7.7.5 The Mather Cube Theorem Revisted
We now study the (strong) universality of diagrams indexed by the $\infty $-category $\Delta ^1 \times \Delta ^1$.
Definition 7.7.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ satisfies the first Mather cube theorem if, for every cubical diagram
7.88
\begin{equation} \label{equation:mather-cube-general} \begin{gathered} \xymatrix@R =50pt@C=50pt{\overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd] \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \end{gathered}\end{equation}
having the property that the back and left faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & \overline{A} \ar [r] \ar [d] & \overline{C} \ar [d] \\ A \ar [r] & B & A \ar [r] & C } \]
are pullback squares and the top and bottom faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & A \ar [r] \ar [d] & B \ar [d] \\ \overline{C} \ar [r] & \overline{D} & C \ar [r] & D } \]
are pushout squares, then the front and right faces
\[ \xymatrix@R =50pt@C=50pt{ \overline{C} \ar [r] \ar [d] & \overline{D} \ar [d] & \overline{B} \ar [r] \ar [d] & \overline{D} \ar [d] \\ C \ar [r] & D & B \ar [r] & D } \]
are also pullback squares.
We say that $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem if, for every cubical diagram (7.88) where the the front, back, left, and right faces are pullback squares and the bottom face is a pushout square, the top face is also a pushout square.
Example 7.7.5.3. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces. Then $\operatorname{\mathcal{S}}$ satisfies both the first and second Mather cube theorem. To prove this, we can assume without loss of generality that (7.88) arises from a commutative diagram in the ordinary category of Kan complexes (Corollary 5.6.5.18). In this case, a face of the diagram (7.88) is a pullback square in $\operatorname{\mathcal{S}}$ if and only if it is a homotopy pullback square (Example 7.6.3.2), and a pushout square in $\operatorname{\mathcal{S}}$ if and only if it is a homotopy pushout square (Example 7.6.3.3). The first Mather cube theorem is now a restatement of Theorem 3.4.4.4, while the second cube theorem is a restatement of Theorem 3.4.3.3.
Example 7.7.5.4. Let $\operatorname{\mathcal{C}}$ be (the nerve of) the category of sets. Then $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem (Exercise 3.4.3.1), but does not satisfy the first Mather cube theorem (Warning 3.4.4.2).
Example 7.7.5.5. The $\infty $-category $\operatorname{\mathcal{QC}}$ satisfies neither the first nor the second Mather cube theorem.
Proposition 7.7.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and finite colimits. Then finite colimits in $\operatorname{\mathcal{C}}$ are universal if and only if $\operatorname{\mathcal{C}}$ satisfies the following pair of conditions:
- $(a)$
Pushouts in $\operatorname{\mathcal{C}}$ are universal: that is, $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem (Remark 7.7.5.2).
- $(b)$
The initial object $\emptyset \in \operatorname{\mathcal{C}}$ is universal: that is, every morphism $C \rightarrow \emptyset $ is an isomorphism (Example 7.7.1.17).
Proof.
According to Corollary 7.7.2.34, finite colimits in $\operatorname{\mathcal{C}}$ are universal if and only if, for every morphism $f: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$, the pullback functor
\[ f^{\ast }: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}_{/C'} \quad \quad X \mapsto C' \times _{C} X \]
preserves finite colimits. By virtue of Corollary 7.6.2.30, this is equivalent to the requirement that $f^{\ast }$ preserves both pushouts and initial objects. The desired result follows by allowing $f$ to vary (and using Corollary 7.7.2.34 again).
$\square$
Variant 7.7.5.7. Let $\kappa $ be an infinite cardinal, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $\kappa $-small colimits. Then $\kappa $-small colimits in $\operatorname{\mathcal{C}}$ are universal if and only if $\kappa $-small coproducts in $\operatorname{\mathcal{C}}$ are universal and $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem. This follows from the proof of Proposition 7.7.5.6, using Exercise 7.6.6.11 in place of Corollary 7.6.2.30.
Proposition 7.7.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and finite colimits. Then finite colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the following conditions are satisfied:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ satisfies the first and second Mather cube theorems.
- $(b)$
The initial object $\emptyset \in \operatorname{\mathcal{C}}$ is universal: that is, every morphism $C \rightarrow \emptyset $ is an isomorphism.
Proof.
Choose an uncountable cardinal $\lambda $ for which $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, and let
\[ \operatorname{Tr}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{< \lambda } \quad \quad C \mapsto \operatorname{\mathcal{C}}_{/C} \]
be a contravariant transport representation for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. Then finite colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the functor $\operatorname{Tr}$ preserves finite limits (Remark 7.7.2.18). By virtue of Corollary 7.6.2.30, this is equivalent to the requirement that $\operatorname{Tr}$ preserves pullback squares and final objects. The case of pullback squares is a reformulation of condition $(a)$ (Remark 7.7.5.2), and the case of final objects is a reformulation of condition $(b)$ (Example 7.7.2.16).
$\square$
Corollary 7.7.5.9. Let $\kappa $ be an infinite cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $\kappa $-small colimits. Then $\kappa $-small colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the following conditions are satisfied:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ satisfies both Mather cube theorems.
- $(b)$
In the $\infty $-category $\operatorname{\mathcal{C}}$, $\kappa $-small coproducts are universal.
Proof.
By virtue of Proposition 7.7.5.8, we may assume without loss of generality that finite colimits in $\operatorname{\mathcal{C}}$ are universal, so that $\operatorname{\mathcal{C}}$ satisfies both Mather cube theorems. Choose an uncountable cardinal $\lambda $ for which $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, and let
\[ \operatorname{Tr}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{< \lambda } \quad \quad C \mapsto \operatorname{\mathcal{C}}_{/C} \]
be a contravariant transport representation for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. Then $\kappa $-small colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the functor $\operatorname{Tr}$ preserves $\kappa $-small limits (Remark 7.7.2.18). By virtue of Exercise 7.6.6.11, this is equivalent to the requirement that $\operatorname{Tr}$ preserves $\kappa $-small products: that is, that $\kappa $-small coproducts in $\operatorname{\mathcal{C}}$ are strongly universal. Using Corollary 7.7.4.11, we see that this is equivalent to condition $(b)$.
$\square$
Corollary 7.7.5.10. Small colimits are strongly universal in the $\infty $-category $\operatorname{\mathcal{S}}$.
Proof.
By virtue of Example 7.7.5.3, the $\infty $-category $\operatorname{\mathcal{S}}$ satisfies both Mather cube theorems. Consequently, to show that small colimits in $\operatorname{\mathcal{S}}$ are strongly universal, it will suffice to show that small coproducts in $\operatorname{\mathcal{S}}$ are universal (Corollary 7.7.5.9). This follows from Corollary 7.7.4.12.
$\square$