nLab colimits of equifibered transformations -- section

Colimits of equifibered transformations

Proposition

(equifibered natural transformations of (∞,1)-colimits in an (∞,1)-topos)

Let $\mathbf{H}$ be an (∞,1)-topos. For $\mathcal{I}$ a small (∞,1)-category, write $\mathcal{I}^\rhd$ for the result of adjoining a terminal object (the shape of cocones under $\mathcal{I}$-shaped diagrams), and let

$$X^\rhd \overset{f^\rhd}{\Rightarrow} Y^\rhd \;\colon\; \mathcal{I}^\rhd \longrightarrow \mathbf{H}$$

be a natural transformation between two $\mathcal{I}^\rhd$-shaped diagrams (∞-functors), with

$$X \overset{f}{\Rightarrow} Y \;\colon\; \mathcal{I} \longrightarrow \mathbf{H}$$

denoting its restriction away from the cocone tip.

If

• $Y^\rhd$ is an (∞,1)-colimit diagram,

and

• $f$ is a cartesian natural transformation,

then the following are equivalent:

1. $X^\rhd$ is an (∞,1)-colimit diagram,

2. $f^\rhd$ is a cartesian natural transformation.

(Rezk 10, 6.5, Lurie, Theorem 6.1.3.9 (4))

Example

Let $\mathcal{I} = \Delta^{op}$ be the opposite of the simplex category, so that $\mathcal{I}^{\rhd} = \Delta_+^{op}$ is the opposite of the augmented simplex category.

Let

$$X_\bullet, Y_\bullet \colon \Delta^{op} \longrightarrow \mathbf{H}$$

be groupoid objects and write

$$\array{ X_0 \\ \big\downarrow \\ \mathcal{X} } \phantom{AAAAA} , \phantom{AAAAA} \array{ Y_0 \\ \big\downarrow \\ \mathcal{Y} }$$

for the corresponding effective epimorphisms into their (∞,1)-colimits.

Then Prop. implies that the following are equivalent:

1. a morphism of groupoid objects $X_\bullet \overset{f_\bullet}{\Rightarrow} Y_\bullet$ is a cartesian natural transformation;

2. the corresponding transformation of effective epimorphisms

   $$\array{ X_0 &\overset{f_0}{\longrightarrow}& Y_0 \\ \big\downarrow &\swArrow& \big\downarrow \\ \mathcal{X} &\underset{ \underset{\longrightarrow}{\lim}f }{\longrightarrow}& \mathcal{Y} }$$

   is an (∞,1)-pullback square.