category theory

Contents

Idea

The lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. A number of elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

Often it is useful to think of lifting properties as a expressing a kind of qualitative negation (“Quillen negation”): The morphisms with the left/right lifting property against those in a class $P$ tend to be characterized by properties opposite of those in $P$. For example, a morphism in Sets is surjective iff it has the right lifting property against the archetypical non-surjective map $\varnothing \to \{*\}$, and injective iff it has either left or right lifting property against the archetypical non-injective map $\{x_1,x_2\}\to \{*\}$. (For more such examples see at separation axioms in terms of lifting properties.)

Definition

Definition

A morphism $i$ in a category has the ‘’left lifting property’‘ with respect to a morphism $p$, and $p$ also has the ‘’right lifting property’‘ with respect to $i$, sometimes denoted $i\,\,⧄\,\, p$ or $i\downarrow p$, iff the following implication holds for each morphism $f$ and $g$ in the category:

• if the outer square of the following diagram commutes, then there exists $h$ completing the diagram, i.e. for each $f:A\to X$ and $g:B\to Y$ such that $p\circ f = g \circ i$ there exists $h:B\to X$ such that $h\circ i = f$ and $p\circ h = g$.

This is sometimes also known as the morphism $i$ being ‘’weakly orthogonal to’‘ the morphism $p$; however, ‘’orthogonal to’‘ will refer to the stronger property that whenever $f$ and $g$ are as above, the diagonal morphism $h$ exists and is also required to be unique.

For a class $C$ of morphisms in a category, its ‘’left weak orthogonal’‘ or its ‘’left Quillen negation’‘ $C^{⧄ \ell}$ with respect to the lifting property, respectively its ‘’right weak orthogonal’‘ and its ‘’right Quillen negation’‘ $C^{⧄r}$ is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class $C$. In notation,

$C^{⧄ \ell} := \{ i \mid \forall p\in C, i\,\,⧄\,\, p\} \,\,\, C^{⧄ r} := \{ p \mid \forall i\in C, i\,\,⧄\,\, p\}, \,\,\, C^{⧄ lr} := (C^{⧄ l})^{⧄ r}$

Taking the Quillen negation of a class $C$ is a simple way to define a class of morphisms excluding isomorphisms from $C$, in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right Quillen negation $\{\emptyset \to \{*\}\}^{⧄ r}$ of the simplest non-surjection $\emptyset\to \{*\},$ is the class of surjections. The left and right Quillen negation of $\{x_1,x_2\}\to \{*\},$ the simplest non-injection, are both precisely the class of injections, $\{\{x_1,x_2\}\to \{*\}\}^{⧄ \ell} = \{\{x_1,x_2\}\to \{*\}\}^{⧄ r} = \{ f \mid f \text{ is an injection } \}.$

It is clear that $C^{⧄\ell r} \supset C$ and $C^{⧄ r\ell} \supset C$. The class $C^{⧄ r}$ is always closed under retracts, pullbacks, (small) products (whenever they exist in the category) and composition of morphisms, and contains all isomorphisms of C. Meanwhile, $C^{⧄ \ell}$ is closed under retracts, pushouts, (small) coproducts and transfinite composition (filtered colimits) of morphisms (whenever they exist in the category), and also contains all isomorphisms.

Examples of lifting properties

Decyphering notation in most of the examples below leads to standard definitions or reformulations. The intution behind most examples below is that the class of morphisms consists of simple or archetypal examples related to the property defined.

Elementary examples

Sets

In Set,

• $\{\emptyset\to \{*\}\}^{⧄ r}$ is the class of surjections,

• $(\{a,b\}\to \{*\})^{⧄ r}=(\{a,b\}\to \{*\})^{⧄ \ell}$ is the class of injections.

Modules

In the category RMod of modules over a commutative ring $R$,

• $\{0\to R\}^{⧄ r}, \{R\to 0\}^{⧄ r}$ is the class of surjections, resp. injections,

• A module $M$ is projective, resp. injective, iff $0\to M$ is in $\{0\to R\}^{⧄ \ell r}$, resp. $M\to 0$ is in $\{R\to 0\}^{⧄ rr}$.

Groups

In the category Grp of groups,

• $\{\mathbb{Z} \to 0\}^{⧄ r}$, resp. $\{0\to \mathbb{Z}\}^{⧄ r}$, is the class of injections, resp. surjections (where $\mathbb{Z}$ denotes the infinite cyclic group),

• A group $F$ is a free group iff $0\to F$ is in $\{0\to \mathbb{Z} \}^{⧄ r\ell},$

• A group $A$ is torsion-free iff $0\to A$ is in $\{ n \mathbb{Z} \to \mathbb{Z} : n\ge0 \}^{⧄ r},$

• A subgroup $A$ of $B$ is pure? iff $A \to B$ is in $\{ n\mathbb{Z}\to \mathbb{Z} : n\ge0 \}^{⧄ r}.$

• $(*\to 1)^{⧄ l}$ is the class of retracts

• $(1\to *)^{⧄ r}$ is the class of split homomorphisms

• $(0\longrightarrow \mathbb{Z})^{⧄ r}$ is the class of surjections

• $(\mathbb{Z}\to 1)^{⧄ r}$ is the class of injections

• a group $F$ is free iff $1\to F$ is in $(0\longrightarrow \mathbb{Z})^{⧄rl}$

• a group $A$ is Abelian iff $A\to 1$ is in $( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄ r}$

• group $G$ can be obtained from $H$ by adding commutation relations, i.e.~the kernel of $H\to G$ is generated by commutators $[h_1,h_2]$, $h_1,h_2\in H$, iff $H\to G$ is in $( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄rl}$

• subgroup $H$ of $G$ is the normal span of substitutions in words $w_1,..,w_i$ of the free group $\mathbb{F}_n$ iff $G \to G/H$ is in $( \mathbb{F}_n \to \mathbb{F}_n/\le\!w_1,...,w_i\!\ge)^{⧄rl}$

• $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell l}$ is the class of homomorphisms whose kernel is perfect

For a finite group $G$, in the category of finite groups,

• $\{0\to {\mathbb{Z}}/p{\mathbb{Z}}\} \,\,⧄\,\, G\to 1$ iff the order of $G$ is prime to $p$,

• $G\to 1 \in (0\to {\mathbb{Z}}/p{\mathbb{Z}})^{⧄ rr}$ iff $G$ is a $p$-group,

• $H$ is nilpotent iff the diagonal map $H\to H\times H$ is in $(1\to *)^{⧄ \ell r}$ where $(1\to *)$ denotes the class of maps $\{ 1\to G : G \text{ arbitrary}\},$

• a finite group $H$ is soluble? iff $1\to H$ is in $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}.$

Moreover,

• $\{0\to G : G\,\,\text{ arbitrary}\}^{⧄ \ell r}$ is the class of subnormal subgroups

• $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}$, and is the class of subgroups $H\leq G$ such that there is a chain of subnormal subgroups $H=G_0 \vartriangleleft G_1 \vartriangleleft \ldots \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ is Abelian, for $i=0,...,n-1$.

• $\{1 \to S\}^{⧄ \ell r}$ is the class of subgroups $H\leq G$ such that there is a chain of subnormal subgroups $H=G_0 \vartriangleleft G_1 \vartriangleleft \ldots \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ embeds into $S$, for $i=0,...,n-1$.

• $(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄r}$ is the class of homomorphisms whose kernel has no elements of order $p$

• $(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄rr}$ is the class of surjective homomorphisms whose kernel is a $p$-group

In algebraic topology and in model categories

Lifting properties are paramount in homotopy theory and algebraic topology. In “abstract homotopy theory” lifting properties are encoded in the structures of model categories, whose defintion revolves all around compatible classes of weak factorization systems. In particular:

Serre fibrations of topological spaces

The classical model structure on topological spaces $Top_{Qu}$ is controlled by the following lifting properties:

consider let $C_0$ be the class of maps $S^n\to D^{n+1}$, embeddings of the boundary $S^n=\partial D^{n+1}$ of a ball into the ball $D^{n+1}$. Let $WC_0$ be the class of maps embedding the upper semi-sphere into the disk. $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$ are the classes of Serre fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. Hovey, Model Categories, Def. 2.4.3, Th.2.4.9

Hurewicz fibrations of topological spaces

A map $f:U\to B$ has the ‘’path lifting property’‘ iff $\{0\}\to [0,1] \,\,⧄\,\, f$ where $\{0\} \to [0,1]$ is the inclusion of one end point of the closed interval into the interval $[0,1]$.

A map $f:U\to B$ has the homotopy lifting property iff $X \to X\times [0,1] \,\,⧄\,\, f$ where $X\to X\times [0,1]$ is the map $x \mapsto (x,0)$.

Kan fibrations of simplicial sets

The classical model structure on simplicial sets $sSet_{Qu}$ is controlled by the following lifting properties:

Let $C_0$ be the class of boundary inclusions $\partial \Delta[n] \to \Delta[n]$, and let $WC_0$ be the class of horn inclusions $\Lambda^i[n] \to \Delta[n]$. Then the classes of Kan fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$. (Model Categories, Def. 3.2.1, Th.3.6.5)

Degreewise surjections of chain complexes

A model structure on chain complexes is controlled by the following lifting properties:

• Let Ch($R$) be the category of chain complexes over a commutative ring $R$. Let $C_0$ be the class of maps of form $\cdots\to 0\to R \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow{\operatorname{id}} R \to 0 \to 0 \to \cdots,$ and $WC_0$ be $\cdots \to 0\to 0 \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow{\operatorname{id}} R \to 0 \to 0 \to \cdots.$ Then $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations. (Model Categories, Def. 2.3.3, Th.2.3.11)

Topology

Many elementary properties in general topology, such as compactness, being dense or open, can be expressed as iterated Quillen negation of morphisms of finite topological spaces in the category Top of topological spaces. This leads to a concise, if useless, notation for a number of properties. Items below use notation for morphisms of finite topological spaces defined in the page on separation axioms in terms of lifting properties, and some examples are explained there in detail.

Uniform spaces

In the category of uniform spaces or metric spaces with uniformly continuous maps.

• A space $X$ is complete iff $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, X\to \{0\}$ where $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}}$ is the obvious inclusion between the two subspaces of the real line with induced metric, and $\{0\}$ is the metric space consisting of a single point,

• A subspace $i:A\to X$ is closed iff $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, A\to X.$

In topological spaces

The following lifting properties are calculated in the category of (all) topological spaces.

Iterated lifting properties

• $(\emptyset\longrightarrow \{o\})^{⧄r}$ is the class of surjections

• $(\emptyset\longrightarrow \{o\})^{⧄r}$ is the class of maps $A\longrightarrow B$ where $A\neq \emptyset$ or $A=B$

• $(\emptyset\longrightarrow \{o\})^{⧄rr}=\{\{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}=\{\{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}$ is the class of subsets, i.e. injective maps $A\hookrightarrow B$ where the topology on $A$ is induced from $B$

• $(\emptyset\longrightarrow \{o\})^{⧄lr}$ is the class of maps $\emptyset\longrightarrow B$, $B$ arbitrary

• $(\emptyset\longrightarrow \{o\})^{⧄lrr}$ is the class of maps $A\longrightarrow B$ which admit a section

• $(\emptyset\longrightarrow \{o\})^{⧄l}$ consists of maps $f:A\longrightarrow B$ such that either $A\neq \emptyset$ or $A=B=\emptyset$

• $(\emptyset\longrightarrow \{o\})^{⧄rl}$ is the class of maps of form $A\longrightarrow A\sqcup D$ where $D$ is discrete

• $(\emptyset\longrightarrow \{o\})^{⧄rll}$ is the class of maps $A\to B$ such that each connected subset of $B$ intersects the image of $A$; for “nice” spaces it means that the map $\pi_0(A)\to \pi_0(B)$ is surjective, where “nice” means that connected componets are both open and closed.

• $(\emptyset\longrightarrow \{o\})^{⧄rllr}$ is the class of maps of form $A\to A\sqcup B$ where $A\sqcup B$ denotes the disjoint union of $A$ and $B$.

• $\{ \{z\leftrightarrow x\leftrightarrow y\rightarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄l} = \{\{c\}\longrightarrow \{o\rightarrow c\}\}^{⧄lr}$ is the class of closed inclusions $A\subset B$ where $A$ is closed

• $\{ \{z\leftrightarrow x\leftrightarrow y\leftarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}$ is the class of open inclusions $A\subset B$ where $A$ is open

• $\{ \{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄l}$ is the class of closed maps $A\longrightarrow B$ where the topology on $A$ is pulled back from $B$

• $\{ \{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄l}$ is the class of open maps $A\longrightarrow B$ where the topology on $A$ is pulled back from $B$

• $(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄l}$ is the class of maps with dense image

• $(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}=\{ \{z\leftrightarrow x \leftrightarrow y\rightarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}$ is the class of closed subsets $A \subset X$, $A$ a closed subset of $X$

• $\{ \{z\leftrightarrow x \leftrightarrow y\leftarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}$ is the class of open subsets $A \subset X$, $A$ a open subset of $X$

• $(\{a\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}$ is the class of subsets $A \subset X$ such that $A$ is the intersection of open subsets containing $A$

• $((\{a\}\longrightarrow \{a \searrow b\})^{⧄r}_{\le 4})^{⧄lr}$ is roughly the class of proper maps

Separation axioms

Here follows a list of examples of well-known properties defined by iterated Quillen negation starting from maps between finite topological spaces, often with less than 5 elements. See at separation axioms in terms of lifting properties for more on the following.

• a space $K$ is non-empty iff $K\longrightarrow \{o\}$ is in $(\emptyset\longrightarrow \{o\})^{⧄l}$

• a space $K$ is empty iff $K \longrightarrow \{o\}$ is in $(\emptyset\longrightarrow \{o\})^{⧄ll}$

• a space $K$ is $T_0$ iff $K \longrightarrow \{o\}$ is in $(\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄r}$

• a space $K$ is $T_1$ iff $K \longrightarrow \{o\}$ is in $(\{a{ \searrow}b\}\longrightarrow \{a=b\})^{⧄r}$

• a space $X$ is Hausdorff iff for each injective map $\{x,y\} \hookrightarrow X$ it holds $\{x,y\} \hookrightarrow {X} \,⧄\, \{ {x} { \searrow} {o} { \swarrow} {y} \} \longrightarrow \{ x=o=y \}$

• a non-empty space $X$ is regular (T3) iff for each arrow $\{x\} \longrightarrow X$ it holds $\{x\} \longrightarrow {X} \,⧄\, \{x{ \searrow}X{ \swarrow}U{ \searrow}F\} \longrightarrow \{x=X=U{ \searrow}F\}$

• a space $X$ is normal (T4) iff $\emptyset \longrightarrow {X} \,⧄\, \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}$

• a space $X$ is completely normal iff $\emptyset\longrightarrow {X} \,⧄\, [0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\}$ where the map $[0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\}$ sends $0$ to $0$, $1$ to $1$, and the rest $(0,1)$ to $x$

• a space $X$ is hereditary normal iff $\emptyset \to X ⧄ \{ x \leftarrow au \leftrightarrow u' \leftarrow u \leftarrow uv \rightarrow v \rightarrow v'\leftrightarrow bv \rightarrow x \} \longrightarrow \{ x \leftarrow au \leftrightarrow u' = u \leftarrow uv \rightarrow v = v'\leftrightarrow bv \rightarrow x \}$

• a space $X$ is path-connected iff $\{0,1\} \longrightarrow [0,1] \,⧄\, {X} \longrightarrow \{o\}$

• a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow K$ it holds $\{x,y\} \hookrightarrow {K} \,⧄\, {X} \longrightarrow \{o\}$

• A map $X\longrightarrow Y$ is a quotient iff $X\to Y \,\,⧄\,\, \{o \rightarrow c\}\longrightarrow \{o\leftrightarrow c\}$

• For every pair of disjoint closed subsets of $X$, the closures of their images of $Y$ do not intersect, if $X\to Y \,\,⧄\,\, \{x\leftarrow o\rightarrow y\}\longrightarrow \{x=o=y\}$

• A topological space $X$ is extremally disconnected iff $\emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{u\rightarrow a=b\leftarrow v\}$

• A topological space $X$ is zero-dimensional iff $\emptyset\to X \,\,⧄\,\, \{a\leftarrow u,v\rightarrow b\}\longrightarrow \{a\leftarrow u=v\rightarrow b\}$

• A topological space $X$ is ultranormal iff $\emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{a\leftarrow u=v\rightarrow v\}$

• $\{\bullet\}\longrightarrow A$ is in $(\emptyset\longrightarrow \{o\})^{⧄rll}$ iff $A$ is connected

• $Y$ is totally disconnected iff $\{\bullet\}\xrightarrow y Y$ is in $(\emptyset\longrightarrow \{o\})^{⧄rllr}$ for each map $\{\bullet\}\xrightarrow y Y$ (or, in other words, each point $y\in Y$).

• a Hausdorff space $K$ is compact iff $K\longrightarrow \{o\}$ is in $((\{o\}\longrightarrow \{o{ \searrow}c\})^{⧄r}_{\le5})^{⧄lr}$

• a Hausdorff space $K$ is compact iff $K\longrightarrow \{o\}$ is in $$\{\, \{a\leftrightarrow b\}\longrightarrow \{a=b\},\, \{o{ \searrow}c\}\longrightarrow \{o=c\},\, \{c\}\longrightarrow \{o{ \searrow}c\},\,\{a{ \swarrow}o{ \searrow}b\}\longrightarrow \{a=o=b\}\,\,\}^{⧄lr}$$

• a topological space $X$ is compactly generated iff $\varnothing\longrightarrow X$ is in $\bigcup(\{\{0,1\}\to\{0=1\}\}\cup\{\varnothing \to K \,\,:\,\, K\,\, \text{ compact}\}\big)^{⧄rl}$

• a space $D$ is discrete iff $\emptyset \longrightarrow D$ is in $(\emptyset\longrightarrow \{o\})^{⧄rl}$

• a space $D$ is codiscrete iff ${D} \longrightarrow \{o\}$ is in
$(\{a,b\}\longrightarrow \{a=b\})^{⧄rr}= (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄lr}$

• a space $K$ is connected or empty iff $K\longrightarrow \{o\}$ is in $(\{a,b\}\longrightarrow \{a=b\})^{⧄l}$

• a space $K$ is totally disconnected and non-empty iff $K\longrightarrow \{o\}$ is in $(\{a,b\}\longrightarrow \{a=b\})^{⧄lr}$

• a space $K$ is connected and non-empty iff for some arrow $\{o\}\longrightarrow K$ it holds that $\{o\}\longrightarrow K$ is in $(\emptyset\longrightarrow \{o\})^{⧄rll} = (\{a\}\longrightarrow \{a,b\})^{⧄l}$

• A topological space $X$ has Lebesgue dimension at most $n$ iff for each finite set $I$ $\emptyset\to X \,\,⧄\,\, \{ (F,J): 1\leq |F|\leq n+1, F\subset J\subset I\}\longrightarrow \{ J: 1\leq |J|, J\subset I\}$ where the order on the domain $\{ (F,J): 1\leq |F|\leq n+1, F\subset J\subset I\}$ is given by $(F,J)\to (F',J')$ iff $F\subset F'$ and $J\subset J'$.

• A topological space $X$ has Lebesgue dimension at most $n$ iff for each closed subset $A$ of $X$ $A\to X \,\,⧄\,\, \mathbb{S}^n\to \{o\}$ where $\mathbb{S}^n$ denotes the $n$-sphere.

Model theory

In model theory, a number of the Shelah’s divining lines, namely $NOP, NSOP, NSOP_i, NTP, NTP_i$, and $NATP$ are expressed as Quillen lifting properties of form

$A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top$

where $\top$ is the terminal object, and $M$ is a situs associated with a model and a formula, and $A$ and $B$ are objects of combinatorial nature, in the category of simplicial objects in the category of filters.