nLab
lift

Contents

Contents

Idea

Intuitively, the lifting property is a negation in category theory: Taking the class of morphisms having the lifting property with respect to each morphism in a class PP is a simple, and unexpectedly common, way to define a class of morphisms excluding non-isomorphisms from PP, in a way which is useful in a diagram chasing computation. A number of elementary notions may be expressed by iteratively using the lifting property starting from a list of simple or archetypal (counter)examples.

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.

A useful intuition is to think that the lifting property as a kind of negation: taking the class of morphisms having the lifting property with respect to each morphism in a class PP is a simple way to define a class of morphisms excluding non-isomorphisms from PP, in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, a map is surjective iff it has the right lifting property with respect to the simplest non-surjection {*}\emptyset\to \{*\}, and injective iff it has either left or right lifting property with respect to {x 1,x 2}{*}\{x_1,x_2\}\to \{*\}, the simplest non-injection.

Definition

A morphism ii in a category has the ‘’left lifting property’‘ with respect to a morphism pp, and pp also has the ‘’right lifting property’‘ with respect to ii, sometimes denoted ipi\,\,⧄\,\, p or ipi\downarrow p, iff the following implication holds for each morphism ff and gg in the category:

  • if the outer square of the following diagram commutes, then there exists hh completing the diagram, i.e. for each f:AXf:A\to X and g:BYg:B\to Y such that pf=gip\circ f = g \circ i there exists h:BXh:B\to X such that hi=fh\circ i = f and ph=gp\circ h = g.

This is sometimes also known as the morphism ii being ‘’weakly orthogonal to’‘ the morphism pp; however, ‘’orthogonal to’‘ will refer to the stronger property that whenever ff and gg are as above, the diagonal morphism hh exists and is also required to be unique.

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

C :={ipC,ip}C r:={piC,ip},C lr:=(C l) r 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 CC is a simple way to define a class of morphisms excluding isomorphisms from CC, in a way which is useful in a diagram chasing computation.

Thus, in the category Set of sets, the right Quillen negation {{*}} r\{\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}{*},\{x_1,x_2\}\to \{*\}, the simplest non-injection, are both precisely the class of injections, {{x 1,x 2}{*}} ={{x 1,x 2}{*}} r={ff is an injection }.\{\{x_1,x_2\}\to \{*\}\}^{⧄ \ell} = \{\{x_1,x_2\}\to \{*\}\}^{⧄ r} = \{ f \mid f \text{ is an injection } \}.

It is clear that C rCC^{⧄\ell r} \supset C and C rCC^{⧄ r\ell} \supset C. The class C rC^{⧄ 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 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,

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

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

Modules

In the category RMod of modules over a commutative ring RR,

  • {0R} r,{R0} r\{0\to R\}^{⧄ r}, \{R\to 0\}^{⧄ r} is the class of surjections, resp. injections,

  • A module MM is projective, resp. injective, iff 0M0\to M is in {0R} r\{0\to R\}^{⧄ \ell r}, resp. M0M\to 0 is in {R0} rr\{R\to 0\}^{⧄ rr}.

Groups

In the category Grp of groups,

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

  • A group FF is a free group iff 0F0\to F is in {0} r,\{0\to \mathbb{Z} \}^{⧄ r\ell},

  • A group AA is torsion-free iff 0A0\to A is in {n:n0} r,\{ n \mathbb{Z} \to \mathbb{Z} : n\ge0 \}^{⧄ r},

  • A subgroup AA of BB is pure? iff ABA \to B is in {n:n0} r.\{ n\mathbb{Z}\to \mathbb{Z} : n\ge0 \}^{⧄ r}.

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

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

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

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

  • a group FF is free iff 1F1\to F is in (0) rl(0\longrightarrow \mathbb{Z})^{⧄rl}

  • a group AA is Abelian iff A1A\to 1 is in (𝔽 2×) r( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄ r}

  • group GG can be obtained from HH by adding commutation relations, i.e.~the kernel of HGH\to G is generated by commutators [h 1,h 2][h_1,h_2], h 1,h 2Hh_1,h_2\in H, iff HGH\to G is in (𝔽 2×) rl( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄rl}

  • subgroup HH of GG is the normal span of substitutions in words w 1,..,w iw_1,..,w_i of the free group 𝔽 n\mathbb{F}_n iff GG/HG \to G/H is in (𝔽 n𝔽 n/w 1,...,w i) rl( \mathbb{F}_n \to \mathbb{F}_n/\le\!w_1,...,w_i\!\ge)^{⧄rl}

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

For a finite group GG, in the category of finite groups,

  • {0/p}G1\{0\to {\mathbb{Z}}/p{\mathbb{Z}}\} \,\,⧄\,\, G\to 1 iff the order of GG is prime to pp,

  • G1(0/p) rrG\to 1 \in (0\to {\mathbb{Z}}/p{\mathbb{Z}})^{⧄ rr} iff GG is a p p -group,

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

  • a finite group HH is soluble? iff 1H1\to H is in {0A:A abelian} r={[G,G]G:G arbitrary } r.\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}.

Moreover,

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

  • {0A:A abelian} r={[G,G]G:G arbitrary } r\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}, and is the class of subgroups HGH\leq G such that there is a chain of subnormal subgroups H=G 0G 1G n=GH=G_0 \vartriangleleft G_1 \vartriangleleft \ldots \vartriangleleft G_n =G such that G i+1/G iG_{i+1}/G_{i} is Abelian, for i=0,...,n1i=0,...,n-1.

  • {1S} r\{1 \to S\}^{⧄ \ell r} is the class of subgroups HGH\leq G such that there is a chain of subnormal subgroups H=G 0G 1G n=GH=G_0 \vartriangleleft G_1 \vartriangleleft \ldots \vartriangleleft G_n =G such that G i+1/G iG_{i+1}/G_{i} embeds into SS, for i=0,...,n1i=0,...,n-1.

  • (/p0) rlr(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄rlr} is the class of homomorphisms whose kernel has no elements of order pp

  • (/p0) rr(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄rr} is the class of surjective homomorphisms whose kernel is a pp-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 QuTop_{Qu} is controlled by the following lifting properties:

consider let C 0C_0 be the class of maps S nD n+1S^n\to D^{n+1}, embeddings of the boundary S n=D n+1S^n=\partial D^{n+1} of a ball into the ball D n+1D^{n+1}. Let WC 0WC_0 be the class of maps embedding the upper semi-sphere into the disk. WC 0 ,WC 0 r,C 0 ,C 0 rWC_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:UBf:U\to B has the ‘’path lifting property’‘ iff {0}[0,1]f\{0\}\to [0,1] \,\,⧄\,\, f where {0}[0,1]\{0\} \to [0,1] is the inclusion of one end point of the closed interval into the interval [0,1][0,1].

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

Kan fibrations of simplicial sets

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

Let C 0C_0 be the class of boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n], and let WC 0WC_0 be the class of horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n]. Then the classes of Kan fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, WC 0 ,WC 0 r,C 0 ,C 0 rWC_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(RR) be the category of chain complexes over a commutative ring RR. Let C 0C_0 be the class of maps of form 0R00RidR00,\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 0WC_0 be 0000RidR00.\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 ,WC 0 r,C 0 ,C 0 rWC_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 XX is complete iff {1/n} n{0}{1/n} nX{0}\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, X\to \{0\} where {1/n} n{0}{1/n} n\{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}\{0\} is the metric space consisting of a single point,

  • A subspace i:AXi:A\to X is closed iff {1/n} n{0}{1/n} nAX.\{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

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

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

  • ({o}) rr={{xyc}{x=y=c}} l={{xyc}{x=y=c}} l(\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 ABA\hookrightarrow B where the topology on AA is induced from BB

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

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

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

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

  • {{zxyc}{z=xy=c}} lrl={{c}{oc}} lr\{ \{z\leftrightarrow x\leftrightarrow y\rightarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄lrl} = \{\{c\}\longrightarrow \{o\rightarrow c\}\}^{⧄lr} is the class of closed inclusions ABA\subset B where AA is closed

  • {{zxyc}{z=xy=c}} lrl\{ \{z\leftrightarrow x\leftrightarrow y\leftarrow c\}\longrightarrow\{z=x\leftrightarrow y=c\} \}^{⧄lrl} is the class of open inclusions ABA\subset B where AA is open

  • {{xyc}{xy=c}} lrl\{ \{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄lrl} is the class of closed maps ABA\longrightarrow B where the topology on AA is pulled back from BB

  • {{xyc}{xy=c}} lrl\{ \{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x\leftrightarrow y=c\} \}^{⧄lrl} is the class of open maps ABA\longrightarrow B where the topology on AA is pulled back from BB

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

  • ({b}{ab}) lr={{zxyc}{z=xy=c}} l(\{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 AXA \subset X, AA a closed subset of XX

  • {{zxyc}{z=xy=c}} l\{ \{z\leftrightarrow x \leftrightarrow y\leftarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l} is the class of open subsets AXA \subset X, AA a open subset of XX

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

  • (({a}{ab}) 4 r) lr((\{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 KK is non-empty iff K{o}K\longrightarrow \{o\} is in ({o}) l (\emptyset\longrightarrow \{o\})^{⧄l}

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

  • a space KK is T 0T_0 iff K{o}K \longrightarrow \{o\} is in ({ab}{a=b}) r (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄r}

  • a space KK is T 1T_1 iff K{o}K \longrightarrow \{o\} is in ({ab}{a=b}) r (\{a{ \searrow}b\}\longrightarrow \{a=b\})^{⧄r}

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

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

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

  • a space XX is completely normal iff X[0,1]{0x1}\emptyset\longrightarrow {X} \,⧄\, [0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\} where the map [0,1]{0x1}[0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\} sends 00 to 00, 11 to 11, and the rest (0,1)(0,1) to xx

  • a space XX is hereditary normal iff X{xauuuuvvvbvx}{xauu=uuvv=vbvx} \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 XX is path-connected iff {0,1}[0,1]X{o}\{0,1\} \longrightarrow [0,1] \,⧄\, {X} \longrightarrow \{o\}

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

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

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

  • A topological space XX is extremally disconnected iff XY{ua,bv}{ua=bv}X\to Y \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{u\rightarrow a=b\leftarrow v\}

  • A topological space XX is ultranormal iff XY{ua,bv}{au=vv}X\to Y \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{a\leftarrow u=v\rightarrow v\}

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

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

  • a Hausdorff space KK is compact iff K{o}K\longrightarrow \{o\} is in (({o}{oc}) 5 r) lr((\{o\}\longrightarrow \{o{ \searrow}c\})^{⧄r}_{\le5})^{⧄lr}

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

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

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

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

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

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

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

Model theory

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

A B M A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top

where \top is the terminal object, and MM is a situs associated with a model and a formula, and AA and BB are objects of combinatorial nature, in the category of simplicial objects in the category of filters.

Last revised on October 16, 2021 at 14:29:51. See the history of this page for a list of all contributions to it.