nLab Cartesian morphism



Category theory

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



Given a functor p:XYp\colon X \to Y between categories one may ask for each morphism f:y 1y 2f\colon y_1 \to y_2 that, if given a lift of its target

X y^ 2 p Y y 1 f y 2 \array{ X &&& && \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 }

there be a universal lift of ff

X y^ 1 f^ y^ 2 p Y y 1 f y 2. \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 ff, but the universal one is essentially unique, as usual for anything having a universal property. Specifically, f^\hat f in XX is essentially uniquely determined by its target y^ 2\hat y_2 and its image f=p(f^)f = p(\hat f) in YY, and is called a cartesian morphism. A morphism which is cartesian relative to p op:X opY opp^{op}\colon X^{op}\to Y^{op} is called opcartesian or cocartesian.

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

f *:X y 2X y 1 f^* : X_{y_2} \to X_{y_1}

between the fibers of pp over y 1y_1 and y 2y_2.

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

This has its analog in higher categories.


In categories

Traditional definition


(cartesian morphism)

Let p:XYp : X \to Y be a functor. A morphism f:x 1x 2f : x_1 \to x_2 in the category XX is strongly cartesian with respect to pp (nowadays often just cartesian), or (strongly) pp-cartesian if for every xXx'\in X, for every h:xx 2h:x'\to x_2 and every u:p(x)p(x 1)u:p(x')\to p(x_1) such that p(h)=p(f)up(h) = p(f) u, there exists a unique v:xx 1v:x'\to x_1 such that h=fvh = f v and u=p(v)u = p(v):

x !v h x 1 f x 2pp(x) u p(h) p(x 1) p(f) p(x 2) \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 YY (involving p(f)p(f) as above) and all lifts through pp of its 2-horn to XX (involving ff 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 pp: let X xX_x denote p 1(y)p^{-1}(y), the set of objects living over yy; and, for f:yyf : y \to y', x:X yx : X_y and x:X yx' : X_{y'}, let X f(x,x)X_f(x,x') denote p 1(f)X(x,x)p^{-1}(f) \cap X(x,x'), the set of morphisms from xx to xx' that pp maps to ff.

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

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

Δ {1,2} f Λ 2[2] N(X) p Δ[2] N(Y) \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 ff, a unique lift σ\sigma

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



(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)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:xx f:x\to x^\prime is called locally pp-Cartesian (relative to a functor XYX\to Y) if it is Cartesian with respect to the projection functor p pf:X pf2p_{pf}:X_{pf}\to \mathbf{2} where X pfX_{pf} is the (homotopy) pullback of XX along the functor c pf:2Yc_{pf}:\mathbf{2}\to Y classifying the arrow pf:pxpx pf:px\to px^\prime in YY.


(Grothendieck fibration)

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

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


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


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 2X/x_2 by the overcategory of XX over the object x 2x_2;

  • Y/p(x 2)Y/p(x_2) the corresponding overcategory of YY over p(x 2)p(x_2);

  • X/fX/f the category whose objects

    Obj(X/f)={ a x 1 f x 2} Obj(X/f) = \left\{ \array{ && a \\ &\swarrow && \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right\}

    are objects aa of XX equipped with morphisms to x 1x_1 and x 2x_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)Y/p(f).


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

ϕ:X/fX/x 2× Y/p(x 2)Y/p(f) \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 2Y/p(x 2)X/{x_2} \to Y/p(x_2) along the projection Y/p(f)Y/p(x 2)Y/p(f) \to Y/p(x_2) induced by the commutativity of

X/f ϕ 2 Y/p(f) ϕ 1 X/x 2 Y/p(x 2) \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.


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

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

Notice that objects in the pullback category are compatible pairs

(( a x 1 f x 2)X/ x 2,( b p(x 1) p(f) p(x 2))Y/p(f)). \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 ϕ\phi being surjective on object means that every such pair is in the image of some object

( a g x 1 f x 2)X f, \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 gg and gg'. 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/fX/f connecting them

a g h a g x 1 f x 2 \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 ϕ\phi to the identity morphism in the pullback category. But in particular this maps to the morphism

a h a x 2 \array{ && a \\ && \downarrow^{\mathrlap{h}} & \searrow \\ && a \\ & && \searrow & \downarrow \\ &&&& x_2 }

in X/x 2X/{x_2} and evidently is the identity there if and only if hh is the identity. Hence this maps also to the identity in the pullback category if and only if hh is the identity. So hh must be the identity. So if two lifts of an object through the surjective equivalence ϕ\phi exist, they must already be equal. Hence the surjective equivalence ϕ\phi is even an isomorphism on objects and hence an isomorphism of categories.

In (,1)(\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


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

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

is a pullback square for every aXa \in X.

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


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

is a pullback square.


The definition of being pp-cartesian can be described as a pullback square of hom-functors. By the contravariant (∞,1)-Grothendieck construction, X(,x)X(-, x) classifies the right fibration, X /xXX_{/x} \to X, and Y(p,p(x))Y(p-, p(x)) classifies the pullback of Y /p(x)YY_{/p(x)} \to Y along pp; that is by the comma category (pp(x))Y(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 ff is pp-cartesian, so the proposition follows.


If f:xyf : x \to y, then X /fX /xX_{/f} \to X_{/x} is an equivalence of (∞,1)-categories, and composing its inverse with X /fX /xX_{/f} \to X_{/x} is the dependent sum f !:X /xX /yf_! : 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 XX relative to a morphism p:XYp : 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 (,1)(\infty,1)-categories.

Let p:XYp : X \to Y be a morphism of simplicial sets. Let f:x 1x 2f : x_1 \to x_2 be an edge in XX, i.e. a morphism f:Δ 1Xf : \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/fX/f, X/x 2X/{x_2}, S/p(f)S/p(f) and S/p(x 2)S/p(x_2) in generalization of the categories considered in the above definition of cartesian morphisms in categories.


(cartesian edge in a simplicial set)

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

Then a morphism f:xyf : x \to y in XX is pp-cartesian if the induced morphism

X /fX /y× Y /p(y)Y /p(f) 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


The morphism f:xyf : x \to y as above, for p:XYp : X \to Y an inner fibration, is pp-cartesian precisely if for all n2n \geq 2 and all right outer horn inclusions

Δ {n1,n} f Λ[n] n X p Δ[n] Y \array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \Delta[n] &\to& Y }

(with Λ[n] n\Lambda[n]_n the nnth horn of the nn-simplex) such that the last edge of the horn is the given edge ff, a lift σ\sigma

Δ {n1,n} f Λ[n] n X σ p Δ[n] Y \array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}} \\ \Delta[n] &\to& Y }


This is HTT remark


This means that an inner fibration p:XYp : X \to Y with a collection of pp-cartesian morphisms in XX 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.


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

Hom X(a,x) Hom X(a,f) Hom X(a,y) Hom Y(p(a),p(x)) Hom Y(p(a),p(f)) Hom Y(p(a),p(y)) \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).


This is HTT, prop.

In sSetsSet-categories

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


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

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

is an acyclic Kan fibration.


If CC and DD are enriched in Kan complexes and if FF is hom-wise a Kan fibration, then

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

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

    C(a,x) C(a,f) C(a,y) D(F(a),F(x)) C(F(a),F(f)) D(F(a),F(y)) \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.


This is HTT, prop.


Pullback along Cartesian morphisms


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

A f B g h C k D \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 pp (meaning that p(g)=Id p(A)p(g) = Id_p(A) and p(h)=Id(B)p(h) = Id(B)) and if the two horizontal morphisms are pp-Cartesian morphisms, then this square is a pullback square.



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

is another cone over CDBC \to D \leftarrow B, then its image under pp is

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

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

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

which gives a unique filler QAQ \to A by the fact that ff is Cartesian.

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

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


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

Since kk is Cartesian, these two fillers must be equal. This means that the morphism QAQ \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


For CC a category, a morphism in CC is cartesian with respect to the terminal functor C*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:


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

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


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

The second statement is HTT, prop.

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


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]


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

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

See also the references at Grothendieck fibration.

Last revised on December 28, 2023 at 11:40:03. See the history of this page for a list of all contributions to it.