nLab
cartesian morphism

Idea

Given a functor $p:X\to Y$ between categories one may ask for each morphism $f:y_1\to y_2$ if given a lift of its target

$$\begin{array}{cccccc}X& & & & & {\hat{y}}_2\\ {\downarrow }^p\\ Y& & & y_1& \stackrel{f}{\to }& y_2\end{array}$$\array{ X &&& && \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 }

there is a universal lift of $f$

$$\begin{array}{cccccc}X& & & {\hat{y}}_1& \stackrel{\hat{f}}{\to }& {\hat{y}}_2\\ {\downarrow }^p\\ Y& & & y_1& \stackrel{f}{\to }& y_2\end{array}.$$\array{ X &&& \hat y_1 &\stackrel{\hat f}{\to}& \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 } \,.

There may also be other lifts of $f$, but the universal one is essentially unique, as usual for anything having a universal property. Specifically, $\hat{f}$ in $X$ is essentially uniquely determined by its source ${\hat{y}}_1$ and its image $f=p(\hat{f})$ in $Y$, and is called a cartesian morphism. A morphism which is cartesian relative to $p^{\mathrm{op}}:X^{\mathrm{op}}\to Y^{\mathrm{op}}$ is called opcartesian or cocartesian.

If there are enough cartesian morphisms in $Y$, they may be used to define functors

$$f^{*}:X_{y_2}\to X_{y_1}$$f^* : X_{y_2} \to X_{y_1}

between the fibers of $p$ over $y_1$ and $y_2$.

David Roberts: There would surely be an anafunctor version of this, that would require no choices whatsoever. It is unlikely that I would be able to find time to write this up, so my plea goes out to those in the know…

I imagine that there would then be an $(\mathrm{\infty }\mathrm{,1})$-version using whatever passes as anafunctors in that setting (dratted memory, failing at the first gate)

Mike Shulman: Yes, there would surely be such a version. (-: The simplest way would be to take the specifications $\mid f^{*}\mid$ for the anafunctor $f^{*}$ to be the cartesian morphisms over $f$, with domain and codomain giving the functions $\sigma$ and $\tau$. Unique factorization would give you the values of morphisms.

This way a functor $p:X\to Y$ with enough Cartesian morphisms – called a Cartesian fibration or Grothendieck fibration – determines and is determined by a fiber-assigning functor $Y\to {\mathrm{Cat}}^{\mathrm{op}}$.

This has its analog in higher categories.

Definition

In categories

Let $p:X\to Y$ be a functor. A morphism $f:x_1\to x_2$ in the category $X$ is cartesian with respect to $p$, or $p$-cartesian. If it satisfies the following property:

$$\begin{array}{c}x\prime \\ {\downarrow }^{\exists !}& {\searrow }^{\forall h}\\ x_1& \stackrel{f}{\to }& x_2\end{array}\stackrel{p}{\mapsto }\begin{array}{c}p(x\prime )\\ {\downarrow }^{\forall g}& {\searrow }^{p(h)}\\ p(x_1)& \stackrel{p(f)}{\to }& p(x_2)\end{array}$$\array{ x' \\ \downarrow^{\mathrlap{\exists!}} & \searrow^{\mathrlap{\forall h}} \\ x_1 &\stackrel{f}{\to}& x_2 } \;\;\; \;\;\; \stackrel{p}{\mapsto} \;\;\; \;\;\; \array{ p(x') \\ \downarrow^{\mathrlap{\forall g}} & \searrow^{\mathrlap{p(h)}} \\ p(x_1) &\stackrel{p(f)}{\to}& p(x_2) }

If for every morphism in $Y$ there is a lift through $p$ that is a cartesian morphism, one says that $p$ is a Grothendieck fibration.

This may equivalently be expressed as follows:

let

• $X/x_2$ by the overcategory of $X$ over the object $x_2$;

• $Y/p(x_2)$ the corresponding overcategory of $Y$ over $p(x_2)$;

• $X/f$ the category whose objects

  $$\mathrm{Obj}(X/f)=\left\{\begin{array}{ccc}& & a\\ & \swarrow & & \searrow \\ x_1& & \stackrel{f}{\to }& & x_2\end{array}\right\}$$Obj(X/f) = \left\{ \array{ && a \\ &\swarrow && \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right\}

  are objects $a$ of $X$ eqipped with morphisms to $x_1$ and $x_2$ such that the obvious triangle commutes, and whose morphisms are morphisms between these tip objects such that all diagrams in sight commute.

• similarly $Y/p(f)$.

Then the condition that $f$ is cartesian with respect to $p$ is equivalently the condition that the functor

$$X_f\to X_{x_2}{\times }_{Y_{p(x_2)}}Y/p(f)$$X_f \to X_{x_2} \times_{Y_{p(x_2)}} Y/p(f)

into the pullback of the obvious projection $X_{x_2}\to S/p(x_2)$ along the projection $S/p(f)\to S/p(x_2)$ is a surjective equivalence.

This definition in terms of pullbacks is the one that straightforwardly generalizes to higher category theory.

In $(\mathrm{\infty }\mathrm{,1})$-categories

There is a notion of cartesian edge in a simplicial set $X$ relative to a morphism $p:X\to Y$ of simplicial sets. In the case that these simplicial sets are quasi-categories – i.e. simplicial set incarnations of (∞,1)-categories – this yields a notion of cartesian morphisms in $(\mathrm{\infty }\mathrm{,1})$-categories.

Let $p:X\to Y$ be a morphism of simplicial sets that may be the quasi-category incarnation of an infinity-functor of (∞,1)-categories. Let $f:x_1\to x_2$ be an edge in $X$, i.e. a morphism $f:{\Delta }^1\to X$.

Recall the notion of over quasi-category obtained from the notion of join of quasi-categories. Using this we obtain simplicial sets $X/f$, $X/x_2$, $S/p(f)$ and $S/p(x_2)$ in generalization of the categories considered in the above definition of cartesian morphisms in categories.

Assume that $p$ is an inner Kan fibration of simplicial sets.

Then $f$ in $X$ is $p$-cartesian if the induced morphism

$$X_{/f}\to X_{/y}{\times }_{Y_{/p(y)}}{Y}_{/p(f)}$$X_{/f} \to X_{/y} \times_{Y_{/p(y)}} Y_{/p(f)}

is acyclic Kan fibration.

This is def 2.4.1.1 in HTT.

This is equivalent to the condition that for all horn inclusions

$$\begin{array}{c}{\Delta }^{\{n-1,n\}}\\ \downarrow & {\searrow }^f\\ \Lambda [n{]}_n& \to & X\\ \downarrow & & \downarrow \\ \Delta [n]& \to & Y\end{array}$$\array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow && \downarrow \\ \Delta[n] &\to& Y }

(with $\Lambda [n{]}_n$ the $n$th horn of the $n$-simplex) such that the last edge of the horn is the given egde $f$, a lift $\sigma$

$$\begin{array}{c}{\Delta }^{\{n-1,n\}}\\ \downarrow & {\searrow }^f\\ \Lambda [n{]}_n& \to & X\\ \downarrow & {}^{\sigma }\nearrow & \downarrow \\ \Delta [n]& \to & Y\end{array}$$\array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow &{}^\sigma\nearrow& \downarrow \\ \Delta[n] &\to& Y }

exists. (See HTT remark 2.4.1.4.)

References

For the 1-categorical case see the references at Grothendieck fibration.

The $(\mathrm{\infty }\mathrm{,1})$-categorical version is in section 2.4 of

• Jacob Lurie, Higher Topos Theory