nLab Cartesian morphism

Idea

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

$$\array{ X &&& && \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 }$$

there be a universal lift of $f$

$$\array{ X &&& \hat y_1 &\stackrel{\hat f}{\to}& \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 } \,.$$

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 target $\hat y_2$ and its image $f = p(\hat f)$ in $Y$, and is called a cartesian morphism. A morphism which is cartesian relative to $p^{op}\colon X^{op}\to Y^{op}$ is called opcartesian or cocartesian.

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

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

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

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 $Y \to Cat^{op}$.

This has its analog in higher categories.

Definition

In categories

Traditional definition

Definition

(cartesian morphism)

Let $p : X \to Y$ be a functor. A morphism $f : x_1 \to x_2$ in the category $X$ is strongly cartesian with respect to $p$ (nowadays often just cartesian), or (strongly) $p$-cartesian if for every $x'\in X$, for every $h:x'\to x_2$ and every $u:p(x')\to p(x_1)$ such that $p(h) = p(f) u$, there exists a unique $v:x'\to x_1$ such that $h = f v$ and $u = p(v)$:

$$\array{ \forall x' \\ \downarrow^{\mathrlap{\exists! v}} & \searrow^{\mathrlap{\forall h}} \\ x_1 &\stackrel{f}{\to}& x_2 } \;\;\; \;\;\; \stackrel{p}{\mapsto} \;\;\; \;\;\; \array{ p(x') \\ \downarrow^{\mathrlap{\forall u}} & \searrow^{\mathrlap{p(h)}} \\ p(x_1) &\stackrel{p(f)}{\to}& p(x_2) }$$

In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-horn to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.

We can make this definition slightly more explicit by working with the fibres of $p$: let $X_x$ denote $p^{-1}(y)$, the set of objects living over $y$; and, for $f : y \to y'$, $x : X_y$ and $x' : X_{y'}$, let $X_f(x,x')$ denote $p^{-1}(f) \cap X(x,x')$, the set of morphisms from $x$ to $x'$ that $p$ maps to $f$.

Then for $x : X_y$, $x' : X_{y'}$ and $f : y' \to y$, a morphism $\bar f : X_f(x',x)$ is cartesian iff for any $u : y'' \to y'$, $x'' : X_{y''}$ and $h : X_{f \circ u}(x'', x)$, there is a unique $\bar u : X_u(x'',x')$ with $h = \bar f \circ \bar u$. This can be expressed in the following diagram, where the upper objects and morphisms live in $X$ over their corresponding data in $Y$:

If we pass to the nerve $N(X)$ and $N(Y)$ of the categories, then in terms of diagrams in sSet this means that the morphism $f : x \to y$ is $p$-cartesian precisely if for all horn inclusions

$$\array{ \Delta^{\{1,2\}} \\ \downarrow & \searrow^f \\ \Lambda_2[2] &\to& N(X) \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \Delta[2] &\to& N(Y) }$$

such that the last edge of the 2-horn is the given edge $f$, a unique lift $\sigma$

$$\array{ \Delta^{\{1,2\}} \\ \downarrow & \searrow^f \\ \Lambda_2[2] &\to& N(X) \\ \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}} \\ \Delta[2] &\to& N(Y) }$$

exists.

Remark

(weak/local Cartesian morphisms)
There is a weaker universal property, originally devised by Grothendieck and Gabriel, where one requires the above lifting property only for $u = id_{p(x_1)}$. Morphisms satisfying this universal property have in recent years been called locally Cartesian morphisms, although historically they have been called simply Cartesian, or sometimes weak Cartesian. For the case of Grothendieck fibered categories the notion of weak Cartesian morphisms already coincides with that of actual Cartesian morphisms.

Equivalently, a morphism $f:x\to x^\prime$ is called locally $p$-Cartesian (relative to a functor $X\to Y$) if it is Cartesian with respect to the projection functor $p_{pf}:X_{pf}\to \mathbf{2}$ where $X_{pf}$ is the (homotopy) pullback of $X$ along the functor $c_{pf}:\mathbf{2}\to Y$ classifying the arrow $pf:px\to px^\prime$ in $Y$.

Definition

(Grothendieck fibration)

If for every morphism in $Y$ and every lift of its target there is at least one lift which has as its target the chosen one and is a $p$-cartesian morphism in the strong sense, one says that $p$ is a fibered category (also called Grothendieck fibration). Equivalently,

• ($p$ is prefibered category) for every morphism in $Y$ and every lift of its target there is at least one lift through $p$ which has as its target the chosen one and is a $p$-cartesian morphism in the weak sense

and

• the composition of every two composable $p$-cartesian morphisms (in the weak sense) is a $p$-cartesian morphism (in the weak sense).

Reformulations

We discuss equivalent reformulations of the above definition of Cartesian morphism that lend themselves better to generalization to higher category theory.

For the following, we need this notation: 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

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

  are objects $a$ of $X$ equipped 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 for $Y/p(f)$.

Proposition

The condition that $f \in Mor X$ is a Cartesian morphism with respect to $p : X \to Y$ is equivalent to the condition that the functor

$$\phi : X/f \to X/{x_2} \times_{Y/{p(x_2)}} Y/p(f)$$

into the (strict) pullback of the obvious projection $X/{x_2} \to Y/p(x_2)$ along the projection $Y/p(f) \to Y/p(x_2)$ induced by the commutativity of

$$\array{ X/f &\stackrel{\phi_2}{\to}& Y/p(f) \\ {}^{\mathllap{\phi_1}}\downarrow && \downarrow \\ X/{x_2} &\to& Y/{p(x_2)} }$$

is a surjective equivalence, and this in turn is equivalent to it being an isomorphism of categories.

Proof

It is immediate to see that $\phi$ being an isomorphism of categories is equivalent to the condition that $f$ is a Cartesian morphism. We discuss that just the condition that $\phi$ is a surjective equivalence already implies that it is an isomorphism of categories.

So assume now that $\phi$ is a surjective equivalence.

Notice that objects in the pullback category are compatible pairs

$$\left( \left( \array{ && a \\ &&& \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right) \in X/_{x_2} \;\;\;\;\;\;\,,\;\;\;\;\;\; \left( \array{ && b \\ & \swarrow && \searrow \\ p(x_1) &&\stackrel{p(f)}{\to}&& p(x_2) } \right) \in Y/p(f) \right) \,.$$

We have that $\phi$ being surjective on object means that every such pair is in the image of some object

$$\left( \array{ && a \\ &{}^{\mathllap{g}}\swarrow&& \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right) \in X_{f} \,,$$

and hence that every filler exists . Assume two such fillers $g$ and $g'$. Then by the fact that an equivalence of categories is a surjection (even an isomorphism) on corresponding hom-sets, it follows that there exists (even uniquely) a morphism in $X/f$ connecting them

$$\array{ && a \\ &{}^{\mathllap{g}}\swarrow & \downarrow^{\mathrlap{h}} & \searrow \\ && a \\ \downarrow & {}^{\mathllap{g'}}\swarrow && \searrow & \downarrow \\ x_1 &&\stackrel{f}{\to}&& x_2 }$$

such that this maps under $\phi$ to the identity morphism in the pullback category. But in particular this maps to the morphism

$$\array{ && a \\ && \downarrow^{\mathrlap{h}} & \searrow \\ && a \\ & && \searrow & \downarrow \\ &&&& x_2 }$$

in $X/{x_2}$ and evidently is the identity there if and only if $h$ is the identity. Hence this maps also to the identity in the pullback category if and only if $h$ is the identity. So $h$ must be the identity. So if two lifts of an object through the surjective equivalence $\phi$ exist, they must already be equal. Hence the surjective equivalence $\phi$ is even an isomorphism on objects and hence an isomorphism of categories.

In $(\infty,1)$-categories

The notion of cartesian morphism generalizes from category theory to (∞,1)-category theory. The definition given above can be rephrased as a pullback relation between homsets, which we can take as an abstract definition

Definition

(cartesian edge in an $(\infty,1)$-category)

Let $p : X \to Y$ be a functor of (∞,1)-categories. Then, a morphism $f : x \to x'$ in $X$ is $p$-cartesian if and only if the commutative square induced by the distributivity of $p$ over composition

is a pullback square for every $a \in X$.

Similarly, the reformulation in terms of slice/cone categories generalizes directly, and is indeed equivalent to the definition given above:

Proposition

Let $p : X \to Y$ be a functor of (∞,1)-categories. Then, a morphism $f : x \to x'$ in $X$ is $p$-cartesian iff the commutative square

is a pullback square.

Proof

The definition of being $p$-cartesian can be described as a pullback square of hom-functors. By the contravariant (∞,1)-Grothendieck construction, $X(-, x)$ classifies the right fibration, $X_{/x} \to X$, and $Y(p-, p(x))$ classifies the pullback of $Y_{/p(x)} \to Y$ along $p$; that is by the comma category $(p \downarrow p(x)) \to Y$. There is a commutative diagram

The bottom-right square and wide rectangle are pullbacks by construction. By the pasting law, the bottom-left square is a pullback, and thus the top-left square is a pullback if and only if the tall rectangle is.

By the (∞,1)-Grothendieck construction, the top-left square is a pullback iff $f$ is $p$-cartesian, so the proposition follows.

Remark

If $f : x \to y$, then $X_{/f} \to X_{/x}$ is an equivalence of (∞,1)-categories, and composing its inverse with $X_{/f} \to X_{/x}$ is the dependent sum $f_! : X_{/x} \to X_{/y}$.

To make this concrete, we can also discuss adaptations of the abstract idea to two different models of (∞,1)-category theory: quasi-categories and sSet categories.

In quasi-categories

We formulate a notion cartesian edge or cartesian morphism 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 $(\infty,1)$-categories.

Let $p : X \to Y$ be a morphism of simplicial sets. 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.

Definition

(cartesian edge in a simplicial set)

Let $p : X \to Y$ be an inner Kan fibration of simplicial sets.

Then a morphism $f : x \to y$ in $X$ is $p$-cartesian if the induced morphism

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

into the pullback in sSet is an acyclic Kan fibration.

This is HTT, def 2.4.1.1.

Proposition

The morphism $f : x \to y$ as above, for $p : X \to Y$ an inner fibration, is $p$-cartesian precisely if for all $n \geq 2$ and all right outer horn inclusions

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

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

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

exists.

This is HTT remark 2.4.1.4.

Remark

This means that an inner fibration $p : X \to Y$ with a collection of $p$-cartesian morphisms in $X$ specified satisfies the same kind of condition as a right fibration , the only difference being that not all right outer horns inclusion are required to have lifts, but only those where the last edge of the horn maps to a cartesian morphism.

In this sense a Cartesian fibration is a generalization of a right fibration.

Proposition

If $p : X \to Y$ is an inner fibration of quasi-categories then a morphism $f : x \to y$ in $X$ is $p$-Cartesian precisely if for all objects $a$ in $X$ the diagram

$$\array{ Hom_X(a,x) &\stackrel{Hom_X(a,f)}{\to}& Hom_X(a,y) \\ \downarrow && \downarrow \\ Hom_Y(p(a), p(x)) &\stackrel{Hom_Y(p(a), p(f))}{\to}& Hom_Y(p(a), p(y)) }$$

of hom-objects in a quasi-category is a homotopy pullback square (in sSet equipped with its standard model structure).

Proof

This is HTT, prop. 2.4.4.3.

In $sSet$-categories

Let $C$ and $D$ be simplicially enriched categories and $F : C \to D$ a sSet-enriched functor.

Definition

A morphism $f : x \to y \in C$ is $F$-cartesian if it is so under the homotopy coherent nerve $N : sSet Cat \to sSet$ in the sense of quasi-categories above, i.e. if

$$N(C)_{/f} \to N(C)_{/y} \times_{N(D)_{/F(y)}} N(D)_{/F(f)}$$

is an acyclic Kan fibration.

Proposition

If $C$ and $D$ are enriched in Kan complexes and if $F$ is hom-wise a Kan fibration, then

• $N(F) : N(C) \to N(D)$ is an inner fibration;

• a morphism $f :x \to y$ in $N(C)$ is an $N(F)$-cartesian morphism precisely if for all objects $a$ in $C$ the diagram

  $$\array{ C(a,x) &\stackrel{C(a,f)}{\to}& C(a,y) \\ \downarrow && \downarrow \\ D(F(a), F(x)) &\stackrel{C(F(a), F(f))}{\to}& D(F(a), F(y)) }$$

  is a homotopy pullback square in sSet equipped with its standard model structure.

Proof

This is HTT, prop. 2.4.1.10.

Properties

Pullback along Cartesian morphisms

Lemma

For $p : \mathcal{E} \to \mathcal{C}$ a functor, if in a diagram

$$\array{ A &\stackrel{f}{\to}& B \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{h}} \\ C &\stackrel{k}{\to}& D }$$

in $\mathcal{E}$ the two vertical morphisms are vertical with respect to $p$ (meaning that $p(g) = Id_p(A)$ and $p(h) = Id(B)$) and if the two horizontal morphisms are $p$-Cartesian morphisms, then this square is a pullback square.

Proof

If

$$\array{ Q &\stackrel{}{\to}& B \\ \downarrow && \downarrow^{\mathrlap{h}} \\ C &\stackrel{k}{\to}& D }$$

is another cone over $C \to D \leftarrow B$, then its image under $p$ is

$$\array{ p(Q) \\ \downarrow & \searrow \\ p(C) &\stackrel{p(k)}{\to}& D } \,.$$

Since $p(f) = p(k)$, another lift of the right horn of this is given by

$$\array{ Q \\ & \searrow \\ A &\stackrel{f}{\to}& B }$$

which gives a unique filler $Q \to A$ by the fact that $f$ is Cartesian.

But this produces now two fillers – namely the original $Q \to C$ and the $Q \to A \to C$ just obtained – of the horn

$$\array{ Q &\to& B \\ && \downarrow \\ C &\stackrel{k}{\to}& D }$$

over

$$\array{ p(Q) \\ \downarrow & \searrow \\ p(C) &\stackrel{p(k)}{\to}& D } \,.$$

Since $k$ is Cartesian, these two fillers must be equal. This means that the morphism $Q \to A$ is a cone morphism and unique as such. Hence the original square is a pullback.

This appears as Elephant, lemma 1.3.3.

Cartesian morphisms and equivalences

Observation

For $C$ a category, a morphism in $C$ is cartesian with respect to the terminal functor $C \to *$ precisely if it is an isomorphism.

In particular all identity morphisms are cartesian.

This is trivial to see. The analog statement holds also for quasi-categories, where it is rather more nontrivial and quite useful:

Proposition

For $C$ a quasi-category, a morphism in $C$ is cartesian with respect to the terminal morphism $C \to *$ precisely if it is an equivalence.

More generally, for $p : X \to Y$ an inner fibration, a morphism $f$ in $X$ is an equivalence precisely if it is $p$-cartesian and $p(f)$ is an equivalence in $Y$.

Proof

The first statement is a proposition of Andre Joyal, slightly reformulated in the language of cartesian morphisms. It appears as HTT, prop 1.2.4.3. A proof appears below HTT, corollary 2.1.2.2.

The second statement is HTT, prop. 2.4.1.5.

Related pages

• A Cartesian morphism is the special case of a strictly final lift of a structured sink when the sink is a singleton.

References

Original reference:

• Alexander Grothendieck, §VI.5 of: Revêtements Étales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1) , LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

Review:

• Angelo Vistoli, Def. 3.1 in: Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry – Grothendieck's FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, math.AG/0412512]

For the 1-categorical case see for instance section B1.3 of

• Peter Johnstone, Sketches of an Elephant

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

• Jacob Lurie, Higher Topos Theory

See also the references at Grothendieck fibration.