nLab pullback-stable colimit

Definition

Let $C$ be a category with pullbacks and colimits of some shape $D$.

Definition

We say that colimits of shape $D$ are stable by base change, or stable under pullback, or that these colimits are universal, if for every functor $G \colon D \to C$ and for all pullback diagrams of the form

$$\array{ \big(\underset{\longrightarrow}{\lim}_D G\big) \times_Z Y &\longrightarrow& \underset{\longrightarrow}{\lim}_D G \\ \big\downarrow && \big\downarrow \\ Y &\underset{f}{\longrightarrow} & Z }$$

the canonical morphism

(1)$$\underset{\underset{d \in D}{\longrightarrow}}{\lim} \big( G(d) \times_Z Y \big) \overset{\sim}{\longrightarrow} \big( \underset{\underset{d \in D}{\longrightarrow}}{\lim} G(d) \big) \times_Z Y$$

is an isomorphism.

This says equivalently that every pullback functor $f^* \colon C/Z \longrightarrow C/Y$ preserves $D$-colimits.

Similar definitions apply for higher categories.

Proposition

The condition in Def. is equivalent to the following:

For every functor $G \colon D \to C$ and every morphism $f : Y \to \underset{\longrightarrow}{\lim}_D G$, if we define $Y(d)$ as the pullback of $f$ along the coprojection $\iota_d : G(d) \to \underset{\longrightarrow}{\lim}_D G$, then the induced morphism $\underset{\underset{d \in D}{\longrightarrow}}{\lim} Y(d) \to Y$ is an isomorphism.

Proof

The implication $\implies$ follows by applying Def. to $Z = \underset{\longrightarrow}{\lim}_D G$ and the identity morphism, The converse implication $\impliedby$ follows by applying the assumption to the projection $(\underset{\longrightarrow}{\lim}_D G) \times_Z Y \to \underset{\longrightarrow}{\lim}_D G$ and the cancelling rule for pullbacks.

Proposition

The condition in Def. is equivalent to the universality condition given at van Kampen colimit.

Proof

Observe that, given a natural transformation $\alpha' : F' \Rightarrow G'$, the diagram

$$\array{ F'(*) &\overset{\alpha'_*}{\longrightarrow}& \underset{\longrightarrow}{\lim}_D G \\ \mathllap{^\id}\big\downarrow && \big\downarrow\mathrlap{^\id} \\ F'(*) &\underset{\alpha'_*}{\longrightarrow} & \underset{\longrightarrow}{\lim}_D G }$$

is a degenerate pullback square, hence there is a canonical isomorphism

$$\underset{\underset{d \in D}{\longrightarrow}}{\lim} \big( G(d) \times_{\underset{\longrightarrow}{\lim}_D G} F'(*) \big) \simeq F'(*) \mathrlap{\,.}$$

But if $\alpha'$ is equifibered, we have $G(d) \times_{\underset{\longrightarrow}{\lim}_D G} F'(*) \simeq F(d)$, hence we get the desired isomorphism $F'(*) \simeq \underset{\longrightarrow}{\lim}_D F$.

Conversely, given a pullback diagram as above, let $F' = f^* \circ G'$ (viewing $G'$ as a functor $D \to C/Z$ and remembering that colimits in $C/Z$ are computed as colimits in $C$) and $\alpha' \colon F' \Rightarrow G'$ the natural transformation induced by the pullback projections, which is equifibered as a consequence of the pasting law for pullbacks. Then $f^* \circ G'$ is a colimiting cocone, which is to say that $f^*$ preserves $colim_D G$.

Examples

Toposes

Example

The stability of all colimits is one of Giraud's axioms that characterize Grothendieck toposes in the 1-categorical context and Grothendieck-Rezk-Lurie (∞,1)-toposes in the higher categorical context.

The fact that colimits are stable in toposes can be seen from the characterization of toposes as left-exact reflective subcategories of presheaf categories as follows:

• First observe that colimits are stable in $C =$ Set.

• Now colimits are stable for $C =$ a presheaf category $[S^{op},Set]$, since colimits in such $C$ are computed objectwise in $Set$. (See limits and colimits by example.)

• Finally, stability of colimits is preserved in left exact reflective subcategories, since the reflector preserves both colimits and pullbacks.

For (∞,1)-toposes, this is HTT, theorem 6.1.0.6 (3) ii).

.

Non-toposes

Generalizing Ex. :

Example

Colimits are stable in any locally cartesian closed category: In that case the pullback functors $f^*$ all have right adjoints.

Conversely, if $C$ is cocomplete with all stable colimits, and the adjoint functor theorem applies to all its slice categories, then it is locally cartesian closed.

Example

Colimits are also stable in any exact infinitary extensive category, since all colimits can be constructed out of coproducts, images, and quotients by equivalence relations, which are all pullback-stable in an exact and infinitary-extensive category.

Example

A counterexample:

• Colimits are not stable $C =$ Ab.

Properties

Relation to commutativity and distributivity

Although stability of colimits appears as a sort of “commutativity” between colimits and pullbacks, it is not literally an instance of commutativity of limits and colimits. It is an example of the latter if the colimit over $D$ of the diagram constant on a single object (such as $Y$) is that single object. For ordinary colimits in category theory this is a mild condition, requiring $D$ to be a connected category; but in higher category theory this becomes an ever stronger condition; for colimits in an (infinity,1)-category it means that the infinity-groupoid generated by $D$ is contractible homotopy type (see this corollary).

It is generally true that (1) is an example of distributivity of limits over colimits; see there.

Related concepts

• van Kampen colimit
• pullback-stability

References

• Jacob Lurie, Higher Topos Theory (Section 6.1.1)