nLab cartesian natural transformation

Definition

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

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 cartesian natural 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.

Colimits of equifibered transformations

Proposition

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

Let $\mathbf{H}$ 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

$$X^\rhd \overset{f^\rhd}{\Rightarrow} Y^\rhd \;\colon\; \mathcal{I}^\rhd \longrightarrow \mathbf{H}$$

be a natural transformation between two $\mathcal{I}^\rhd$-shaped diagrams (∞-functors), with

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

denoting its restriction away from the cocone tip.

If

• $Y^\rhd$ is an (∞,1)-colimit diagram,

and

• $f$ is a cartesian natural transformation,

then the following are equivalent:

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

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

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

Example

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

Let

$$X_\bullet, Y_\bullet \colon \Delta^{op} \longrightarrow \mathbf{H}$$

be groupoid objects and write

$$\array{ X_0 \\ \big\downarrow \\ \mathcal{X} } \phantom{AAAAA} , \phantom{AAAAA} \array{ Y_0 \\ \big\downarrow \\ \mathcal{Y} }$$

for the corresponding effective epimorphisms into their (∞,1)-colimits.

Then Prop. implies that the following are equivalent:

1. a morphism of groupoid objects $X_\bullet \overset{f_\bullet}{\Rightarrow} Y_\bullet$ is a cartesian natural transformation;

2. the corresponding transformation of effective epimorphisms

   $$\array{ X_0 &\overset{f_0}{\longrightarrow}& Y_0 \\ \big\downarrow &\swArrow& \big\downarrow \\ \mathcal{X} &\underset{ \underset{\longrightarrow}{\lim}f }{\longrightarrow}& \mathcal{Y} }$$

   is an (∞,1)-pullback square.

Related pages

• cartesian monad
• club
• polynomial functor
• van Kampen colimit

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: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 $\alpha_x$ 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.)