Lifting Problems
Table of Contents
We given an overview of the different types of lifting problems discussed in “Introduction to Infinity-Categories” by Markus Land:
- anodyne maps and fibrations
1. Abstract Discussion
Let \( \mathcal C \) be a category. A lifting problem in \( \mathcal C \) is a commutative diagram
\begin{equation*} \xymatrix{ A \ar@{->}[d] \ar@{->}[r] & X \ar@{->}[d] \\ B \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}in \( \mathcal C \) where we ask if the dashed arrow exists. We say that two lifting problems are equivalent if a solution of one problem let’s us construct a solution of the other problem (and the other way around).
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( G \colon \mathcal D \rightarrow \mathcal C \) be adjoint functors (\( F \dashv G \)). Then we can transfer to following lifting problems:
\begin{equation*} \xymatrix{ A \ar@{->}[d]_{f} \ar@{->}[r]^{i} & GX \ar@{->}[d]^{G(g)} \\ B \ar@{->}[r]^{j} \ar@{-->}[ru] & GY } \Leftrightarrow \xymatrix{ FA \ar@{->}[d] \ar@{->}[r] & X \ar@{->}[d] \\ FB \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}Let \( F \dashv G \) and \( F' \dashv G' \) be adjunctions with \( (\epsilon, \eta) \) and \( (\epsilon ', \eta ') \) being the unit/counit pairs. Let \( \alpha \colon F \Rightarrow F' \) and \( \beta \colon G' \Rightarrow G \) be natural transformations such that \( \eta ' _{(-)} \circ \alpha _{G'(-)} = \eta _{(-)} \circ F(\beta _{(-)}) \) and \(G(\alpha _{(-)}) \circ \epsilon _{(-)} = \beta _{F'(-)} \circ \epsilon ' _{(-)} \). Then the lifting problem
\begin{equation*} \xymatrix{ A \ar@{->}[d]_{f} \ar@{->}[r]^{G'(j) \circ \epsilon ' _{A}} & G'X \ar@{->}[d]^{\beta _X \times G'(g)} \\ B \ar@{->}[r]^{k \times l} \ar@{-->}[ru] & GX \times_{GY} G'Y } \end{equation*}is equivalent to
\begin{equation*} \xymatrix{ FB \amalg_{FA} F'A \ar@{->}[d]_{\alpha _B \amalg F'(f)} \ar@{->}[r]^{i \amalg j} & X \ar@{->}[d]^{g} \\ F'B \ar@{->}[r]^{\eta ' _Y \circ F'(l)} \ar@{-->}[ru] & Y } \end{equation*}where \(\eta _X \circ F(k) = i \) and \( G(i) \circ \epsilon _B = k\) (note that we only need one of these equations as they are equivalent).
I am sorry that this formulation is so cryptic. Maybe it would be easier to start with a diagram and formulate the corresponding diagram …
Proof.
\begin{align*} \eta '_X \circ F'(t) \circ \alpha_B = & \eta '_X \circ \alpha_{G'X} \circ F(t) \\ = & \eta_X \circ F(\beta_X) \circ F(t) \\ = & \eta_X \circ F(k) \\ = & i \end{align*} \begin{align*} \eta '_X \circ F'(t) \circ F'(f) = & \eta_X ' \circ F'(t \circ f) \\ = & \eta '_X \circ F'G'(j) \circ F'(\epsilon '_A) \\ = & j. \end{align*} \begin{align*} g \circ \eta '_X \circ F'(t) = & \eta '_Y \circ F'G'(g) \circ F'(t) \\ = & \eta '_Y \circ F'(l). \end{align*} \begin{align*} G'(t) \circ \epsilon '_B \circ f = & G'(t) \circ G'F'(f) \circ \epsilon '_A \\ = & G'(j) \circ \epsilon '_A. \end{align*} \begin{align*} \beta _X \circ G'(t) \circ \epsilon '_B = & G(t) \circ \beta _{F'B} \circ \epsilon '_B \\ = & G(t) \circ G(\alpha _B) \circ \epsilon _B \\ = & G(i) \circ \epsilon _B \\ = & k \end{align*} \begin{align*} G'(g) \circ G'(t) \circ \epsilon _B = & G'(\eta '_Y \circ F'(l)) \circ \epsilon _B \\ = & G'( \eta '_Y) \circ G'F'(l) \circ \epsilon _B \\ = & l. \end{align*}1.1. QUESTION Can we find critera for Lemma No description for this link?
Something like: let \(F(-,-) \) be a bifunctor such that \(F(A,-) \) has the right adjoint \(G_A(-) \).
Let \(F(-,-) \colon \mathcal C \times \mathcal C \rightarrow \mathcal D \) and \( G(-,-) \).
Let \(S \) be a set of morphisms in a category \( \mathcal C\). Then we want to study lifting problems of the form
\begin{equation*} \xymatrix{ A \ar@{->}[d]^f \ar@{->}[r] & X \ar@{->}[d] \\ B \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}where \(f \) is a morphism in \(S \).
Let \(S \) be a class of morphisms in a category \(\mathcal C \). Let \(\chi_R(S) \) be the class of morphisms having the RLP with respect to all morphisms in \(S \). Let \(\chi_L(S) \) be the class of morphisms having the LLP with respect to all morphisms in \(S \). Let \(\chi(S) \) be \(\chi_L(\chi_R(S)) \). Note that \(\chi_R(\chi_L(\chi_R(S))) = \chi_R(S) \) and \(\chi_L(\chi_R(\chi_L(S))) = \chi_L(S)\).
Given \(S \) we try to study \(\chi_R(S) \) and \(\chi(S) \).
Let \(S \) be a class of morphisms in a category \(\mathcal C \). We call \(S \) saturated if \(S \) is closed under pushouts (along arbitrary maps), arbitrary coproducts, countable compositions, and retracts. Let \(\bar{S} \) be the smallest saturated set of morphisms containing \(S \).
If we have a solvable lifting problem and we apply one of these operations to the LHS, then we get a solvable lifting problem again. This is expressed by the following lemma.
Let \(S \) be a class of morphisms in a category \(\mathcal C \). Then \(\chi_L(S) \) is saturated.
Hence \(\bar{S} \subseteq \chi(S)\) and we can hope that the other inclusion holds too. Sadly in general this does not hold, but the following proposition gives an important criteria under which the other inclusion holds.
Let \(S = \{ A_i \rightarrow B_i \} _{i \in I} \) be a class of morphisms in \(\sSet \) such that \(A_i \) has only finitely many non-degenerate simplices for every \(i \in I \). Then \(\bar{S} = \chi(S) \).
Proof.
1.2. QUESTION Can we get a similar result for \(\chi_R(S) \) and a ``cosaturation’’ of \(S \)?
2. Anodyne Maps and Fibrations
This sections discusses:
- (inner, left, right) horn inclusions on the LHS
- (inner, left, right) fibration of the RHS
- trivial fibrations
Sine \(\infty \)-categories are defined via lifting problems with inner horn inclusions, it is natural to study lifting problems with horn inclusions.
2.1. TODO Add Joyal fibration (Def 1.3.27)
Let \(S \) be the set of (inner, left, right) horn inclusions. The class of (inner, left, right) fibrations is \(\chi_R(S)\). The class of (inner, left, right) anodyne maps is \(\chi(S) \). The class of trivial fibrations is \(\chi_R(\{\text{boundary inclusions}\}) \). The class of trivial anodyne maps is \(\chi(\{\text{boundary inclusions}\}) \).
First we discuss how the nerve functors interact with these lifting problems.
Let \(F \colon \mathcal C \rightarrow \mathcal D \) be a functor between categories. Then \(\operatorname{N}F \colon \operatorname{N}\mathcal C \rightarrow \operatorname{N}\mathcal D \) is an inner fibration.
Proof.
Let \(F \colon \mathcal C \rightarrow \mathcal D \) be a simplicial functor between simplicial categories such that \(\mathcal C(X,Y) \rightarrow \mathcal D(FX,FY) \) is a Kan fibration (fibration between Kan complexes). Then \(\operatorname{N}F \colon \operatorname{N}\mathcal C \rightarrow \operatorname{N}\mathcal D \) is an inner fibration.
boxtimes operator scalar product operator
Let \(i \colon A \rightarrow B \), \(f \colon X \rightarrow Y \) and \(g \colon S \rightarrow T \) be morphisms. Then the following two lifting problems are equivalent:
\begin{equation*} \xymatrix{ S \ar@{->}[d]_{g} \ar@{->}[r]^{j} & X^B \ar@{->}[d]^{\langle f,i \rangle} \\ T \ar@{->}[r]^{k \times l} \ar@{-->}[ru] & X^A \times_{Y^A} Y^B } \Leftrightarrow \xymatrix{ A \times T \amalg_{A \times S} B \times S \ar@{->}[d]_{i \boxtimes g} \ar@{->}[r]^{a} & X \ar@{->}[d]^{g} \\ B \times T \ar@{->}[r]^{b} \ar@{-->}[ru] & Y} \end{equation*}where \(a \) is induced by \(A \times T \xrightarrow{\id _A \times k} A \times X^A \rightarrow X \) and \(B \times S \xrightarrow{\id_B \times j} B \times X^B \rightarrow X \) and \(b \) is given by \(B \times T \xrightarrow{\id_B \times l} B \times Y^B \rightarrow Y \). Equivalently the following two lifting problems are equivalent:
\begin{equation*} \xymatrix{ A \times T \amalg_{A \times S} B \times S \ar@{->}[d]_{i \boxtimes g} \ar@{->}[r]^{k \amalg l} & X \ar@{->}[d]^{f} \\ B \times T \ar@{->}[r]^{j} \ar@{-->}[ru] & Y } \;\;\;\; \xymatrix{ S \ar@{->}[d]_{g} \ar@{->}[r]^{a} & X^B \ar@{->}[d]^{\langle f,i\rangle} \\ T \ar@{->}[r]^{b} \ar@{-->}[ru] & X^A \times_{Y^A} Y^B } \end{equation*}where \(a \) is given by \(S \xrightarrow{\text{counit}} (B \times S)^B \xrightarrow{l^B} X^B \) and \(b \) is induced by \(T \xrightarrow{\text{counit}} (A \times T)^A \xrightarrow{k} X^A \) and \(T \xrightarrow{\text{counit}} (B \times T)^B \xrightarrow{j^B} Y^B \).
Proof.
Since \(\boxtimes \) is a geometric operation (gluing of two space along an intersection), it is natural to ask if \(i \boxtimes g \) is (inner, left, right) anodyne.
Let \(i \colon A \rightarrow B \) and \(g \colon S \rightarrow T \) be monomorphisms. Then \(i \boxtimes g \) is (inner, left, right) anodyne, if \(i \) or \(g \) is (inner, left, right) anodyne.
Proof.
[Product of fibrations] Let \(f \colon X \rightarrow Y \) be a (inner, left, right) fibration and \(i \colon A \rightarrow B \) a monomorphism. Then
- \(\langle f,i \rangle \) is a (inner, left, right) fibration.
- \(\langle f,i \rangle \) is a trivial fibration, if \(i \) is (inner, left, right) anodyne.
Proof.
[Generators of inner-anodyne maps] The saturated closure of the following classes generate the class of inner-anodyne maps:
- inner horn inclusions .\(\{ \Lambda^n_j \rightarrow \Delta^n \mid n \ge 2 \} \).
- \(\{(K \rightarrow L) \boxtimes (\Lambda^2_1 \rightarrow \Delta^2) \mid K \rightarrow L \text{ monomorphism}\}\).
- \(\{(\partial\Delta^n \rightarrow \Delta^n) \boxtimes (\Lambda^2_1 \rightarrow \Delta^2) \mid n \ge 0\} \).
- \(\{(K \rightarrow L) \boxtimes (\Lambda^n_j \rightarrow \Delta^n ) \mid K \rightarrow L \text{ monomorphism}, 0 \le n, 0 < j < n\}\).
Proof.
[Generators of left-anodyne maps] The saturated closure of the following classes generate the class of left-anodyne maps:
- left horn inclusions .\(\{ \Lambda^n_j \rightarrow \Delta^n \mid n \ge 1, 0 \le j < n \} \).
- \(\{(K \rightarrow L) \boxtimes (\{0\} \rightarrow \Delta^1) \mid K \rightarrow L \text{ monomorphism}\}\).
- \(\{(\partial\Delta^n \rightarrow \Delta^n) \boxtimes (\{0\} \rightarrow \Delta^1) \mid n \ge 0\} \).
- \(\{(K \rightarrow L) \boxtimes (\Lambda^n_j \rightarrow \Delta^n ) \mid K \rightarrow L \text{ monomorphism}, 1 \le n, 0 \le j < n\}\).
[Generators of right-anodyne maps] The saturated closure of the following classes generate the class of inner-anodyne maps:
- right horn inclusions \(\{ \Lambda^n_j \rightarrow \Delta^n \mid n \ge 1, 0 < j \le n \} \).
- \(\{(K \rightarrow L) \boxtimes (\{1\} \rightarrow \Delta^1) \mid K \rightarrow L \text{ monomorphism}\}\).
- \(\{(\partial\Delta^n \rightarrow \Delta^n) \boxtimes (\{1\} \rightarrow \Delta^1) \mid n \ge 0\} \).
- \(\{(K \rightarrow L) \boxtimes (\Lambda^n_j \rightarrow \Delta^n ) \mid K \rightarrow L \text{ monomorphism}, n \ge 1, 0 < j \le n\}\).
[Characterization of trivial anodyne maps] The class of trivial anodyne maps is exactly the class of monomorphisms.
Lastly we collect some common (inner, left, right) anodyne maps and fibrations.
[Spine and general horn inclusions] Let \(S \subseteq [n] \) be non-empty. Then \(\Lambda^n_S \rightarrow \Delta^n \) is
- anodyne, if \(S \neq [n] \),
- left-anodyne, if \(n \notin S \),
- right-anodyne, if \(0 \notin S \),
- inner-anodyne, if \(S \) is not the complement of an interval in \([n] \).
All spine inclusions are inner anodyne maps.
[Examples of fibrations] Cor 1.3.44 Remark 1.3.45
sections of trivial fibrations (Lemma 1.3.46)
2.2. Joins and Slices
join construction
slice construction
Let \(S \) be a simplicial set. Then \(S \star - \colon \sSet \rightarrow \sSet_{S/} \) is left adjoint to \((-) _{-/} \colon \sSet _{S/} \rightarrow \sSet \) and \(- \star S \colon \sSet \rightarrow \sSet _{S/} \) is left adjoint to \((-) _{/-} \colon \sSet _{S/} \rightarrow \sSet \).
Let \(i \colon A \rightarrow B \) be a morphism of simplicial sets. The pushforward \(i_* \colon \sSet _{A/} \rightarrow \sSet _{B/} \) is defined by \((p \colon A \rightarrow X) \mapsto (B \rightarrow X \amalg_A B) \). The pullback \(i^* \colon \sSet _{B/} \rightarrow \sSet _{A/} \) is defined by \((p \colon B \rightarrow X) \mapsto ( p \circ i \colon A \rightarrow X) \).
Let \(i \colon A \rightarrow B \) be a morphism of simplicial sets. Then \(i_* \colon \sSet_{A/} \rightarrow \sSet_{B/} \) is left adjoint to \(i^* \colon \sSet_{B/} \rightarrow \sSet _{A/} \) and \(i_* \colon \sSet _{/A} \rightarrow \sSet _{/B} \) is left adjoint to \(i^* \colon \sSet _{/B} \rightarrow \sSet _{/A} \).
hat * product for both adjoints
We need a result from category theory.
Let \(\mathcal C \) be a category. Then \((Y \amalg_A B) \amalg _{X \amalg_A B} S \cong Y \amalg_X S \).
Proof.
Let \(A \xrightarrow {i} B \xrightarrow{\varphi} X \xrightarrow {f} Y \) and \(S \xrightarrow{g} T \) be morphisms of simplicial sets. Then the following lifting problems are equivalent:
\begin{equation*} \xymatrix{ S \ar@{->}[d]_g \ar@{->}[r] & X_{\varphi/} \ar@{->}[d]^{\langle f,i \rangle_{\varphi /}} \\ T \ar@{->}[r] \ar@{-->}[ru] & X _{\varphi i/} \times_{Y _{f \varphi i /}} Y _{\varphi f/} } \;\;\;\; \xymatrix{ B \ar[r] \ar@/^1.5pc/[rr]^{\varphi}& A \star T \amalg_{A \star S} B \star S \ar@{->}[d]_{i \hat\star g} \ar@{->}[r] & X \ar@{->}[d] \\ & B \star T \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}where the left lifting problem happens in \(\sSet \) and the right lifting problem happens in \(\sSet _{B/} \). Note that this just means that we have a lifting problem in \(\sSet \) where the diagram satisfies the extra condition that all morphisms commute with the canonical morphisms \(B \rightarrow - \). This is only an extra condition for the morphism \(A \star T \amalg_{A \star S} B \star S \rightarrow X \), hence we include \(\varphi \) in the diagram.
Alternative formulation: let \(i \colon A \rightarrow B \), \(f \colon X \rightarrow Y \) and \(g \colon S \rightarrow T \) be morphisms of simplicial sets. Then the following lifting problems are equivalent:
\begin{equation*} \xymatrix{ S \ar@{->}[d]_g \ar@{->}[r] & X_{\varphi/} \ar@{->}[d]^{\langle f,i \rangle_{\varphi /}} \\ T \ar@{->}[r] \ar@{-->}[ru] & X _{\varphi i/} \times_{Y _{f \varphi i /}} Y _{\varphi f/} } \;\;\;\; \xymatrix{ A \star T \amalg_{A \star S} B \star S \ar@{->}[d]_{i \hat\star g} \ar@{->}[r] & X \ar@{->}[d] \\ B \star T \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}where \(\varphi \colon B \rightarrow X \) is the morphism \(B \rightarrow B \star S \rightarrow X \).
Proof.
Let \(A \xrightarrow{i} B \) and \(S \xrightarrow{g} T \xrightarrow{\psi} X \xrightarrow{f} Y \) be morphisms of simplicial sets. Then the following lifting problems are equivalent:
\begin{equation*} \xymatrix{ A \ar@{->}[d]_i \ar@{->}[r] & X_{/ \psi } \ar@{->}[d] ^{\rangle g,f \langle_{/ \psi}} \\ B \ar@{->}[r] \ar@{-->}[ru] & X _{/ \psi g} \times_{Y _{/ f \psi g}} Y _{/ f \psi} } \;\;\;\; \xymatrix{ S \ar[r] \ar@/^1.5pc/[rr]^{\psi}& A \star T \amalg_{A \star S} B \star S \ar@{->}[d]_{i \hat\star g} \ar@{->}[r] & X \ar@{->}[d]^{f} \\ & B \star T \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}Alternative formulation: let \(i \colon A \rightarrow B \), \(f \colon X \rightarrow Y \) and \(g \colon S \rightarrow T \) be morphisms of simplicial sets. Then the following lifting problems (both in \(\sSet \)) are equivalent:
\begin{equation*} \xymatrix{ A \ar@{->}[d]_i \ar@{->}[r] & X_{/ \psi } \ar@{->}[d] ^{\langle g,f \rangle_{/ \psi }} \\ B \ar@{->}[r] \ar@{-->}[ru] & X _{/ \psi g} \times_{Y _{/ f \psi g}} Y _{/ f \psi} } \;\;\;\; \xymatrix{ A \star T \amalg_{A \star S} B \star S \ar@{->}[d]_{i \hat\star g} \ar@{->}[r] & X \ar@{->}[d]^{f} \\ B \star T \ar@{->}[r] \ar@{-->}[ru] & Y } \end{equation*}where \(\psi \colon T \rightarrow X \) is the morphism \(T \rightarrow A \star T \rightarrow X \).
Proof.
\begin{equation*} \alpha_X := (\id_X \star g) \amalg (T \rightarrow X \star T) \colon g_*(X \star S) \rightarrow X \star T \end{equation*} \begin{equation*} (\beta_X)_n := (- \circ (\id_{\Delta^n} \star g))\colon \Hom _{\sSet _{T/}}(\Delta^n \star T, X) \rightarrow \Hom _{\sSet _{S/}}(\Delta^n \star S, g^*X) \end{equation*} \begin{align*} \eta _X \circ F(\beta _X)(\sigma ) = & \sigma \\ = & p(\sigma ). \end{align*} \begin{align*} \eta '_X \circ \alpha _{G'X}(\sigma ) = & \sigma \\ = & p(\sigma ). \end{align*} \begin{align*} \eta_X \circ F(\beta _X)(\sigma ) = & \eta _X(\sigma) \\ = & p \circ g \circ \sigma \end{align*} \begin{align*} \eta '_x \circ \alpha _{G'X}(\sigma ) = & \eta '_X(g(\sigma )) \\ = & p \circ g \circ \sigma. \end{align*} \begin{align*} \eta _X \circ F(\beta _X)(\sigma ) = & \eta _X(\sigma \circ (\id _{\Delta^n} \star g)) \\ = & \sigma \circ (\id _{\Delta^n} \star g) \circ (\Delta^n \rightarrow \Delta^n \star S) \\ = & \sigma \circ (\Delta^n \rightarrow \Delta ^n \star S) \end{align*} \begin{align*} \eta '_X \circ \alpha _{G'X}(\sigma ) = & \eta '_X(\sigma ) \\ = & \sigma \circ (\Delta^n \rightarrow \Delta^n \star S). \end{align*} \begin{align*} n_X \circ F(\beta _X)(\sigma ) = & n_X((\sigma _1 \circ (\id _{\Delta ^{n_1}} \star g), \sigma _2)) \\ = & \sigma_1 \circ (\id _{\Delta ^{n_1}} \star g) \circ (\id _{\Delta ^{n_1}} \star \sigma _2) \\ = & \sigma _1 \circ (\id _{\Delta ^{n_1}} \star (g \circ \sigma _2)). \end{align*} \begin{align*} \eta '_X \circ \alpha _{G'X}(\sigma _1, \sigma _2) = & \eta '_X(\sigma _1, g \circ \sigma _2) \\ = & \sigma _1 \circ (\id _{\Delta ^{n_1}} \star (g \circ \sigma _2)). \end{align*} \begin{align*} G(\alpha _X) \circ \epsilon _X(\sigma ) = & G(\alpha _X)(\iota_{X \star S} \circ (\sigma \star \id_S)) \\ = & (\id_X \star g) \circ (\sigma \star \id_S) \\ = & \sigma \star g \end{align*} \begin{align*} \beta _{F'X} \circ \epsilon '_X(\sigma ) = & \beta _{F'X}(\sigma \star \id_T) \\ = & (\sigma \star \id _T) \circ (\id _{\Delta^n} \star g) \\ = & \sigma \star g. \end{align*}3. Joyal’s Lifting Theorem
Let \(\mathcal C \rightarrow \mathcal D \) be an inner fibration between \(\infty \)-categories. Let \(\phi \colon \Delta^1 \rightarrow \mathcal C \) be a morphism in \(\mathcal C \). Then for \(n \ge 2 \) the lifting problem
\begin{equation*} \xymatrix{ \Lambda^{\{0,1\}} \ar@{->}[r] & \Lambda^n_0 \ar@{->}[r] \ar@{->}[d] & \mathcal C \ar@{->}[d] \\ & \Delta^n \ar@{->}[r] \ar@{-->}[ru] & \mathcal D } \end{equation*}where the top composite is \(\phi \) can be solved, if \(\phi \) is an equivalence in \(\mathcal C \).