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.

References

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)

Last revised on February 5, 2017 at 15:30:20. See the history of this page for a list of all contributions to it.