equivalences in/of $(\infty,1)$-categories
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
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.
(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
be a natural transformation between two $\mathcal{I}^\rhd$-shaped diagrams ((∞,1)-functors), with
denoting its restriction away from the cocone tip.
If
and
then the following are equivalent:
$X^\rhd$ is an (∞,1)-colimit diagram,
$f^\rhd$ is a cartesian natural transformation.
(Rezk 10, 6.5, Lurie, Theorem 6.1.3.9 (4))
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
be groupoid objects and write
for the corresponding effective epimorphisms into their (∞,1)-colimits.
Then Prop. implies that the following are equivalent:
a morphism of groupoid objects $X_\bullet \overset{f_\bullet}{\Rightarrow} Y_\bullet$ is a cartesian natural transformation;
the corresponding transformation of effective epimorphisms
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.
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 June 30, 2020 at 15:54:37. See the history of this page for a list of all contributions to it.