A simple visual editor for creating commutative diagrams.╰(*°▽°*)╯
The constructions look like these (see the links above for details):
But in practice, these diagrams may not be helpful to see what the equalisers should be. Now I give proofs for the (co)equalisers in two examples: the connected component of a simplicial set and the sheaf condition.
The connected components [Background]
For definitions and other backgrounds, see Subsection 00G5. For the record, see [P12, DLOR07] for the cosimplicial identities and Tag 000G for simplicial identities. (These identities are used in my proofs.)
Two examples of (co)limits as (co)equalisers
This is about an exercise in [Bass]:
Exercise 19.5. Prove that if is infinte-dimensional, that is, it has no finite basis, then the closed unit ball in is not compact.
Proof. Choose an orthonormal basis , then . This means the sequence is not Cauchy hence has no convergent subsequence.
For a Banach space, by Riesz’s lemma to find a non-Cauchy sequence.
[Bass] Bass, R. F. (2013). Real analysis for graduate students. Createspace Ind Pub.
A question about the philosophy of mathematics for casual reading:
Here we give proofs for two versions of Lusin’s Theorem, one from Exercise 44, Ch2 in Folland’s Real Analysis and the other from the textbook used for my first year undergraduate mathematical analysis course in Beijing. The latter version is a stronger result which in addition discusses the condition for a real-valued function defined on a subset of to be extended to the whole of . A more general result in topology is the Tietze Extension Theorem.
See the full post here: Lusin’s Theorem and Continuous Extension
Here we let denote the Lebesgue measure on .
Lusin’s Theorem (Version 1)[Exercise 2.44, Folland]. Suppose is Lebesgue measurable, is Lebesgue measurable and , there is a compact set such that and is continuous.
Lusin’s Theorem(Version 2)[Huan]. Suppose is Lebesgue measurable and is a Lebesgue measurable extended real valued function with , then , such that , where denotes the space of continuous function on .
Continuous Extension Theorem[Huan]. Suppose , then can be extended to a continuous function on if and only if can be extended to a continuous function on the closure of .
Tietze Extension Theorem. Let be normal and be closed and let be continuous. Then there is a map such that
for all . (Note that in topology, by a map we mean a continuous function. )
This post is about some applications of Krull’s Principal Ideal Theorem and regular local rings in dimension theory and regularity of schemes [Part IV, Vakil], with the aim of connecting the 2018-2019 Warwick course MA4H8 Ring Theory with algebraic geometry. The lecture notes/algebraic references are here: 2018-2019 Ring Theory. Note that the algebraic results included here follow the notes. Alternatively, one can also find them in [Vakil] either as exercises or proved results for which I have included the references.
Besides including results in both their geometric and algebraic statements, I have given proofs to a selection of exercises in Part IV, [Vakil] to illustrate more applications and other connections to the contents in the Ring Theory course. The indexes for exercises follow those in [Vakil].
See here for the full post: Application of Krull’s Principal Ideal Theorem
Please also let me know if you find any errors or have suggestions on any of my posts.
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 then we identify the line segment with and with 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
Proof of the Qustion. Suppose is a map from to itself. The cone for is obtained by pinching the top of the mapping cylinder to a point. As is the cylinder with the bottom pasted to by the map , is with pasted to by the map . So the dunce cap is just defined as
which is homotopic to the identity by a linear homotopy (note that we make the choice of for an easy definition of the homotopy)
So the dunce cap is homotopic to which is contractible.
A question from my classmate:
Question. The Quintic Del Pezzo 3-fold of Degree is the intersection of with a codimension linear subspace. Show that for any point , 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 has dimension (it’s isomorphic to ). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of characterising lines on is by an incidence correspondence argument.
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.
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