A Note on Quillen's Model Structure

Note on Model Category

• Note on Model Category
  • Abstract
• Kan Extension
  • The Basic Definitions
    • Categories and Functors
    • Universe
    • Limits and Colimits
  • A note on Kan Extension
    • The Definition
    • Example: Homology from Limits
    • The Construction
    • On Derived Functors
• Simplicial Methods
  • Simplicial Objects
    • Basics
    • The Truncation
  • The Topologies
    • Motivation for Simplicial Methods
    • A Categorical Overview
    • The Pointed Construction
    • Equivalences
    • The Geometric Realisation
    • CW-approximation
  • Cofibrantly Generating Class
    • The (W)FS
    • Quillen’s Small Object Argument
    • Kan Complexes
    • Adjoint Lifting
  • Functors
    • Topological Space for Convenience
    • Hurewicz Map
    • DK Correspondence
    • (Homotopy) Spectra
    • Quillen’s Equivalence
• Quillen’s Model Structure

Kan Extension

In this section, we endeavour to elucidate the universal construction known as the Kan extension. The concept originates from the book by H. P. Cartan and S. Eilenberg in 1956, wherein it was introduced for the computation of derived functors. Further computational aspects are explored in D. Kan’s paper on adjoint functors in 1958. The general notion of extension, attributed to D. Kan, emerged circa 1960, and a modern treatment is presented in Mac Lane’s book Categories for the Working Mathematician.

The Basic Definitions

Categories and Functors

We clarify some of the notations: General categories are written in \mathcal font, e.g., $\mathcal{C}$ for a category, where

• $𝐊 _0 := 𝖮𝖻$ denotes the class of objects and $𝐊 _1 := 𝖬𝗈𝗋$ denotes the class of morphisms,
• $(X, Y)_\mathcal{C}$ or $\mathrm{Hom}_\mathcal{C}(X,Y)$ denotes the set of morphisms in $\mathcal{C}$ from $X$ to $Y$,
• $1_X$ denotes the identity morphism in $(X,X)_\mathcal{C}$,
• the notion of composition is denoted as $f ∘ g$, where $∘$ is of the type $𝐊_1 × _{𝐊_0} 𝐊_1 → 𝐊 _1$.

Specific categories are denoted in \mathbf font, e.g., $𝐒𝐞𝐭𝐬$ for the category of sets, $𝐌𝐨𝐝_k$ for the category of right $k$-modules. A general functor category is denoted as $𝐅𝐮𝐧(\mathcal{C}, \mathcal{D})$ or simply $\mathcal{D}^\mathcal{C}$, where

• a general functor $F$ is in italic font,
• a general natural transformation $θ = \{θ_X\}_{X ∈ 𝖮𝖻}$ is a Greek letter in italic font,
• the $\mathrm{Hom}$-object in the functor category is denoted as $𝐍𝐚𝐭[F,G]$.

When it is valid to regard the superclass of categories as a category itself, the vertical and horizontal compositions of natural transformations are denoted by $∘$ and $∗$, respectively. For instance, in the following diagram of natural transformations

$θ _2 ∘ θ _1$ (of the type $𝐊_2× _{𝐊_1}𝐊_2$) denotes the vertical composition of natural transformations, whilst $θ _2 ∗ θ _1$ (of the type $𝐊_2× _{𝐊_0}𝐊_2$) denotes the horizontal composition.

Universe

The conception of a universe is introduced by A. Grothendieck in this appendix of SGA4, which enables one to circumvent the set-theoretic paradoxes, notably Russell’s paradox. The assumption of a universe $𝔘$ specifies what is small, large, or uncontrollably large. Alternative approaches to avoiding such paradoxes exist; see M. A. Shulman’s note for further details. The following definition is adapted from a note by D. Murfet.

A universe is denoted in \mathfrak-font. It is a set, satisfying the following (overdetermined) properties:

1. If $x ∈ A$ and $A ∈ 𝔘$, then $x ∈ 𝔘$;
2. If $x ∈ 𝔘$ and $y ∈ 𝔘$, then $\{x,y\} ∈ 𝔘$;
3. If $x ∈ 𝔘$, then the powerset $\mathcal{P}(x) ∈ 𝔘$;
4. For any $I ∈ 𝔘$ such that $\{x_i\}_{i ∈ I} ⊆ 𝔘$, one has $⋃_{i ∈ I}x_i ∈ 𝔘$.

We define the small, large, and uncontrollably large sets (relative to the universe $𝔘$) as follows:

We do not omit the prefix $𝔘$-, since the terminology small object has a different meaning.

Throughout, we adopt the axiom ZFCU, rather than NBGC.

In conclusion, there exists a universe that accommodates the standard set-theoretic operations. Therefore, one may disregard the set-theoretic paradoxes and concentrate on the desired universe. The ensuing discussion is conducted under the assumption that the universe $𝔘$ is fixed; we underscore the selected universe when necessary.

Limits and Colimits

Before we proceed, it is advisable to be acquainted with the following phenomenon:

This likely arises from the principle of associativity, as the restriction operates on the right. Let $\mathcal{C}^→ = 𝐅𝐮𝐧𝐜𝐭 (→ , \mathcal{C})$ denote the morphism category, wherein the collection of objects is $𝖬𝗈𝗋(\mathcal{C})$ and the morphisms are commutative squares.

When we generalise $→$ to a $𝔘$-small diagram $I$, the functor $1_∙$ assigns each object $X$ to an $X$-valued diagram of type $I$. The bi-adjoints (if they exist) of $1_∙: \mathcal{C} → \mathcal{C}^I$ coincide with the definition of colimit and limit functor.

Let $F : \mathcal{C} → \mathcal{D}$ be a functor, and $X_∙ : I → \mathcal{C}$ be an $I$-diagram in $\mathcal{C}$ (as an object in the functor category $\mathcal{C}^I$ assigning $i$ to $X_i$). We shall frequently adopt the following notation.

• (Preserving). Say $F$ preserves (or commutes with) the limit of $X_∙$, if the induced cone $F(\varprojlim _I X) → FX_∙$ is already the limit.
• (Reflecting). Say $F$ reflects the limit of $X_∙$, if, if the induced cone $F(Y) → FX_∙$ is the limit, then $Y$ is the limit of $X_∙$ in the category $\mathcal{C}$.
• (Creating). Say $F$ creates the limit of $X_∙$, if the limit $\varprojlim _IFX$ exists in $\mathcal{D}$ and the entire cone is an image of the unique limit cone in $\mathcal{C}$ (in short, the limit in $\mathcal{C}$ is uniquely determined by that in $\mathcal{D}$).

The terminologies are clearly explained in the following example.

We summarise some special functors that preserve, reflect, or create limits and colimits.

1. A left exact/right exact/exact functor is defined to preserve all finite limits/finite colimits/finite limits and colimits.
2. A fully faithful functor reflects all limits and colimits.
3. Concrete forgetful functors usually create small limits and filtered colimits, e.g., the forgetful functor from $𝐌𝐨𝐝_k$ to $𝐒𝐞𝐭𝐬$, from $𝐆𝐫𝐩𝐬$ to $𝐒𝐞𝐭𝐬$, from $𝐂𝐇$ (the category of compact Hausdorff spaces) to $𝐒𝐞𝐭𝐬$, from sheaves to presheaves over algebraic structures, and from $C(\mathcal{B})$ the category of chain complexes to $\mathcal{B}^ℤ$, etc.

A note on Kan Extension

The Definition

The ideal of Kan extension is simple: to find the biadjoints of the pullback functor $φ ^∗$. Both limits and colimits are special cases of Kan extensions. We use $φ_!$ and $φ_∗$ to denote the functorial construction of left and right Kan extension, respectively. The desired adjoints are $φ_! ⊣ φ^! = φ^∗ ⊣ φ_∗$ (we shall not discuss the sheaf theory here, the notion is not misleading).

It is straightforward to demonstrate the universal property using diagrams. For example, the existence of $φ _! F$ signifies that $(F, φ ^∗ (-))$ is representable, as illustrated below.

Example: Homology from Limits

We show an example of Kan extension in theory of group representation. Any group $G$ is considered as a category $BG$ with a single object, and the morphisms are the group elements. A representation of $G$ in $𝐌𝐨𝐝 _k$ is exactly a functor $F : G → 𝐌𝐨𝐝 _k$.

We leave the discussion of higher dirived (co)limits to the reader.

The Construction

We shall discuss the construction and existence of Kan extensions. Let $φ : I → J$ be a functor between $𝔘$-small diagrams, and let $F : I → \mathcal{C}$ be a functor from the diagram to the category $\mathcal{C}$.

Hence, when $\mathcal{C}$ is complete/cocomplete, the right/left Kan extension always exists.

On Derived Functors

The concept of derived functors was (probably) initially discovered by A. Grothendieck, albeit without a functorial formulation; a formal statement is attributed to J.-L. Verdier in his doctoral thesis in 1967. We follow P. R. Deligne’s definition, which is, in my view, the most accessible. Subsequently, we provide an explanation in terms of Kan extension.

Broadly speaking, derived categories serve as a universal invariant of a category. For instance, the invariants of an algebra $A$, including $Z(A)$ the centre, $K_i(A)$ the $K$-groups, and $HH^∙ (A,A)$ the Hochschild cohomologies, are defined over $D(A)$. Consequently, derived functors induce the morphisms between these invariants.

For the construction, let $F : \mathcal{A} → \mathcal{B}$ be an additive functor between additive categories, which induces the triangular functors $C(\mathcal{A}) → C(\mathcal{B})$ between chain complexes, and $K(\mathcal{A}) → K(\mathcal{B})$ between the homotopy categories of chain complexes. When the categories are Abelian, the collection of quasi-isomorphisms forms a multiplicative system for the homotopy category (or the chain complex category), which yields a relatively straightforward construction of the localisation $Q_\mathcal{A} : C(\mathcal{A}) → D(\mathcal{A})$. Since $F$ need not preserve quasi-isomorphisms, the functor $C(\mathcal{A}) → C(\mathcal{B}) → D(\mathcal{B})$ does not, in general, extend along the localisation $Q_\mathcal{A}$. It is futile to seek $D(\mathcal{A}) → D(\mathcal{B})$ which renders the square commutative; Kan extensions appear to be the most suitable replacement.

We demonstrate, through exercises, that Deligne’s construction of right derived functors is equivalent to the left Kan extension of the localisation, and clarify how a well-known criterion for the existence of right derived functors is a special case of the existence of Kan extensions.

When $\mathcal{A}$ possesses enough projectives, one may revise the aforementioned construction of $𝐑 F$.

1. One may restrict the complexes to be bounded below, and work with $D^+ \mathcal{A}$ instead;
2. One may introduce $K_{\mathrm{h \ inj}}$, the homotopy injective (or $K$-injective) complexes, and replace the injective resolution with the $K$-injective resolution.

In order to combine Kan extension with derived functors, it is necessary to verify the following.

In the computation of derived categories, there are usually suitable conditions such that $\varinjlim _{X \xrightarrow[∈ S]{s} Y} F(Y)$ is straightforward to compute.

Simplicial Methods

This note represents merely the tip of the iceberg concerning simplicial methods in model categories, which constitute a powerful tool in homotopy theory as well as the fundamental language of $∞$-categories. The aim of this note is to provide a foundation for further study of model structures. For a more comprehensive treatment of this subject, the reader is referred to kerodon.

Simplicial Objects

Before proceeding, it is illuminating to consider a perspective from F. Lawvere. Broadly speaking, to assert that $A$ is a $k$-algebra is to say that $A$ is a ring object in $𝐌𝐨𝐝_k$. Analogously, one may say that a simplicial set is a simplicial object in $𝐒𝐞𝐭𝐬$ and thenceforth focus on the simplicial structure.

Basics

It is straightforward to summarise all automatic exchange relations of the face and degeneracy maps, and conclude that any morphism in $Δ$ may be uniquely written as a composition of $$\begin{equation} \underset{n _k > \cdots > n_1}{\underbrace{d^{n_k} ∘ \cdots ∘ d^{n_1}}} ∘ \underset{m _1 < \cdots < m_l}{\underbrace{s^{m_1} ∘ \cdots ∘ s^{m_l}}}. \end{equation}$$

The Truncation

An $n$-simplex is an element $σ ∈ X_n ≃ (Δ ^n , X)$, denoted by $(σ , [n])$ or simply $σ$. The (combinatorial) dimension $\dim σ$ is defined as the largest $m$ such that $σ$ factors through $Δ ^m$. The simplex $(σ, [n])$ is said to be non-degenerate if and only if $\dim σ = n$. The boundary $∂ Δ ^n$ assigns to $[m]$ the set of non-surjective morphisms in $Δ ^n ([m]) = ([m], [n])$, or, equivalently, is given by the colimit $\mathrm{coeq}\left(∐ _{0 ≤ i < j ≤ n}Δ ^{n-2} ⇉ ∐ _{0 ≤ k ≤ n} Δ ^{n-1}\right)$.

The purpose of the $n$-coskeleton is to fill all chambers of dimension $≥ n$, so that all $π _{≥ n}$ vanish. In the language of Kan extension, the inclusion $φ : Δ _{ ≤ n} → Δ$ yields the restriction $φ ^! = φ ^∗$. One demonstrates the existence of left and right Kan extensions, $φ_!$ and $φ _!$, as $𝐒𝐞𝐭𝐬$ is bi-complete. The adjoint triple $L ⊣ M ⊣ R$ induces the adjoint endofunctor $ML ⊣ MR$. In this case, $φ ^!φ _!$ and $φ ^∗ φ _∗$ are the $n$-skeleton and $n$-coskeleton functors, respectively.

The Topologies

Motivation for Simplicial Methods

What is the origin of the concept of simplicial methods? How should one understand simplicial objects in a concrete, particularly geometric manner? The introduction of topological ideas provides a framework to address these questions.

A Categorical Overview

The usual category of topological spaces, $𝐓𝐨𝐩$, consists of topological spaces as its objects, and continuous maps as the morphisms. $𝐓𝐨𝐩$ possesses equalisers (with subspace topology) and arbitrary products (with product topology); in particular, the limit exists. $𝐓𝐨𝐩$ also possesses coequalisers (with quotient topology) and arbitrary coproducts (with coproduct topology); in particular, the colimit exists.

By adjoints, the forgetful functor commutes with limits and colimits. Hence, the underlying set of the (co)limit of topological spaces is computed elementwise.

The category $(𝐓𝐨𝐩, ×, \{∗\})$ is symmetric monoidal. The $\mathcal{HOM}$-functor occasionally exists for special subcategories. The Hom space $(X, Y)_{𝐓𝐨𝐩}$ is equipped with the compact-open topology wherein the topological basis is characterised by $$\begin{equation} \mathcal{O}_{U, K} = \{f ∣ f(K) ⊆ U\}_{K \ \text{is compact,}\ U \ \text{is open}}. \end{equation}$$ We denote the topological Hom object by $\mathcal{HOM}(X,Y)$. When $Y$ is locally compact Hausdorff, there is a natural isomorphism (currying): $$\begin{equation} (-, \mathcal{HOM}(Y, ?))_{𝐓𝐨𝐩} ≃ (- × Y, ?)_{𝐓𝐨𝐩} ,\quad φ ↦ [(t,y) ↦ (φ(t))(y)]. \end{equation}$$

We generally prefer a convenient category of topological spaces in which the internal $\mathcal{HOM}$ exists; moreover, such a category ought to include fundamental objects such as CW complexes, and be bicomplete (not necessarily inherited from $𝐓𝐨𝐩$). $k$-spaces or compactly generated Hausdorff spaces (with proper mappings as the morphisms) are frequently employed, as the inclusion admits left adjoints ($k$-functor or Stone-Čech compactification, respectively).

The Pointed Construction

The definition of $π$-groups necessitates base points, which in turn requires the introduction of pointed topological spaces $𝐓𝐨𝐩 _+$ defined as the category of topological spaces equipped with a distinguished point.

We remark upon certain categorical properties of $𝐓𝐨𝐩_+$. The most distinguished property is that $𝐓𝐨𝐩$ is pointed, meaning that the initial and terminal objects coincide (we refer to $0 := \{∗\}$ as the zero object). In general, the limit of the system $\{(X_i, x_i)\}_{i ∈ I}$ is $\varprojlim_I X_i$ with the base point $(x_i)_{i ∈ I}$. The colimit of $\{(X_i, x_i)\}$ is a quotient of $\varinjlim_I X_i ∈ 𝐓𝐨𝐩$ obtained by identifying all $x_i$ as the new base point.

The base point makes the cateogory algebraic, as one can define fibre sequence (analogue of exact sequence). There is a functor $(-)_+ : 𝐓𝐨𝐩 → 𝐓𝐨𝐩 _+$ sending $X$ to $X ⊔ 0$ with the coproduct topology. $(-)_+$ is the left adjoint of the inclusion $𝐓𝐨𝐩_+ → 𝐓𝐨𝐩$; there is no adjoint triple here since neither $(-)_+$ preserves terminal objects nor the forgotful functor preserve initial objects. It is highlighted that $(-)_+$ preserves the symmetric monoidal structure, where

• $(X ⊔ Y)_+ ≃ X_+ ∨ Y_+$, where $P ∨ Q$ is the usual coproduct in $𝐓𝐨𝐩 _+$;
• $(X × Y)_+ ≃ X_+ ∧ Y_+$, where $P ∧ Q := \frac{P × Q}{P ∨ Q}$ is the smash product.

The functor $(-)_+$ is strongly monoidal, which carries the monoidal structure. We restrict $(-)_+$ onto the $𝐋𝐂𝐇$ and obtain $(-)_+ : 𝐋𝐂𝐇 → 𝐋𝐂𝐇_+$, which also admits a right adjunction. The category $𝐋𝐂𝐇$ is closed symmetric monoidal with internal Hom space $\mathcal{HOM}_{𝐋𝐂𝐇} (X,Y)$ endowed with the compact-open topology. Analogously, $\mathcal{HOM}_{𝐋𝐂𝐇 _+ }(X_+, Y_+)$ is also the internal Hom of $𝐋𝐂𝐇 _+$ whose base point is the zero morphism $0_{Y,X}$.

Another well-known monoidal functor from $𝐋𝐂𝐇$ to $𝐓𝐨𝐩 _+$ is the functor of one-point-compactification, whose essential image is precisely the pointed compact topological spaces. We do not examine this further here.

Equivalences

We clarify several types of equivalences of topological spaces, the (classical) homotopy equivalence, the weak homotopy equivalence.

A suspension functor for pointed topological space is defined as $Σ (-):= S_∙ ∧ -$, for $S_∙$ the pointed unit circle. We define $Σ ^{n+1} := Σ ^n ∘ Σ$ recursively, and $S_∙ ^n := Σ ^n (\{∗ \}_+)$ as the pointed unit $n$-sphere. The right adjoint of $Σ$ is the known as the loop space $Ω (-)$, defined by internal Hom.

We remark that the weak homotopy equivalence is no longer a equivalence relation, e.g., the totally disconnected topology v.s. the discrete topology. We denote the collection of weak homotopy equivalences as $𝖶𝖾𝗊$ (weak equivalences in sense of Quillen), and the localised category as $𝖧𝗈 \ 𝐓𝐨𝐩 := 𝐓𝐨𝐩 [𝖶𝖾𝗊^{-1}]$.

We focus on the category $𝖧𝗈 \ 𝐓𝐨𝐩$. A general morphism in $𝖧𝗈 \ 𝐓𝐨𝐩$ is a finite zig-zag diagram in the sense of Gabriel-Zisman localisation. We wish that there exists a subcategory $\mathcal{X} ⊆ 𝐓𝐨𝐩$ such that every $X ∈ 𝐓𝐨𝐩$ is associated with a replacement $X' ∈ \mathcal{X}$ via a canonical weak homotopy equivalence $X' → X$. Moreover, for any $X'$ and $Y'$ in $\mathcal{X}$, the hom-set $(X', Y')_{𝖧𝗈 \ 𝐓𝐨𝐩}$ is a quotient of $(X', Y')_{𝐓𝐨𝐩}$.

Such a replacement is known as a CW-approximation, the construction of which is explained by iteratively attaching cells. We shall explain this in the language of simplicial sets. Before that, we introduce the geometric realisation.

The Geometric Realisation

We introduce the adjunction between $𝐓𝐨𝐩 ⇆ 𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}}$ in this section, where the localised adjunction is the well-known Quillen equivalence. To begin with, we introduce the functor of geometric realisation $|-| : 𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}} → 𝐓𝐨𝐩$ which is expected to satisfy the following requirements:

1. (regularity). $|Δ ^n|$ is $\{x ∈ ℝ_{≥ 0} ^{n+1} ∣ ∑ x_i = 1\}$;
2. (colimit-preserving). $| - |$ commutes with colimits, or even is a left adjoint;
3. ($×$-preserving). It is difficult to assert that $| - |$ preserves limits; but it is expected that $| - |$ preserves the monoidal structure of $𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}}$, i.e., $|X × Y| ≃ |X| × |Y|$.

The geometric realisation labels all possible $|Δ^n|$ with $X_n$, and quotients by the relation generated by the face and degeneracy maps. We omit the verification of functoriality, since we shall construct its right adjoint $\operatorname{Sing}$. To see how $\operatorname{Sing}$ is supposed to appear, we consider the following natural isomorphism by the Yoneda lemma: $$\begin{equation} (|Δ ^n |, T)_{𝐓𝐨𝐩 } ≃ (Δ ^n , (|Δ ^- |, T)_{𝐓𝐨𝐩 })_{𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}}}. \end{equation}$$

We remark some properties of this adjunction.

1. $\operatorname{Sing}$ is faithful in general, but not full. For instance, $\operatorname{Sing}$ never distinct the different disconnected topologies over the same underlying set.
2. $| - |$ is does not preserve the monoidal structure. It is pointed out by C. H. Dowker an example that the canonical morphism $|X| × |Y| → |X × Y|$ is not a homeomorphism in general.

CW-approximation

The term CW complexes was introduced by J. H. C. Whitehead in 1946, providing a means to approximate topological spaces in a straightforward manner when one is concerned solely with the homotopy groups. We refer to the appendix by A. Hatcher as an elementary introduction to CW complexes. The following are more categorical in nature.

We shall first present a basic idea: the term CW denotes Closure-finiteness and Weak topology, wherein

• Closure-finiteness signifies that each cell intersects only finitely many other cells (thus it is compactly generated, and locally contractible);
• Weak topology refers to the topology related to $π_∙$-groups.

The goal of this section is to

• show the existence of CW-approximation, and the functoriality is due to counit $|\operatorname{Sing} (X)| → X$ serves as such a CW-approximation for any $X ∈ 𝐓𝐨𝐩$;
• prove the Whitehead’s theorem, which states that the weak homotopy equivalence is equivalent to homotopy equivalence for CW complexes.

The following well-known theorem due to J. H. C. Whitehead (also appearing in combinatorical homotopy) is useful. The proof concerns a series of theory in algebraic topology, such as the cellular approximation, mapping cone and mapping cylinder, and the relative homotopy groups. We state only the main theorem and a sketch of the proof.

We remark that the Whitehead’s theorem provides a (functorial) bi-fibrant replacement of objects, serving as an important technique in the study of homotopy category for model structures.

Cofibrantly Generating Class

Cofibrantly generated model structure is ubiquitous in the study of model categories. In this chapter, we do not introduce the concept of model category in general, but introduce the cofibrantly generating class and the weak factorisation system instead. The ideal traces back to D. Quillen in 1967.

The (W)FS

We begin 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 (orthogonality) and the (functorial) factorisation property. The writer is not certain about its origin, but it was probably introduced by G. M. Kelly in 1980. A nice introduction can be found in the CatLab of A. Joyal.

Unwinding the definition of (W)FS, we adopt the symbol $S^⟂$ ($S^⋔$) to denote the class of morphisms having (weak) right lifting property with respect to $S$.

We provide a simple example in $𝐒𝐞𝐭𝐬$.

The simplicial language is a functorial way to describe some categorical operations. For instance, we see how it rephrase the functorial factorisation for (W)FS.

The case is much easier for FS. It is clear that there is an one-to-one correspondence between FS and idempotent monads (where $μ : R^2 ⇒ R$ is a natural isomorphism, even an identity by MacLane’s assumption). We show the partial converse for a creterion on functorial WFS.

Quillen’s Small Object Argument

We enumerate the relevant set-theoritic definitions in this section, and prove Quillen’s small object argument without hesitation. Simplicial and topological explanations are given in the proceeding sections.

The notion small is over-used, this does not the same as $𝔘$-small introduces a priori. When there is a class of morphisms $S$ which is considered to be relatively small to some ordinal $α$, it is possible to construction a WFS by a reasonable induction. The construction is not canonical since it depends on the choice of $α$.

We firstly explain the notion of small based on the cofinality of ordinals. We identify an ordinal (ususally a limit oridnal) $γ$ as a poset category with $𝖮𝖻(γ ) = \{β ∣ β < γ \}$. A $γ$-sequence in $\mathcal{C}$ is a functor $γ → \mathcal{C}$. Say $γ$ is $γ '$-filtered if $γ$ has stronger confinality than $γ '$, i.e., for any inclusion $ι : γ ' → γ$ of posets, the limit oridinal $\varinjlim ι$ is strictly smaller than $γ$. We remark that

• It suffices to define $γ$-filtration where $γ'$ is a cardinal;
• For a finite cardianl $κ$, the $κ$-filtered oridnals are exactly the limit oridnals.
• For any cardinal $κ$ (as a limit oridnal), the least $κ$-filtered ordinal is precisely the successor cardinal of $κ$.

For either $𝐒𝐞𝐭𝐬$ or $𝐌𝐨𝐝_k$, an general object $X$ is a $𝔘$-small colimit (controlled by $κ$) of compact objects (i.e., $(K, -)$ preserves filtered colimits, for instance $|K|$ is finitely presented). Now $(X, - )$ is $κ$-small, as $κ$-filtered colimits commutes with limits controlled by $κ$. In particular, any set or modules are small.

The following small object argument demonstrates the weak orthogonality and functoriality of the pair $(S_{cell}, S_{inj})$. We show later that $S_{cof}$ is the completion of $S_{cell}$ under retractions, thus it returns to the WFS $(^⋔ (S^⋔), S^⋔)$.

Kan Complexes

We provide a concrete exmaple of $S_{inj}$, $S_{cell}$, and $S_{cof}$ in terms of (simplicial) Kan fibrations. The class of Kan fibration ($S_{inj}$) and anodyne morphisms ($S_{cof}$) serve as the class of trivial cofibrations and fibrations in simplicial model category. The corresponding fibrant object are Kan complexes which admits a closed monoidal structure. The topological version is translaed from the adjoint lifting property.

We introduce the basic notions of simplicial complex. Let $\mathcal{I}$ denote a collection of subsets of $[n]$ ($\mathcal{I} ⊆ \mathcal{P}([n])$ for power set notion). This defines a subfunctor $Δ ^\mathcal{I} ⊆ \mathcal{D}^n$ by

$$\begin{equation} Δ ^\mathcal{I} ([m]) := \{f : [m] → [n] ∣ ∃ S ∈ \mathcal{I}, f([m]) ⊆ S\}. \end{equation}$$

In particular, one has the standard simplex $Δ ^{n} = Δ ^{\mathcal{P}([n])}$, and the boundary $∂ Δ ^n = Δ ^{\mathcal{P}([n]) ∖ [n]}$. A simplicial complex is of the form $Δ ^{\mathcal{I}}$ for $\mathcal{I}$ closed under taking subsets. We remark the followings

• intuitively, if an edge belongs to a simplicial complex, then so are its end points;
• any simplex in a simplicial complex has non-overlapping boundaries at every dimensions, e.g., the simplicial circle $Δ ^1 / ∂ Δ ^1$ is never a simplicial complex;
• one may generalise the definition to a good subfunctor of $Δ ^∞$ as how we define $∞$-CW complexes; there is no need at this moment.

We may rephrase some usual construction for categories in simplicial language. We usually view $𝖮𝖻(\mathcal{C})$, $𝖬𝗈𝗋 (\mathcal{C})$, etc, as special subcategory with operations; and may wish to state such operation atomatically. This motivate the definition of nerve, enabling tools from topology (e.g., homotopy groups, cohomology) to analyse categorical structures. We denote $𝐂𝐚𝐭$ the category of $𝔘$-small categories ($2$-category), which is also $𝐂𝐚𝐭$-enriched as the hom-sets are the usual functor categories.

Unlike $|-| ⊣ \operatorname{Sing}$, the functor $\mathrm N$ is fully faithful. The counit morphism tautologically assign the folding of freelised category to itself, which is a natural isomorphism. Unlike general topology, category has no $∞$-path as in topologist’s sine function. This is how $𝐂𝐚𝐭$ better than $𝐓𝐨𝐩$ in some aspect. Whereas one good thing of $𝐓𝐨𝐩$ is that all paths are invertible, which we shall see as follows.

We denote $Λ^n _i := Δ ^{\mathcal{P}([n]) ∖ \{[n], [n]∖ \{i\}\}}$ as the $i$-th horn of $Δ^n$. Graphically, $Λ ^n_i$ is obtained by deleting the interior and the $i$-th face of $Δ^n$. $Λ ^n _i$ is an inner horn, provided $0 < i < n$. The inclusions maps are called (inner) horn inclusions. We show the definition of $∞$-category by the following fact:

We call such those simplicial sets lifting all (inner) horn fillings (weak) Kan complexes. By invertibility of all paths, any singular set of a topological complex are Kan complexes; a general category is a weak Kan complex as only the composition rule works fine. A category with all invertible paths (aka groupoid) is a Kan complex, the converse holds true by combinatoric verification.

Let $S$ denote the horn fillings temporality. The simplicial set $X$ is a Kan complex if and only if $X → \{∅\}$ belongs to $S_{inj}$. We call $S_{inj}$ the class of Kan fibrations. By general theory of lifting property, Kan fibrations are closed under retracts, pullbacks, products, and transfinite co-composition of a tower. The class left lifting class $S_{cof}$ are called the acyclic cofibrations, or anodyne morphisms. By Quillen’s small object argument, $S_{cof}$ is precisely the closure of horn inclusions under isomorphisms, transfinite compositions, coprodcuts and retracts. The terms anodyne or acyclic means to be inoffensive or harmless to homology groups: the transdental induction keeps on filling the holes in the construction of Quillen’s small object argument.

Adjoint Lifting

We introduce the smash product of morphisms and morphism spaces for simplicial comlexes, which shows more closure properties of $S_{inj}$ and $S_{cof}$. Many of the categories in practice are self-enrichend, meaning that every Hom-set is a object in the same category. We may obtain better results analogous of currying (the tensor-hom adjunction). The adjoint lifting serves as a bridge connecting the smash product for cofibrations and morphism spaces for fibrations. We complete this section by showing a criterion (probably initially discovered by A. Joyal, see quasi-categories):

• $X$ is a Kan complex ($(∞,1)$-category), if and only if $\mathcal{HOM}(Δ^1, X) \xrightarrow {(Δ ^0 ↪ Δ ^1) ^∗ }\mathcal{HOM}(Δ^0, X)$ belongs to $𝐌𝐨𝐧𝐨 ^⋔$;
• $X$ is a weak Kan complex ($∞$-category), if and only if $\mathcal{HOM}(Δ^2, X) \xrightarrow {(Λ^2_1 ↪ Δ ^2) ^∗ }\mathcal{HOM}(Λ^2_1, X)$ belongs to $𝐌𝐨𝐧𝐨 ^⋔$.

The product and coproducts of simplicials sets are defined pointwise as they are presheaves over $Δ$. We remark that the product of standard simplicial sets $Δ ^m$ and $Δ ^n$ is of dimension $m+n$ by shuffle construction. The prodcut of degenerated objects are not necessary degenereted, as shown in previous dimensional analysis. One concrete example is $\binom{[001]}{[011]} ∈ Δ ^1 × Δ ^1$.

We remark that

• the usual set of morphism $(K,Y)_{𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}}}$ is $\mathcal{HOM}(K,Y)$ at $0$-th degree;
• by Yoneda, the $(K × -) ⊣ \mathcal{HOM}(K,-)$ is the same thing as the natural isomorphism $\mathcal{HOM}(K × -, ?) → \mathcal{HOM}(-, \mathcal{HOM}(K, ?))$;
• the enriched composition $\mathcal{HOM}(Y, Z) × \mathcal{HOM}(X,Y) → \mathcal{HOM}(X,Z)$ is characterised degree-wise, where the morphisms $g_k : Y × Δ ^k → Z$ and $f_k : X × Δ ^k → Y$ gives $(g ∘ f) _k : X × Δ ^k \xrightarrow{X × \text{diagonal}} X × Δ ^k × Δ ^k → Z$;
• the pull backs are reasonable constructions of morphism spaces, e.g., $\varprojlim \mathcal{HOM}(A_∙ , -)$ is represented by $\mathcal{HOM}(\varinjlim A_∙, -)$ by Yoneda lemma, and $\varprojlim \mathcal{HOM}(-, B_∙)$ is corepresented by $\mathcal{HOM}(-, \varprojlim B_∙)$ by commuting right adjoints.

We pointout a functorial construction of mapping space.

For a fixed morphism $φ$, the left adjoint functor $φ ∧ (-)$ preserves retractions, transfinite compositions, coproducts, and pushouts. When Quillen’s small objects argument holds, $φ ∧ (-)$ preserves the left morphism class of WFS.

The smash product of morphisms is compatible with the usual smash product for pointed simplicial sets, defined as ${X ∏ Y} / {X ∐ Y}$, as explained in the following exercise.

We use smash product to simplify the discription of generating set of horn inclusions. We fix $φ : \{∗\} ↪ Δ ^1$ as the inclusion of $0$, and show that the following collections of morphisms generate the same $S_{cof}$ class:

1. ($S$). the collection of horn inclusions $Λ ^n _i ↪ Δ ^n$ for $0 ≤ i ≤ n$;
2. ($T$). the collection of $φ ∧ f$, for $f$ an arbitrary monomorphism.

Crearly, $i ∧ S_{cof} ⊆ S_{cof}$ for any monomorphism $i$. We apply adjoint lifting to the above equivalent characterisation of $S_{cof}$, and see that $Φ (i, -)$ preserves Kan fibrations for arbitrary monomorphism $i$. In particular, for any simplicial set $X$ and Kan complex $F$, the morphism space $\mathcal{HOM}(X,F)$ is again a Kan complex. Hence we obtain a criterion for Kan complexes:

• $F$ is a Kan complex ($(∞,1)$-category), if and only if the pullback $\mathcal{HOM}(Δ ^1 , F) \xrightarrow {φ ^∗ } F$ lifts all monomorphisms.

This criterion has to do with path-homotopy. When $F^I → F$ lifts all monomorphisms, $F$ has invertible edges.

The above criterion (along with adjoint lifting) shows that for any $i ∈ 𝐌𝐨𝐧𝐨$, $Φ (i, -)$ preserves trivial Kan fibrations. In particular,

• for any simplicial set $X$, the functor $\mathcal{HOM}(X, -)$ preserves weak Kan complexes ($∞$-categories);
• $V$ is a weak Kan-complex ($∞$-category), if and only if the pullback $\mathcal{HOM}(Δ ^2, V) \xrightarrow {ψ ^∗ }\mathcal{HOM}(Λ ^2_1, V)$ lifts all monomorphisms.

Functors

Topological Space for Convenience

We explain the convenient topological category, usually selected as $𝐂𝐆𝐇$ (compactly generated Hausdorff space, or $k$-spaces). We clearify the morphisms ($k$-morphisms) in $𝐂𝐆𝐇$ and the left adjoint functor ($k$-fication) $k : 𝐓𝐨𝐩 → 𝐂𝐆𝐇$ with respect to the inclusion $𝐂𝐆𝐇 → 𝐓𝐨𝐩$ (not full!).

Hurewicz Map

Following the principle $\text{realisation} ⊣ \text{nerve}$, we have constructed the adjunctions

In order to study homologies, we introduce the free-forgetful adjunction of Abelisation $ℤ (-) : 𝐀𝐛 ^{Δ ^{\mathrm{op}}} ⇆ 𝐒𝐞𝐭𝐬 ^{Δ ^{\mathrm{op}}} : U$. The homologies are calculated by the homology groups of the singular chain complex $C(ℤ (X))$. The Hurewitz map points out that the functor $ℤ (-)$ lost some information of $π _∙$ groups, while exactly makes it into $H_∙$. The map says $π _∙ (|U(-)|) ≃ H_∙ (-)$ are the same functor of the type $𝐀𝐛 ^{Δ ^{\mathrm{op}}} → 𝐆𝐫 (𝐀𝐛)$, which generalise the result of $\mathrm{Ab}(π _1) = H_1$.

DK Correspondence

There are also theories revising the assignment $X ↦ C(ℤ (X))$, e.g. the Dold-Kan complex correspondence.

(Homotopy) Spectra

The pointed construction makes topological spaces algebraic. The homotopy spectra $Σ ^∞ : 𝐓𝐨𝐩 _∙ / ∼ → 𝐇𝐨𝐒𝐩𝐞𝐜$ moreover linearise it ($∧ = ∨$ under $Σ ^∞$!).

• the development: due to Lima around 1950s probably, and revolutioned by Adams in stable homotopy theory (1974);
• it has nothing to do with the spectra in algebraic topology, or spectal of operator (the word is over used).

Quillen’s Equivalence

The adjunction $|-| ⊣ \operatorname{Sing}$ descents to an equivalence for there homotopy category. It is acceptable to enlarge $𝐂𝐆𝐇$ to the entire $𝐓𝐨𝐩$.

Quillen’s Model Structure