# nLab equifibered natural transformation

Equifibered natural transformation

category theory

## Applications

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

(∞,1)-category theory

# Equifibered natural transformation

## Definition

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

$\array{ F x & \overset{F f}{\to} & F y\\ ^{\alpha_x}\downarrow & & \downarrow^{\alpha_y} \\ G x & \underset{G f}{\to} & G y}$

is a pullback.

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

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

## Properties

Given a functor $G:C\to D$, if $C$ has a terminal object $1$, then to give a functor $F$ and an equifibered transformation $F\to G$ is equivalent to giving a single object $F1$ and a morphism $F1 \to G1$. The rest of $F$ can then be constructed uniquely by taking pullbacks. This construction is important in the theory of clubs.

In the context of category theory the concept is discussed in

• Aurelio Carboni and Peter Johnstone, Connected limits, familial representability and Artin glueing, Mathematical Structures in Computer Science, Vol. 5 Iss. 4, Cambridge U. Press (December 1995), 441-459.

doi: https://doi.org/10.1017/S0960129500001183. (web)

• Tom Leinster, Higher Operads, Higher Categories, Cambridge University Press 2003. (arXiv link)

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)