Much of this (up to and including cofinality) is taken from Set Theory by Thomas Jech [Jec03], but you can probably find it in any reasonable set theory book. The rest is from Hovey’s book Model Categories [Hov99].

1 Well-ordered sets

A linearly ordered set (P,<) is well-ordered if every nonempty subset of P has a least element. A map f of ordered sets is increasing if x ≥ y ⇒ f(x) ≥ f(y). Lemma 1.1. Suppose that (W,<) is well-ordered. If f : W → W is an increasing function, then f(x) ≥ x for all x ∈ W. Corollary 1.2. The only automorphism of a well-ordered set is the identity. Corollary 1.3. If W_{1} and W_{2} are isomorphic well-ordered sets, there is a unique isomorphism between them.

Given a well-ordered set W and an element u ∈ W, the initial segment given by u is {x ∈ W : x < u}. Lemma 1.4. No well-ordered set is isomorphic to an initial segment of itself. Theorem 1.5. For any well-ordered sets W_{1} and W_{2}, exactly one of the following holds:

1. W_{1} is isomorphic to W_{2}.

2. W_{1} is isomorphic to an initial segment of W_{2}.

3. W_{2} is isomorphic to an initial segment of W_{1}.

If W_{1} and W_{2} are isomorphic well-ordered sets, say that they have the same order type.

2 Ordinals

A set T is transitive if every element of T is a subset of T. (That is, T is a subset of its power set.) A set is an ordinal number (or an ordinal) if it is transitive and well-ordered by ∈. For ordinals α and β, define α < β if α ∈ β.

• 0 = ∅ is an ordinal.

• If α is an ordinal and β ∈ α, then β is an ordinal.

• If α and β are distinct ordinals and α ⊂ β, then α ∈ β.

• If α and β are ordinals, then either α ⊂ β or β ⊂ α.

The class of ordinals, Ord, has these properties:

• Ord is linearly ordered by <.

• For each α ∈ Ord, α = {β: β < α}.

• For each α, α ∪ {α} is an ordinal, and α ∪ {α} = inf{β: β > α}.

So define α + 1 to be α ∪ {α}; this is called the successor to α. If α = β + 1 for some β, then α is called a successor ordinal. If α is not a successor ordinal, then α = sup{β: β < α}, and α is called a limit ordinal.

Theorem 2.1. Every well-ordered set is isomorphic to a unique ordinal number.

Let β be a limit ordinal. A β-sequence is a function whose domain is β:

⟨a_{ξ} : ξ < β⟩.

If ⟨α_{ξ} : ξ < β⟩ is a nondecreasing sequence of ordinals (so ξ < η implies α_{ξ} < α_{η}), then the limit of the sequence is

lim_{ξ→β}α_{ξ} = sup{α_{ξ} : ξ < β}.

Examples of ordinals:

0 = {}
1 = {0}
2 = {0,1}
3 = {0,1,2}
⋮
ω = {0,1,2,...}
ω + 1 = {0,1,2,...;ω}
⋮

3 Cardinals

An ordinal α is a cardinal number if |α| ≠ |β| for all β < α.

If W is well-ordered, there is an ordinal α such that |W | = |α|, so let |W | denote the least ordinal α such that |W | = |α|. (By the axiom of choice, every set can be well-ordered, so we can extend this notation to any set W.)

Note: all infinite cardinals are limit ordinals.

4 Cofinality and κ-filtered ordinals

Let α > 0 be a limit ordinal. An increasing β-sequence ⟨α_{ξ} : ξ < β⟩, for β a limit ordinal, is cofinal in α if lim_{ξ→β} α_{ξ} = α. Similarly, A ⊂ α is cofinal in α if supA = α.

For α an infinite limit ordinal, let cfα be the least limit ordinal β such that there is an increasing β-sequence ⟨α_{ξ} : ξ < β⟩ with lim_{ξ→β} α_{ξ} = α; this is called the cofinality of α. Note that cfα ≤ α always.

From [Hov99, Section 2.1]: given a cardinal κ, an ordinal α is κ-filtered if it is a limit ordinal and, if A ⊆ α and |A| ≤ κ, then supA < α. This holds if and only if the cofinality of α is greater than κ.

For example, if κ is finite, then a κ-filtered ordinal is just an infinite limit ordinal. If κ is infinite, the smallest κ-filtered ordinal is the first cardinal κ_{1} larger than κ.

5 Smallness

(This material is taken from [Hov99, Section 2.1].)

Suppose that C is a category with all small colimits, D is a collection of morphisms of C, A is an object of C and κ is a cardinal. We say that A is κ-small relative to D if, for all κ-filtered ordinals λ and all λ-sequences

X_{0} → X_{1} → ···→ X_{β} → ···,

such that each map X_{β} → X_{β+1} is in D, the map of sets

colim_{β<λ} C(A,X_{β}) → C(A,colim_{β<λ} X_{β})

is an isomorphism. We say that A is small relative to D if it is κ-small for some κ. We say that A is small if it is small relative to the whole category C. We say that A is finite relative to D if it is κ-small relative to D for some finite cardinal κ, and similarly, A is finite if it is finite relative to C. Finite means that maps from A commute with colimits of arbitrary λ-sequences, for any limit ordinal λ.

Examples:

• In the category of sets, every set is small. The finite sets (in the usual sense) are precisely the sets that are finite according to the above definition.

• In the category of R-modules, every module is small. Every finitely presented R-module is finite.

• In the category of topological spaces, the set {0,1} with the indiscrete topology is not small.

References

[Hov99] M. Hovey, Model categories, American Mathematical Society, Providence, RI, 1999.

[Jec03] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded.