nLab
equifibered natural transformation

Equifibered natural transformation

Context

Category theory

(,1)(\infty,1)-Category theory

Equifibered natural transformation

Definition

Let F,G:CDF,G:C\to D be functors. A natural transformation α:FG\alpha:F\to G is equifibered (also called cartesian) if for any morphism f:xyf:x\to y in CC, the naturality square

Fx Ff Fy α x α y Gx Gf Gy\array{ F x & \overset{F f}{\to} & F y\\ ^{\alpha_x}\downarrow & & \downarrow^{\alpha_y} \\ G x & \underset{G f}{\to} & G y}

is a pullback.

The name “equifibered” comes from the fact that since α x\alpha_x is a pullback of α y\alpha_y, they must have isomorphic fibers. (Of course, if CC is not connected, then being equifibered does not imply that all components of α\alpha have isomorphic fibers.)

There is an evident generalization to natural transformations between higher categories.

Properties

  • Given a functor G:CDG:C\to D, if CC has a terminal object 11, then to give a functor FF and an equifibered transformation FGF\to G is equivalent to giving a single object F1F1 and a morphism F1G1F1 \to G1. The rest of FF can then be constructed uniquely by taking pullbacks. This construction is important in the theory of clubs.

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.

References

In the context of category theory the concept is discussed in

In the context of (infinity,1)-categories (with an eye towards (infinity,1)-toposes) the concept is considered in

  • Charles Rezk, p. 9 of Toposes and homotopy toposes (2010) (pdf)

  • Charles Rezk, p. 2 of When are homotopy colimits compatible with homotopy base change? (2014) (pdf)

Last revised on June 30, 2020 at 15:54:37. See the history of this page for a list of all contributions to it.