nLab cartesian natural transformation

Cartesian natural transformation


Category theory

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

Cartesian natural transformation


Let F,G:CDF,G:C\to D be functors. A natural transformation α:FG\alpha:F\to G is 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.

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


  • Given a functor G:CDG:C\to D, if CC has a terminal object 11, then to give a functor FF and an cartesian natural 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


(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 (∞-functors), with

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

denoting its restriction away from the cocone tip.



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 (4))


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.


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.


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:10.1017/S0960129500001183 web)

  • Tom Leinster, Higher Operads, Higher Categories, Cambridge University Press 2003. (arXiv:math/0305049)

  • Mark Weber. Generic morphisms, parametric representations and weakly Cartesian monads. Theory Appl. Categ 13.14 (2004): 191-234.

  • Juergen Koslowski, A monadic approach to polycategories, TAC Vol. 14, 2005, No. 7, pp 125-156.

  • Nicola Gambino and Joachim Kock. Polynomial functors and polynomial monads. Mathematical proceedings of the cambridge philosophical society. Vol. 154. No. 1. Cambridge University Press, 2013.

In the context of (infinity,1)-categories (with an eye towards (infinity,1)-toposes) the concept is called equifibered natural transformations and 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)

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.)

Last revised on December 16, 2022 at 10:46:46. See the history of this page for a list of all contributions to it.