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## Some (easy) definitions:

Appropriated from Wiki:

An *abelian category* is a category that 1) enriched over **Ab**, so every hom-object is an abelian group and morphism-composition is bilinear, 2) admits a zero object and bi (direct) products, 3) admits kernels and cokernels, and 4) has the property that every monomorphism (resp. epi-) is the kernel (in the category-theoretic kernels-are-morphisms sense) (resp. cokernel) of some morphism.

(Examples: fix a ring \(R\); then the category of 1) \(R\)-modules is an abelian category; 2) sheaves of abelian groups on a topological space; 3) abelian groups.)

A sequence of morphisms \(A_1 \overset{f_1}{\to} A_2 \overset{f_2}{\to} A_3 \cdots\) is *exact* when \(\mathrm{im} f_i \simeq \ker f_{i + 1}\) for each \(i\).

A short exact sequence is an exact sequence \(A_1 \overset{f_1}{\to} A_2 \overset{f_2}{\to} A_3\) where \(f_1\) is a monomorphism (resp. \(f_2\), epi); in an abelian category we write

\[0 \to A_1 \overset{f_1}{\to} A_2 \overset{f_2}{\to} A_3 \to 0.\]

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## Climbing Mount Qual:

Summertime’s here, folks.

Besides playing catchup with my general education classes this summer, I’ll be studying for the qualifying exams in the fall. I’ll try to post practice problems at least once every few days.

See you all soon :)

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math textbooks are the best

thanks for the helpful commentary

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I have no idea why I’m using this letter, I can’t even pronounce it.Mathematical Logic lecturer on the letter ‘xi’ (via mathprofessorquotes)

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For example, why is 6/2 = 3? We can take a 6-element set \(S\) with a free action of the group \(G = \mathbb{Z}/(2)\) and identify all the elements in each orbit to obtain a three-element set \(S/G\).John Baez,

*From Finite Sets to Feynman Diagrams*

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Now, given a category C, we may ‘decategorify’ it by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a set (or class) whose elements are isomorphism classes of objects of C. This process is dangerous, because it destroys useful information. It amounts to forgetting which road we took from x to y, and just remembering that we got there. Sometimes this is actually useful, but most of the time people do it unconsciously, out of mathematical naivete. We write equations, when we really should specify isomorphisms. ‘Categorification’ is the attempt to undo this mistake. Like any attempt to restore lost information, it not a completely systematic process. Its importance is that it brings to light previously hidden mathematical structures, and clarifies things that would otherwise remain mysterious. It seems strange and complicated at first, but ultimately the goal is to make things simpler.John Baez,

*From Finite Sets to Feynman Diagrams*

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## There exists a surjection \(2^{N_0} \rightarrow \aleph_1\)

(appropriated from this discussion at math.SE)

Let \(f:\mathbb{N} \rightarrow \mathbb{Q}\) be an enumeration of the rationals; since \(\forall \alpha <_{ON} \omega_1 \exists I \subseteq \mathbb{Q}\) such that \(I \simeq \alpha\) obtain an embedding \(\iota:\omega_1 \rightarrow 2^{\omega}\) by \(\alpha \mapsto f^{-1}(I)\); inverting this embedding and sending everything not in \(\mathrm{range}(\iota)\) to, say, \(0\), gives the required surjection.

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## Every countable ordinal embeds into the rationals

Define by induction the *refinement* \(r(\alpha)\) of an ordinal \(\alpha\): set

\[r_0(\alpha) = \alpha\] and \(r_n(\alpha) = r_{n-1}(\alpha)\) with new elements inserted between every pair of elements \(a_1,a_2\) such that \(\not \exists a_3 : a_1 < a_3 < a_2\); refine the well-ordering of \(r_{n-1}(\alpha)\) to reflect this. Then we have \[r(\alpha) = \bigcup_{n < \omega} r_n(\alpha)\] is a countable dense linear order hence isomorphic to \(\mathbb{Q}\), and \(r_0(\alpha) \hookrightarrow r(\alpha)\) in the obvious way.

To obtain those new elements, take ordered pairs, i.e \(a_3 = \langle a_1,a_2 \rangle\); since we only iterate up to \(\omega\) this is well-defined and we’re fine.

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Of course, maybe it’s as bad to think of this particular instantiation of ordinals as it is bad to think of quotient groups of a group as equivalence relations on cosets of that group. Ordinals are just isomorphism classes of well-ordered sets (equivalently a skeleton of the category of well-ordered sets); they’ve only got wacky properties as von Neumann ordinals because we try to make \(\in\) functorially quack like \(<_{ON} \iff \exists\) an order-embedding.

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Fun fact: every ordinal (except for zero and one, which contain, respectively, nothing and zero) contains only one singleton, and it’s one.

Appropriate, no?

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