S_1$ by the map $f$. So the dunce cap is just $C_f \text{ with } f: S^1\to S^1$ defined as

$f(e^{2\pi i t})= \begin{cases} e^{2\pi i (3t)}, 0\leq t\leq 2/3\\ e^{2\pi i(2- 3t)}, 2/3\leq t\leq 1. \end{cases}$

which is homotopic to the identity by a linear homotopy (note that we make the choice of $f$ for an easy definition of the homotopy)

$H(e^{2\pi i t},s)= \begin{cases} e^{2\pi i (3t(1-s)+st}, 0\leq t\leq 2/3\\ e^{2\pi i[(2- 3t)(1-s)+st]}, 2/3\leq t\leq 1. \end{cases}$.

So the dunce cap is homotopic to $C_{id}\simeq D^2$ which is contractible.

On A Question about Quintic Del Pezzo 3-fold of Degree 5

A question from my classmate:

Question. The Quintic Del Pezzo 3-fold $V(5)$ of Degree $5$ is the intersection of $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$ with a codimension $3$ linear subspace. Show that for any point $p\in V(5)$, there is a line not passing through that point.

Proof. I posted my answer on MO here: https://mathoverflow.net/a/354809/144294. Sasha gave a short answer in the same post using the fact that the Hilbert scheme of lines on $V(5)$ has dimension $2$ (it’s isomorphic to $\mathbb{P}^2$). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of $\mathbb{P}^9$ characterising lines on $V(5)$ is $2$ by an incidence correspondence argument.

[Soft Question] Why higher category theory?

Here is a MO post asking a question that I’ve had in mind for a while: Why higher category theory?

Having studied some elementary topos before, I have been interested in higher topos since I attended a summer school lecture last year. Besides formal generalisations, I expect to see applications which provide new results or meaningful insights. Though the meaning of “new” and “useful” very much diverse between different mathematical cultures.

Here are some important applications of higher categories in K-theory (added on Mar 05, 2020), suggested by my supervisor Schlichting.

Dustin-Mathew-Morrow, Algebraic K-theory and descent for blow-ups

Nikolaus-Scholze, On topological cyclic homology

Clausen-Mathew-Morrow, K-theory and topological cyclic homology of henselian pairs

Antieau-Gepner-Heller, K-theoretic obstructions to bounded t-structures

Blumberg-Gepner-Tabuada, A universal characterization of higher algebraic K-theory

Barwick, On exact infinity-categories and the Theorem of the Heart

Characterisation of Quasicoherent Sheaves by Distinguished Inclusions

This is my work on the six exercises Exercise 13.3.D-13.3.I in Section 13.3.3 of Vakil’s notes. We look at a useful characterisation of quasicoherent sheaves in terms of distinguished inclusions and prove some properties in reasonable circumstances (quasicompact and quasiseparated).

Here is the pdf file: Characterisation of quasicoherent sheaves

Generic Freeness and Chevalley’s Theorem I

This post and the next is my work on Exercises 7.4 A-7.4.O of section 7.4 in Vakil’s note. We discuss Chevalley’s Theorem and prove it using Grothendieck’s Generic Freeness Lemma.

We first discuss some properties of constructible sets and then we prove Grothendieck’s generic freeness lemma following a sequence of exercises in Vakil’s notes.  Then we use Generic Freeness to prove Chevalley’s Theorem. Though there are more direct ways to prove it, such as the proof we did in Thursday’s lecture (06/02/2020) in Applied Scheme Theory (proof of Theorem 2.2.9 in Algebraic Geometry II by Mumford and Oda). We only use Generic Freeness here as we will use it again in the future for generic flatness. Note that except proposition 1.2 the rests of the first section on the properties of constructible sets are not needed later.

We will discuss some applications of Chevalley’s Theorem including its implication of Hilbert’s Nullstellensatz in the next post: Generic Freeness and Chevalley’s Theorem II (Applications). I divided the post into two parts since the next part about applications is to be continued. The pdf file below contains the full article so far. Later more will be added into the next post.

Note that except proposition 1.2 the rests of the first section on the properties of constructible sets are not needed later.

View the full article as pdf here: Generic Freeness and Chevalley’s Theorem[16/02/2020]

As we remarked that the proof of Lemma 1.1 applied to the case that $M_{i+1}/M_{i}$ projective is the induction step of the proof of a generalised result. We include this result here. It’s not related to another part of the post so I add it as an appendix.

The reference for this proof is the lecture notes for the course Ring Theory at Warwick, 2018: Ring Theory Lecture notes 2018.

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.

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 $\mathrm{Spec}((S_\bullet)_{fg})_0$.
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.