CT1

ZCC
July 22, 2025

Lecture 1

Abstract

During the initial few lectures, we follow RHA-1 and RHA-2.

The main theorem of this lecture is Eklof-Trlifaj’s theorem, which states that a cotorsion pair over $C(R)$ cogenerated by a set is complete; our proof is anagolous to Quillen’s small object argument. We subsequently provide some criterion on determining whether a cotorsion pair is cogenerated by a set.

On Orthogonality

Cotorsion theory is usuall established over a category with good extension functor, e.g., triangulated categories, exact categories or simply Abelian categories. A generalisation is external triangulated categories. We denote $\mathcal{A}$ as a general Abelian category.

$⟂$ for Objects

We shall firstly explain the terminologies.

($⟂$). Say $X⟂ Y$, provided $\mathrm{Ext}^1(X,Y) = 0$. It is straightforward to generalise such notion to classes, e.g.,

  • $\mathcal{X} ⟂ \mathcal{Y}$ if $\mathrm{Ext}^1(X , Y) = 0$ for arbitrary $X ∈ \mathcal{X}$ and $Y ∈ \mathcal{Y}$.
  • $\mathcal{X}^⟂$ consists of object $Y$ such that $\mathcal{X} ⟂ \{Y\}$;
  • $^⟂ \mathcal{Y}$ consists of object $X$ such that $\{X\} ⟂ \mathcal{Y}$.

Show the Galois connection, e.g., $({}^⟂ (\mathcal{X}^⟂ ))^⟂ = \mathcal{X} ^⟂$.

We adopt the notion of triangulated categories. For classes $\mathcal{X}$ and $\mathcal{Y}$, we set

Verify carefully the followings identities for either triangulated categories or exact categories:

  1. $\mathrm{Cone}(𝒳 , \mathrm{Cone}(𝒴 , 𝒵 )) = \mathrm{Cone}(𝒴 ∗ 𝒳 , 𝒵 )$;
  2. $\mathrm{coCone}(\mathrm{coCone}(𝒳 ,𝒴 ), 𝒵 ) = \mathrm{coCone}(𝒳 , 𝒵 ∗ 𝒴 )$;
  3. $\mathrm{Cone}(𝒳 , \mathrm{coCone}(𝒴, 𝒵 )) = \mathrm{coCone}(\mathrm{Cone}(𝒳 , 𝒴 ), 𝒵 )$;
  4. $𝒳 ∗ (𝒴 ∗ 𝒵 ) = (𝒳 ∗ 𝒴 )∗ 𝒵$;

and

  1. $\mathrm{coCone}(𝒳 ,𝒵 )∗ 𝒴 ⊆ \mathrm{coCone}(𝒳 ∗ 𝒴 , 𝒵 )⊇ \mathrm{coCone}(𝒴 ,\mathrm{Cone}(𝒳 ,𝒵 ))$;
  2. $𝒴 ∗ \mathrm{Cone}(𝒳 , 𝒵 ) ⊆ \mathrm{Cone}(𝒳 , 𝒴 ∗ 𝒵 ) ⊇ \mathrm{Cone}(\mathrm{coCone}(𝒳 ,𝒵 ),𝒴 )$.

One observes that Noether’s isomorphism and octagon axiom play a significant rôle.

(Cotorsion theory). Let $\mathcal{C}$ and $\mathcal{F}$ be full-subcategories of $\mathcal{A}$. Say $(\mathcal{C}, \mathcal{F})$ is a cotorsion theory, provided $\mathcal{C}^⟂ = \mathcal{F}$ and $\mathcal{C} = {}^⟂ \mathcal{F}$.

Show that the following are cotorsion theories:

  1. $({}^⟂ \mathcal{X}, (^⟂ \mathcal{X})^⟂)$, the cotorsion theory generated by a class $\mathcal{X}$,
  2. $({}^⟂ (\mathcal{X}^⟂), \mathcal{X}^⟂)$, the cotorsion theory cogenerated by a class $\mathcal{X}$,
  3. $(𝖮𝖻, 𝐈𝐧𝐣)$ and $(𝐏𝐫𝐨𝐣 , 𝖮𝖻)$, we say $(𝐏𝐫𝐨𝐣, 𝖮𝖻 , 𝐈𝐧𝐣)$ a cotorsion triple. Note that the cotorsion $∞$-ple exists over Forbenius categories.

$⋔$ for Morphisms

Recall that

In short, morphisms are the first-class citizens in category theory. Once we generalise the ideal of $\mathrm{Ext}$ to morphisms, we see that it is indeed a weak orthogonality.

The followings has to do with the concept of factorisation systems (FS), which provides a way of viewing category $\mathcal{C}$ as a compositional product of two classes of morphisms $\mathcal{L}$ and $\mathcal{R}$. The key requirements are the lifting property (weak orthogonality) and the (functorial) factorisation property.

($⋔$, $⟂$). Say $f ⋔ g$, provided any commutative square $(α , β ) : f → g$ admits a antidiagonal dashed morphism $s$ making the entire diagram commutes. In diagram,



Once $s$ is unique, we write $f ⟂ g$.

We also generalise such notion to classes.

Show that $𝐌𝐨𝐧𝐨 ⟂ 𝐄𝐩𝐢$ and $𝐄𝐩𝐢 ⋔ 𝐌𝐨𝐧𝐨$ over $𝐒𝐞𝐭𝐬$. Indeed the former one is FS and the latter one can be modified into several functorial WFS. Note that any set map $f : X → Y$ admits

  • a graph factorisation $X \xrightarrow{e_1} X ⊔ Y \xrightarrow {(f,1)} Y$, and
  • a co-graph factorisation $X \xrightarrow{\binom{1}{f}} X × Y \xrightarrow {p_2} Y$.

We introduce a lemma connecting $⟂_{𝖮𝖻}$ and $⋔_{𝖬𝗈𝗋}$.

(Weak extension lifting lemma). For object $X$ and a class of objects $\mathcal{Y}$, one has

$$\begin{equation} \{X\} ⟂ \mathcal{Y} \quad ⟷ \quad \{0 → X\} ⋔ \mathrm{coCone}^{-1}(\mathcal{Y}). \end{equation}$$

Here $f ∈ \mathrm{coCone}^{-1}(\mathcal{Y})$ iff $\mathrm{coCone}(f) ⊆ \mathcal{Y}$.

$(→)$. Once $X ⟂ Y$, any $Y ∈ \mathrm{coCone}(p)$ gives the surjection $\mathrm{Hom}(X, p)$ (by les). Hence $(0 → X) ⋔ p$. $(← )$. Conversely, the $⋔$-condition implies the surjection $(X, p)$ for any $p ∈ \mathrm{coCone}^{-1}(\mathcal{Y})$. Hence any ses $Y → E \xrightarrow q X$ splits as $\mathrm{coCone}(q) ⊆ \mathcal{Y}$. Now $\mathrm{Ext}^1(X,Y) = 0$.

We discuss the closure property of $(-)^⋔$ and ${}^⋔(-)$ with the following theorem.

Let $\mathcal{S}$ be arbitrary class of morphisms. Then ${}^⋔ \mathcal{S}$ is closed under retracts (1), pushouts (2), coproducts (3), and transfinite compositions (4).

We show (1), (2), (3) and (4) as follows.

(1). Let $f$ be a retract of $F ∈ {}^⟂ \mathcal{S}$. Write $f \overset {(a,a')} ↪ F \overset {(b,b')} ↠ f$. For any commutative square $(m,n) : f ⇒ g$ with $g ∈ \mathcal{S}$, there exists $s$ giving $F ⋔ g$. Hence $sa'$ gives $f ⋔ g$.



(2). For arbitrary pushout diagram $θ : f → f'$ and $f ⋔ g$, we show that $f' ⋔ g$. For any $(m,n) ; f' ⇒ g$, there is $s$ making all solid arrows commutes. The morphism $t$ is obtained by $(s,m)$ with the pushout diagram. Finally, the universal property shows $g ∘ t = n$. Hence $t$ is the desired lifting morphism.



(3). Given a collection of $f_λ ∈ {}^⋔ \mathcal{S}$, we take $(m_λ , n_λ) : f_λ ⇒ g$ with lifting morphisms $s_λ$. We claim that

  • $((m_λ ∘ e_λ),( n_λ ∘ e_λ)) : ∐ f_λ ⇒ g$ is commutative with lifting morphism $s_λ ∘ e_λ$.

It is straightforward to verify the adjoint lifting $$\begin{equation} L(f) ⋔ g \quad ↔ \quad f ⋔ R(g)\qquad (L ⊣ R). \end{equation}$$

By construction, we have the lifting of functors in $𝐅𝐮𝐧𝐜𝐭 (Λ , \mathcal{C})$:



(4). A transfinite composition is characterised by a colimit-preserving functor $(X,f) ∈ 𝐅𝐮𝐧𝐜𝐭 (α , \mathcal{C})$ for $α$ an ordinal. To be explicit,

$$\begin{equation} X_0 \xrightarrow {f_{1,0}} X_1 \xrightarrow {f_{2,1}} X_2 \xrightarrow {f_{3,2}} \cdots X_γ \xrightarrow {f_{γ+1, γ}} X_{γ+1} → \cdots. \end{equation}$$

Here for limit ordinal $γ$, we have $X_γ = X_{(\varinjlim _{α < γ }α)}= \varinjlim _{α < γ} X_α$. Assume each $f_∙$ belongs to $^⋔ g$, we show that any $(m,n) : f_{α , 0} ⇒ g$ admits some liftings. By transfintie induction, we can inductively construct the lifting morphisms $s_δ$.

(From $s_δ$ to $s_{δ +1}$). Assume $s_δ$ is constructed, we construct $s_{δ +1}$ by $f_{δ +1 , δ }⋔ g$ as follows:



(For limit oridnal $δ$). We show the limit morphism $$\begin{equation} s_δ ∈ (X_δ , A) ≃ \varprojlim_{β < δ } (X_{β } , A) ∋ (s_β)_{β < δ} \end{equation}$$ has lifting property. Notice that the counit $\varprojlim_I (?)^I → ?$ is an isomorphism. The commutativity of upper left square is equivalently verified by adjoint lifting:



Here the commutative of the right triangle is due to our induction. The commutativity lower right part, $g ∘ s_δ = n$, is due to the universal property of colimits.

Show the following closure properties for $⟂ _{𝖮𝖻 }$: the class $^⟂ \mathcal{Y}$ is closed (if $∃$) under retracts, extensions, coproducts, and transfinite filtrations.

  • Say $X$ is a transfinite filtration of $\mathcal{X}$, when there is a chain of subobjects $X_0 ⊆ X_1 ⊆ \cdots ⊆ X_γ⊆ \cdots$ such that
    • $X_0 = 0$;
    • $(X_{γ + 1} / X_γ) ∈ \mathcal{X}$;
    • $X_∙$ presetves colimits;
  • Hint: $X ∈ {}^⟂ \mathcal{Y}$ iff $(0 → X) ∈ {}^⟂ (\mathrm{coCone}^{-1}(\mathcal{Y}))$.

We omit the dual version of the above propositions.

We complete this subsection with a well-known theorem.

(Extension lifting). If $\mathcal{C} ⟂ \mathcal{F}$, then $\mathrm{Cone}^{-1}(\mathcal{C}) ⋔ \mathrm{coCone}^{-1}(\mathcal{F})$.

By a diagram chasing over bi-complex.

Complete Cotorsion Pair

Say a cotorsion pair $(\mathcal{C}, \mathcal{F})$ is comlete, provided $\mathrm{coCone}(\mathcal{F},\mathcal{C}) = \mathrm{Cone}(\mathcal{F},\mathcal{C}) = 𝖮𝖻 (\mathcal{A})$.

We write $X → X_F → X_C$ or $X^F → X^C → X$ for a general ses’s. There is no functoriality for such $\mathcal{C}$ or $\mathcal{F}$ replacements.

In fact, these conditions are overdetermined.

(Wakamatsu (若松) lemma). Let $(\mathcal{C}, \mathcal{F})$ be a cotorsion pair. Consider the conditions:

  1. $\mathrm{Cone}(\mathcal{F},\mathcal{C}) = 𝖮𝖻 (\mathcal{A})$;
  2. for any object $X$, there exists $X ↪ F$ for $F ∈ \mathcal{F}$;
  3. $\mathrm{coCone}(\mathcal{F},\mathcal{C}) = 𝖮𝖻 (\mathcal{A})$.

We claim that $(1).+(2). ↔ (1).+(3).$.

We show $(1).+(2). ↔ (1).+(3).$, as the other direction is trivial. To show every $X$ lies in $\mathrm{coCone}(\mathcal{F}, \mathcal{C})$, we take $X ↪ F$ and the approximation $(F/X)$. We pullback $r_1$ and $c_3$ in the following diagram:



Now $E ∈ \mathcal{F}$ by vanishing of $\mathrm{Ext}^1(c_2, -)$; $r_2$ is what our desire.

We ask under what conditions $(-)^C$ and $(-)^F$ are functorial. The replacement $X^C → X$ is also known as the $\mathcal{C}$-precover or right $\mathcal{C}$ approximation. Since

Hence, any morphism $C → F$ factors through $F^C → F$, where $F^C ∈ \mathcal{C} ∩ \mathcal{F}$. This yields

Set $ω := \mathcal{C} ∩ \mathcal{F}$. Then $\mathrm{Hom}_{\mathcal{A} / ω}(\mathcal{C}, \mathcal{F}) = 0$.

The inclusion $\mathcal{C} /ω ↪ \mathcal{A} / ω$ admits a right adjoint $(-)^C$.

We show $p_∗ : (C, X^C)_{\mathcal{A} / ω} → (C,X)_{\mathcal{A} / ω}$.
The surjection is clear. The long es $(C,X^C)_\mathcal{A} → (C,X)_\mathcal{A} → \mathrm{Ext}^1(C, X^F) = 0$ shows $p_∗$ is surjective in $\mathcal{A}$. Hence $p_∗$ is also surjective in $\mathcal{A} / ω$.
We show the injectivity, i.e., $[f] = [0]$ when $[pf] = 0$. By assumption, $pf$ factors through some $N ∈ ω$. Hence, there exists $s : N → X^C$ making $↻$ commutative:



The long se shows $(f - sa) ∈ \ker p_∗ = \operatorname{im} i_∗$. Since $(f-sa)$ factors extends along some $C → X^F$, the morphism vanishes in $\mathcal{A}/ω$. Hence $[f] = [f-sa] + [sa] = [0]$.

We conclude that



There is a similar lemma (with a long proof) of adjunction lemma for complete cotorsion pairs on chain complexes. For any complete cotorsion pair $(\mathcal{C}, \mathcal{F})$, there exists a right adjoint $(-)^C$ to the inclusion $K(\mathcal{C}) ↪ K(R)$. These two lemma coincides when $ω = C_{ac}(R)$ (see here for equivalent definitions of $K(\mathcal{A})$).

Small Objects

Ordinals and Cardinals

We highlight a subtle point in the study of well-ordered sets (by the axiom of choice, every set may be well-ordered). A well-ordered set $(X, ≤)$ is totally ordered; that is, for any $i, j ∈ X$, one has $i ≤ j$ or $j ≤ i$. Zorn’s lemma demonstrates that an infinite strictly descending chain is prohibited; specifically, there is no $\cdots ≨ i_2 ≨ i_1 ≨ i_0$ in a well-ordered set. In particular, $(ℚ, ≤)$ is not well-ordered.

We define ordinals and cardinals in the sense of J. von Neumann. For simplicity, we introduce neither ordinal arithmetic nor universe.

(Ordinal). The collection of ordinals refers to (a skeleton of) the category of well-ordered sets under the usual set-theoretic axioms favoured by algebraists. Inductively, any ordinal is obtained by the following procedures:

  1. $0 = ∅$ is the initial ordinal;
  2. for any ordinal $α$, the successor ordinal is defined as $α^+ := α ⊔ \{α\}$;
  3. if there exists a set of oridnals $S$, such that $0 ∈ S$ and $(β ∈ S) → (β ^+ ∈ S)$, then we define $α = ⋃ S$ as a limit ordinal.

The existence of limit ordinal is clear. If we assume $n^+ = n+1$ for $n ∈ ℕ$, then $ω = ℕ$ is the first limit oridnal.

Show that a limit ordinal is never a successor ordinal. Hint: there is no such set $X ∈ X$ by axiom of regularity.

Note that $β ≨ α$ iff $β ∈ α$ as sets. We do not employ ordinal arithmetic throughout these lectures, as it may be confusing to beginners.

There are instances where $α ≠ β$, yet there exists a bijection of sets ($|α| = |β|$). For example, we denote $ω = ⋃_{n ∈ ℕ} n$ as the first infinite ordinal, and $|ω| = |ω^+|$.

We demonstrate a technique termed transfinite induction (how ordinals operate). The proposition states:
For any $M := ∐ _{i ∈ I} M_i$ where each $M_i$ is countably generated, $\{f_n\}_{n ∈ ℕ} ⊆ \mathrm{End}(M)$. Then there exists a transfinite filtration $0 = M_0 ⊆ M_1 ⊆ \cdots M_γ ⊆ \cdots$ of $M$ such that

  1. $\{(f_n)|_{M_γ}\}_{n ∈ ℕ} ⊆ \mathrm{End}(M_γ)$;
  2. $M_{γ} = M_γ ⊕ (\text{countably many}\ M_i \text{'s})$.

Set $P_γ := (∃ \ \text{a construction for } \{M_α\}_{α ≤ γ})$. We show via transfinite induction that

  1. $P_0$ is true;
  2. $P_γ → P_{γ +1}$;
  3. $⋀ _{α < γ}(P_α) → P_γ$ for $γ$ a limit ordinal.

Step 1 and Step 3 are clear. We show Step 2 with $P_γ → P_{γ + 1}$ (when $M_γ ⊊ M$). Upon selecting an arbitrary summand $M^0$ with $M^0 ∩ M_γ = 0$, the countably generated image $∑ _{n ∈ ℕ }f_n(M^0)$ is covered by countably many $M_i$’s. Denote their direct sums as $M^1$. Consider $∑ _{n ∈ ℕ }f_n((M^1 ⊕ M^0)$ and obtain $M^3$. Now $M_{γ +1} := (⨁_{n ∈ ℕ} M^n) ⊕ M_γ$ is the desired construction.

Projective modules are coproducts of countably generated projective modules.

Any projective module $P$ is of the form $f_0(∐ _λ R)$ for $(f_0)^2 = f_0$. Taking $M = ∐ _λ R$ and $\{f_n\}_{n ∈ ℕ} = \{1_M, f_0\}$ yields the $F^{(ω)}$-filtration of $M$. Since $f_0(-)$ preserves coproducts, $P = f_0(M)$ is the coproduct of summands of $F^{(ω)}$, i.e., the coproduct of countably generated projective modules.

Let $C^0([0,1], ℝ)$ be the ring of functions. Show that $\{f ∣ \mathrm{supp}(f) ⊆ (0 , 1]\}$ is an indecomposable projective module which is not finitely generated (but/hence countably generated).

(Cardinal). An ordinal $α$ yields the cardinal $\min \{β ∣ |β | = |α|\}$.

The above is well-defined, i.e., $\{β ∣ |β | = |α|\}$ is a set with a unique least element.

Let $X$ denote the under lying set of $α$. Zorn’s lemma ensures the existence of well-ordered structure over $X$, and all these well-ordered structures forms a set controlled by $\mathrm{Hom}_{𝐒𝐞𝐭𝐬 }(\mathrm{Hom}_{𝐒𝐞𝐭𝐬 }(X,X) × \mathrm{Hom}_{𝐒𝐞𝐭𝐬 }(X,X), 2)$ (by definition of binary relation). Hence $\{β ∣ |β | = |α|\}$ is a set.
We can easily check by transfinite induction that any ordinal $γ$ is well-orderd. Indeed for any ordinal $γ$, there is no countably descending chain $γ ∋ γ_1 ∋ \cdots$. Hence, all non-empty subset of $γ$ has a minimal element. Since $γ$ is totoally ordered, the minimal element in unique.

An ordinal $α (≠ 0)$ is a cardinal iff $|β| ≠ |α|$ for any $β ∈ α$.

We typically use $λ$, $μ$, $κ$ to denote cardinals, or $|X|$ for $X$ a set.

(Cardinal vs Sets). Convince yourself that, for any sets $S$ and $T$, the following cardinal arithmetic leads to no contradiction:

  1. $|S| + |T| := |S ⊔ T|$;
  2. $|S| × |T| := |S × T|$;
  3. $|S|^{|T|} := |S^T| := |\mathrm{Hom}(T,S)|$.

Note that

  • the universal property of coproduct gives $λ ^{μ + κ} = λ ^μ λ ^κ$,
  • the universal property of product gives ${μ × κ}^λ = μ^ λ κ^λ$,
  • the tensor-hom adjunction gives $λ^{μ × κ} = (λ^μ) ^κ$.

$λ ⋅ λ = λ$ for any infinite cardinal $λ$.

Suppose there exists a least counterexample among infinite cardinals $κ$ (clearly $κ ≠ ω$). Now consider the function

$$\begin{equation} f : κ × κ → κ,\quad (m,n) ↦ \max (m,n). \end{equation}$$

Observe that the graph of $f$ is larger than the codomain of $f$ ($|\mathrm{Graph}(f)| = κ ⋅ κ > κ$).

Now equip $\mathrm{Graph}(f) = \{(f(x,y), x,y)\} ⊆ κ × κ × κ$ with the lexicographic order, which we learned the case $κ = 10$ in primary school. There exists a point $(α,β,γ) ∈ \mathrm{Graph}(f)$ such that $|\{x ∈ \mathrm{Graph}(f) ∣ x < (α,β,γ)\}| = κ$. Now

$$\begin{equation} κ ≤ \underset{\text{domain of truncated } \ f} {\underbrace{|α| × |α|}} \xlongequal{\text{by assumptions}} |α| < κ. \end{equation}$$

A contradiction.

Show that $\max (λ , μ) = λ + μ = λ ⋅ μ$ for any $λ ≥ ω$ and $μ ≠ 0$.

The above identity simplifies cardinal arithmetic.

Cofinality, and Small

Let $ω' := ⋃ _{n ∈ ℕ} (ω^{n\ + \text{'s}}) ∪ ⋃_{n ∈ ℕ} n$ be the second limit ordinal, which resembles a juxtaposition of two $ℕ$’s. Clearly, one may find an inclusion $i : ω ↪ ω '$ such that $\varinjlim_{ω '} X_∙ ≃ \varinjlim _{ω} X_∙$ for any limit calculation problems.

This signifies that $ω$ is a cofinal subsystem of $ω '$ via $i$. Informally, $ω '$-induction is simplified via some $ω$-induction.

What if certain kinds of $α ↪ β$ fail to be cofinal?

For instance, $1$ is not cofinal for any limit ordinal $α$. Hence, any choice of element in the set $x ∈ ⋃ _{γ < α}X_γ$ implies $x ∈ X_γ$ for some $γ < α$.

For ordinals $α$ and $β$, say $α$ is $β$-filtered provided $\sup (\operatorname{im}(f)) ∈ β$ for any order-preserving morphism (more than inclusions) $f : α → β$.

We slightly simplify the definition that $α$ is $β$-filtered.

  1. WLOG, we assume $α$ to be a limit ordinal. Clearly, the definition is trivial when $α = γ^+$.
  2. It is anodye to assume $β$ to be a cardinal. That is, $β$-filtered ordinals coincides $β '$-filtered ordinals if then have the same underlying cardinal $κ$. WLOG suppose $β ' = κ$.
    1. Assume $α$ is $β$-filtered. Any order preserving morphism $κ → α$ extends to $β ↠ κ → α$ by compressing all $\{γ ∣ κ < γ\}$. Hence $α$ is $κ$-filtered.
    2. Assume $α$ is $κ$-filtered. It suffices to construct an order preserving surjection $p : κ ↠ β$. Let $φ : κ → β$ be bijection of underlying sets. Set $p(0) = φ (0)$, $p(γ^+) = \max \{p(γ), φ (γ ^+)\}$ for successors, and $p (γ) = \sup_{δ < γ} \{p(δ)\}$ for limit ordinal $γ$.

Show that it is anodyne to define a limit ordinal $α$ is cofinal to a set $X$.

Large limit oridnals may have small cofinality. Let $ℵ _0 = ω$ denote the first infinite cardinal. There exists the least cardinal, $ℵ _1$, larger then $ℵ _0$. We define $ℵ _∙ : ω → \text{cardinals}$. Again, there is a least cardinal larger than any $ℵ _k$, denoted as $ℵ _{ω}$. Now, the functor $ℵ _∙ : ω → ℵ _ω$ satisfies

$$\begin{equation} \varinjlim {}^{fil}_{k ∈ ω } ℵ _k = \sup \{ℵ _k ∣ k ∈ ω\} = ℵ _ω ;\quad \text{but} \ ℵ _ω ≫ ω ! \end{equation}$$

A slogan: $α$ is $κ$-filtered if and only if $α$ has better cofinality than $∀ β ∈ κ$.

Show that

  • the first $κ$-filtered ordinal ($\widetilde κ$) is precisely the least cardinal larger than $κ$. Hint: $κ ⋅ κ < \widetilde κ$, and $\widetilde κ$ is a successor cardinal.
  • not every cardinal larger than $κ$ is $κ$-filtered.

Hence, there exist non-$ω$-filtered ordinals. One may consult this interesting example for deeper understanding.

The following corollary motivates the concept of small.

Let $α$ be $κ$-filtered. For any set $X$ with $|X| = κ$, and

  • for any $α$-transfinite composition of morphisms, i.e., a colimit-preserving functor $Y_∙ : α → \mathcal{C}$.

Then, any set map $φ : X → \varinjlim Y_∙$ factors through some $Y_γ$.

For any $x ∈ X$, $φ |_x : x → \varinjlim Y_∙$ factors through some $Y_{γ _x}$. Select $x ↦ Y_{γ _x}$ for arbitrary $x ∈ X$, which defines a map $γ _∙ : X → α$. By definition, $\sup (\operatorname{im}(γ _∙ )) ∈ α$. Hence, $φ$ factors through $Y_{\sup (\operatorname{im}(γ _∙ ))}$.

(Small object). Say $X$ is $κ$-small if, for any $κ$-filtered ordinals and $α$-transfinite compositions, the following is an isomorphism:

$$\begin{equation} \varinjlim _{γ < α } (X, Y_γ ) \xrightarrow ∼ (X, \varinjlim _{γ < α } Y_γ ). \end{equation}$$

We have demonstrated that any set $X$ is $|X|$-small.

Demonstrate that $X ∈ C(R)$ is $\max (|X|, |R|, ω)$-small. Hint: It suffices to show for graded modules, and thus for $𝐌𝐨𝐝 _R$. Note that $M ∈ 𝐌𝐨𝐝_R$ is $|R| ⋅ |X|$-small by tracking the image of cyclic modules. The final answer is $|X| ⋅ |R| ⋅ ω = \max (|X|, |R|, ω)$.

It appears that in good categories, every object is small (with respect to some cardinal). The following example shows that, in certain problematic categories, finite objects may not be small.

We begin with two kinds of topological spaces.

  • A Sierpiński space consists of $2$ points and $3$ open sets, which appears as $\boxed {x \ \boxed y}$.
  • A topological space over a limit ordinal $α$ has base set $α^+$ and open sets $\mathcal{O}_γ : = \{x ∣ x > γ\}$. For $α = ω$, the space appears as $\boxed{0 \ \boxed{1 \ \boxed{2 \ \boxed {\cdots ω }}}}$.

For any ordinal $α$, consider the $α$-transfinite composition

$$\begin{equation} \begin{matrix}\boxed{0 \ \boxed{1 \ \boxed{2 \ \boxed {\cdots α }}}}\\\boxed{0 \ \boxed{1 \ \boxed{2 \ \boxed {\cdots α }}}}\end{matrix} ↠ \boxed{0 \ \begin{matrix}\boxed{1 \ \boxed{2 \ \boxed {\cdots α }}}\\ \boxed{1 \ \boxed{2 \ \boxed {\cdots α }}}\end{matrix}} ↠ \boxed{0 \ \boxed {1 \ \begin{matrix}\boxed{2 \ \boxed {\cdots α }}\\ \boxed{2 \ \boxed {\cdots α }}\end{matrix}}} ↠ \cdots \end{equation}$$

Consider the continuous map from the Sierpiński space to the colimit

$$\begin{equation} ϕ : \boxed{\begin{matrix}x \\ \boxed y\end{matrix}} → \boxed{0 \ \boxed{1 \ \boxed{2 \ \boxed {\cdots \begin{matrix}α\\α \end{matrix} }}}}, \quad \binom{x}{y} ↦ \binom{α _{↑}}{α _{↓}}. \end{equation}$$

Since $ϕ$ never factors through any object in the composition; if it did, then there would exist some open neighbourhood of $α _↑$ whose preimage is $\{x\}$ (not open).

Demonstrate that the compact objects in topological spaces are precisely the finite discrete spaces.

Eklof-Trlifaj Lemma

We present the Eklof-Trlifaj lemma with a proof analogous to Quillen’s small object argument, albeit considerably simpler.

Recall that the cotorsion pair cogenerated by a class of objects $\mathcal{X}$ is denoted by $(^⟂(\mathcal{X}^⟂ ), \mathcal{X}^⟂ )$.

We begin with the following deduction: a cotorsion theory in $C(R)$ is cogenerated by a set of objects $\mathcal{X}$ if and only if it is cogenerated by a single object $S:= (∐ \mathcal{X})$:

(Eklof-Trlifaj lemma). For any object $S ∈ C(R)$, the cotorsion pair cogenerated by $S$ is complete.

Let $𝐅𝐢𝐥 (S)$ denote the class of objects generated by $S$ under transfinite filtration. Clearly, $𝐅𝐢𝐥 (S) ⊆ {}^⟂ (S^⟂ )$. It suffices to show that $\mathrm{coCone}(S^⟂ , 𝐅𝐢𝐥 (S)) = C(R)$. In this case, the $S^⟂$-term appears as a colimit of the transfinite $S$-filtration: $$\begin{equation} M = M_0 ⊆ M_1 ⊆ \cdots ⊆ M_γ ⊆ \cdots \quad (γ < α) \end{equation}$$ We begin with short exact sequences $0 → K \xrightarrow i P \xrightarrow p S → 0$ where $P$ is projective. The construction of $M_γ$ is via a universal pushout



Here $Φ _γ ∈ (K^{∐ (K, M_γ )}, M_γ ) ≃ (K,M_γ)^{(K, M_γ )} ∋ 1_{(K,M_γ)}$. In particular,

  • for any $g : K → M_γ$, the morphism $f_γ ∘ g : K → M_{γ +1}$ extends along $i : K → P$.

Let $α$ be $(|K|, |R|, ω )$-filtered. It remains to show that $M_α ∈ {}^⟂ S$. Equivalently, $(i, M_α)$ is surjective. For any $g : K → M_α = \varinjlim _{γ < α } M_γ$, we observe that $g$ factors through some $M_β$. The universal pushout yields the dashed arrow.



The composition $P → M_{β +1} → M_α$ is our desired morphism.

Construct the dashed arrow. Hint: $$\begin{equation} g' = K \xrightarrow {ι _{g'}} K^{∐ (L, M_β)} \xrightarrow {Φ _β} M_β. \end{equation}$$

We do not require any of $\mathcal{S}$, $\mathcal{S}^⟂$, or $^⟂ (\mathcal{S}^⟂ )$ to be closed under $Σ ^±$. We only use that, $S$ is a $\max(|S|, |R|, ω)$-small object.

We remark that Eklof-Trlifaj lemma holds in Grothendieck category with enough projectives.

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