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colimits of equifibered transformations -- section

Colimits of equifibered transformations

Colimits of equifibered transformations

Proposition

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

Let H\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 f Y : H 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 ((∞,1)-functors), with

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

denoting its restriction away from the cocone tip.

If

and

then the following are equivalent:

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

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

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

Example

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

Let

X ,Y :Δ opH X_\bullet, Y_\bullet \colon \Delta^{op} \longrightarrow \mathbf{H}

be groupoid objects and write

X 0 𝒳AAAAA,AAAAAY 0 𝒴 \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 f Y X_\bullet \overset{f_\bullet}{\Rightarrow} Y_\bullet is a cartesian natural transformation;

  2. the corresponding transformation of effective epimorphisms

    X 0 f 0 Y 0 𝒳 limf 𝒴 \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.

Created on June 30, 2020 at 15:52:44. See the history of this page for a list of all contributions to it.