category theory

# Contents

## Idea

In any context it is of interest to ask which kind of morphisms

$\array{ E \\ \downarrow^{\mathrlap{p}} \\ C }$

arise as pullbacks along a classifying morphism $S_p : C \to U$ to some universal object $U$ of some universal morphism

$\array{ \hat U \\ \downarrow^{\mathrlap{p_{univ}}} \\ U } \,.$

The Grothendieck construction describes this in the context of Cat: a morphism $p : E \to C$ of categories – i.e. a functor – is called a fibered category or Grothendieck fibration if it is encoded in a pseudofunctor/2-functor $S_p : C^{op} \to Cat$.

The reconstruction of $p$ from the pseudofunctor $S_p$ is the Grothendieck construction

$\int \;\; : \;\; Func(C^{op}, Cat) \to Cat/C$

which is a 2-functor from the 2-category of pseudofunctors $C^{op} \to Cat$ to the overcategory of Cat over $C$.

The essential image of this functor consists of Grothendieck fibrations and this establishes an equivalence of 2-categories

$\int : Func(C^{op}, Cat) \stackrel{\simeq}{\to} Fib(C)$

between 2-functors $C^{op} \to Cat$ and Grothendieck fibrations over $C$.

When restricted to pseudofunctors with values in Grpd $\subset$ Cat this identifies the Grothendieck fibrations in groupoids

$\int \;\;:\;\; Func(C^{op}, Grpd) \stackrel{\simeq}{\to} FibGrpd(C) \,.$

This equivalence notably allows one to discuss stacks equivalently as pseudofunctors or as groupoid fibrations (in each case satisfying a descent condition with respect to a Grothendieck topology on $D$).

The Grothendieck construction is one of the central aspects of category theory, together with the notions of universal constructions such as limit, adjunction and Kan extension. It is expected to have suitable analogs in all sufficiently good contexts of higher category theory. Notably there is an (∞,1)-Grothendieck construction in (∞,1)-category theory.

## Definition

Let Cat be the 2-category of categories, functors and natural transformations. In line with the philosophy of generalized universal bundles, the “universal Cat-bundle” is $Cat_{*,\ell} \to Cat$. Here $Cat_{*,\ell}$ denotes the (2-)category of “lax-pointedcategories, also known as the “lax slice” of $Cat$ under the terminal category $*\in Cat$. Its objects are pointed categories, i.e. pairs $(A,a)$ where $A$ is a category and $a$ is an object of $A$, and its morphisms $(A,a) \to (B,b)$ are pairs $(f,\gamma)$ where $f\colon A\to B$ is a functor and $\gamma\colon f(a)\to b$ is a morphism in $B$. The projection $Cat_{*,\ell} \to Cat$ is just the forgetful functor.

Then if $F\colon C \to Cat$ is a pseudofunctor from a category $C$ to $Cat$, the Grothendieck construction for $F$ is the (strict) 2-pullback $p : \int F \to C$ of $Cat_{*,\ell} \to Cat$ along $F$:

$\array{ \int F &\to& Cat_{*,\ell} \\ {}^{p}\downarrow && \downarrow \\ C &\overset{F}{\to}& Cat } \,.$

This means that

• the objects of $\int F$ are pairs $(c,a)$, where $c \in Obj(C)$ and $a \in Obj(F(c))$

• and morphisms in $\int F$ are given by pairs $(c \overset{f}{\to} c', F(f)(a) \overset{\alpha}{\to} a')$. This may be visualized as

$\int F = \left\{ \array{ && {*} \\ & {}^a\swarrow &\seArrow^{\alpha}& \searrow^{a'} \\ F(c) && \stackrel{F(f)}{\to} && F(c') \\ \\ c &&\stackrel{f}{\to}&& c' } \right\} \,.$

This extends to a 2-functor between bicategories

$\int \;\; : \;\; [C, Cat] \to Cat/C$

from pseudofunctors on $C$ to the overcategory of Cat over $C$.

The more commonly described version of this construction works instead on contravariant pseudofunctors, i.e. pseudofunctors $C^{op}\to Cat$. In this case we use instead the “universal $Cat$-cobundle” $(Cat_{*,c})^{op} \to Cat^{op}$, where $(Cat_{*,c})$ is the colax slice, whose objects are again pointed categories $(A,a)$, but whose morphisms $(A,a) \to (B,b)$ are pairs $(f,\gamma)$ where $f\colon A\to B$ and $\gamma\colon b \to f(a)$. Now the 2-pullback

$\array{ \int F &\to& (Cat_{*,c})^{op} \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ C &\stackrel{F}{\to}& Cat^{op} } \,.$

describes a 2-functor

$\int \quad\colon\quad [C^{op},Cat] \to Cat/C.$

In this case,

• the objects of $\int F$ are again pairs $(c,a)$, where $c \in Obj(C)$ and $a \in Obj(F(c))$, but

• the morphisms in $\int F$ from $(c,a)$ to $(c',a')$ are pairs $(c \overset{f}{\to} c', a \overset{\alpha}{\to} F(f)(a'))$.

## Properties

### As an oplax colimit

The Grothendieck construction on $F : C \to Cat$ is equivalently the oplax colimit of $F$ (e.g Gepner-Haugseng-Nikolaus 15). That means that for each category $X$ there is an equivalence of categories

$Lax(F, \Delta X) \simeq [{\textstyle \int} F, X]$

that is natural in $X$, where $\Delta X$ is the constant functor with value $X$. (See oplax colimit for an explanation of why lax natural transformations appear in the definition of an oplax colimit.)

A lax natural transformation $\alpha$ from $F$ to $\Delta X$ is given by

• for each object $c$ of $C$, a functor $\alpha_c \colon F c \to X$, and
• for each morphism $m \colon c \to d$ in $C$, a natural transformation $\alpha_m \colon \alpha_c \Rightarrow \alpha_d \circ m_*$ (writing $m_* = F m$),

such that $\alpha_{1_c}$ is the isomorphism $F 1_c \cong 1_{F c}$ given by pseudofunctoriality of $F$, and that if $m \colon c \to d$, $n \colon d \to e$ is a composable pair in $C$, then $\alpha_{n m}$ is equal to the obvious pasting of $\alpha_m$ and $\alpha_n$.

We want to show that to each such lax transformation there corresponds an essentially unique functor $\int F \to X$. So firstly, given $\alpha$ as above, let $A$ be the functor that sends $x \in F c$ to $\alpha_c x$, and acts on arrows as

$(m \colon c \to d, f \colon m_* x \to y) \quad \mapsto \quad \alpha_c x \overset{\alpha_m x}{\to} \alpha_d m_* x \overset{\alpha_d f}{\to} \alpha_d y$

That $A$ is a functor follows from the coherence properties of $\alpha$ with respect to identities and composition in $C$.

Conversely, if $A \colon \int F \to X$ is a functor, we get a lax transformation $\alpha$ as follows:

• For each $c \in C$, $\alpha_c$ is the restriction of $A$ to the category $F c$, which is the subcategory of $\int F$ whose objects are those of $F c$ and whose morphisms are those with first component an identity morphism. This clearly makes $\alpha_c$ a functor.
• For each $m \colon c \to d$ in $C$, $\alpha_m$ has components $\alpha_c x \to \alpha_d m_* x$ given by $A$‘s value at the morphism $(m,1_{m_* x})$. This is a natural transformation because, if $k \colon x \to x'$ is a morphism in $F c$, then both sides of the naturality square are the value of $A$ at the morphism $(m, m_*k)$.

As one might expect, the coherence conditions on the resulting $\alpha$ follow from the functoriality of $A$.

It is then easy to check that these two mappings form a bijection between the objects of $Lax(F, \Delta X)$ and $[\int F, X]$.

As for the morphisms involved, the modifications between lax transformations and the natural transformations between functors, it is straightforward to show that these are in bijective correspondence too. Hence we have shown that the above equivalence holds.

By inspecting the above proof, it is easy to see that the lax transformation associated to a functor $\int F \to X$ is a pseudonatural transformation if and only if the functor inverts (i.e. sends to an isomorphism) each member of the class $S$ of morphisms of $\int F$ whose second component is an identity. (These are in fact the opcartesian morphisms with respect to the projection $\int F \to C$.) The localization $\int F[S^{-1}]$ is therefore the (weak) 2-colimit of $F$:

$Ps(F, \Delta X) \simeq [{\textstyle \int} F, X]_{S^{-1}} \simeq [{\textstyle \int} F[S^{-1}], X]$

This last result appears in SGA4 Exposé VI, Section 6.

### The equivalence between fibrations and pseudofunctors

One can characterize the image of the Grothendieck construction as consisting precisely of those objects in $Cat/C$ that are Grothendieck fibrations.

We recall the definition of the bicategory of Grothendieck fibrations and pseudofunctors and and then state the main equivalence theorem.

#### The bicategory of pseudofunctors.

A pseudofunctor from a 1-category $C$ to a 2-category (bicategory) $A$ is nothing but a (non-strict) 2-functor between bicategories, with the ordinary category regarded as a special bicategory.

We write $[C^{op}, A]$ for the 2-functor 2-category from the opposite category of $C$ to $A$ (the $op$ here is just convention):

#### The bicategory of fibrations

###### Definition

A functor $p : E \to C$ is a Grothendieck fibration if for every object $e \in E$ and every morphism $f : c \to p(e)$ in $C$ there is a morphism $\hat f : \hat c \to e$ in $E$ that lifts $f$ in that $p(\hat f) = f$ and which is a Cartesian morphism.

A morphism of Grothendieck fibrations $F : (p : E \to C) \to (p' : E' \to C)$ is

• a functor $F : E \to E'$

• such that

• $F$ sends Cartesian morphisms to Cartesian morphisms;

• the diagram

$\array{ E &&\stackrel{F}{\to}&& E' \\ & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p'}} \\ && C }$

in Cat commutes (strictly).

• a 2-morphism between morphism $\eta : F \to F'$ is a natural transformation of the underlying functors, that also makes the obvious diagram 2-commute, i.e. such that $p' \cdot \eta$ is trivial.

Compositions are those induced from the underlying functors and natural transformations.

This defines the 2-category of Grothendieck fibrations

$Fib(C) \hookrightarrow Cat/C$

over $C$, being a 2-subcategory of the overcategory of Cat over $C$.

###### Remark

Cartesian lifts are not required to be unique, but are automatically unique up to a unique vertical isomorphism connecting their domains.

#### Statement of the equivalence

###### Proposition

The Grothendieck construction factors through Grothendieck fibrations over $C$

$\int : [C^{op}, Cat] \to Fib(C) \hookrightarrow Cat/C$

and establishes an equivalence of bicategories

$\int : [C^{op}, Cat] \stackrel{\simeq}{\to} Fib(C) \,.$

In fact, it is more than that: it is an equivalence of strict 2-categories, in the sense of strict 2-category theory, i.e. an equivalence of $Cat$-enriched categories.

When restricted to pseudofunctors that factor through Grpd $\hookrightarrow Cat$ it factors through fibrations in groupoids

$\int : [C^{op}, Grpd] \to Fib_{Grpd}(C)\hookrightarrow Cat/C$

and establishes a similar equivalence

$[C^{op}, Grpd] \simeq Fib_{Grpd}(C) \,.$
###### Proof

This can be verified by straightforward albeit somewhat tedious checking. Details are spelled out in section 1.2 of

The statement itself is theorem 1.3.6 there, all definitions and lemmas are on the pages before that.

#### Model category version

For the case of pseudofunctors with values in groupoids, there is a model category version of the Grothendieck construction discussed in

There the statement of the above equivalence is the statement that the Grothendieck equivalence exhibits a Quillen equivalence between suitable model category structures on functors from and to $C$.

This model category incarnation of the Grothendieck construction generalizes to a model category presentation of the (∞,1)-Grothendieck construction.

### Adjoints to the Grothendieck construction

The Grothendieck construction functor

$\int : [C^{op}, Cat] \to Cat/C$

has a left and a right adjoint functor.

Restricted to Grothendieck fibrations and fibrations in groupoids, both of these exhibit the above equivalences as adjoint equivalences. Notice that much of the traditional literature discusses (just) the right adjoint.

The left adjoint is the functor

$L : (p : E \to C) \mapsto ( (-)/p : C^{op} \to Cat)$

that assigns to a functor $p$ the presheaf which sends $c \in C$ to the comma category $c/p$ with objects given by pairs $(e, c \to p(e))$ and morphisms by commutative triangles

$\array{ && c && \\ & \swarrow && \searrow & \\ p(e_1) &&\to&& p(e_2) }$

i.e.

$L(E \stackrel{p}{\to}C) : c \mapsto c/p \,.$

This functor may equivalently be expressed as follows.

##### In terms of a cone construction

For given $(E \stackrel{p}{\to} C)$ consider the (3,1)-pushout

$\array{ E &\hookrightarrow& E^{\triangleright} \\ \downarrow^{\mathrlap{p}} &\swArrow& \downarrow \\ C &\to& K(p) }$

of (2,1)-categories , where $K^{\triangleright}$ is $K$ with one terminal object $v$ adjoined (a join of categories). (Here $E$, $C$ and $E^{\triangleright}$ are 1-catgeories regarded trivially as $(2,1)$-categories and where $K(p)$ will in general be a (2,1)-category with nontrivial 2-morphisms).

###### Claim

We have

$c/p \simeq Hom_{K(p)}(c,v) \,.$

And hence the left adjoint to the Grothendieck construction may be realized as the assignment that sends $p : E \to C$ to the pseudofunctor

$L(p) := Hom_{K(p)}(-, v) : C^{op} \to Cat \,.$
###### Proof

It is convenient to compute the weak pushout by embedding the situation from Cat into the bigger context of (∞,1)-categories and using the model of that provided by sSet: the model structure for quasi-categories. This also facilitates the generalization of the argument from 1-categories to higher categories.

So consider equivalently the weak pushout diagram

$\array{ N(E) &\hookrightarrow& N(E)^{\triangleright} \\ \downarrow^{\mathrlap{N(p)}} &\swArrow& \downarrow \\ N(C) &\to& N(K(p)) }$

of quasi-categories, where $N(-)$ is the nerve operation and where $N(E)^{\triangleright} = N(E) \star *$ is the join of simplicial sets of $N(E)$ with the point.

By the general yoga of homotopy colimits (see there for details) we know that this $\infty$-pushout here may be computed as an ordinary pushout in the 1-category sSet if the pushout diagram $N(C) \leftarrow N(E) \to N(E)^{\triangleright}$ has the property that

• all three objects are cofibrant;

• at least one of the two morphisms is a cofibration

in the model structure for quasi-categories $sSet_{Joyal}$.

But this is trivially verified since the cofibrations in $sSet_{Joyal}$ are just the monomorphisms in sSet: the degreewise injective maps of simplicial sets. So every object in $sSet_{Joyal}$ is cofibrant and the inclusion $N(E) \hookrightarrow N(E)^{\triangleright}$ is a cofibration.

(The same conclusion would hold for the same simple reasons in the standard model structure on simplicial sets $sSet_{Quillen}$.)

From this it follows that simply because we passed from categories to their nerves, the computation of the weak pushout reduces to the computation of an ordinary pushout (one may think of passing to nerves as providing a cofibrant replacement: since in the nerve all composition of k-morphisms is “freed”, the nerve is a suitably “puffed up” version of a category that is suitable for computing $\infty$-pushouts).

So we are reduced to computing the ordinary pushout

$\array{ N(E) &\hookrightarrow& N(E)^{\triangleright} \\ \downarrow^{\mathrlap{N(p)}} && \downarrow \\ N(C) &\to& Q }$

in sSet. The fibrant replacement of $Q$ is then the nerve of the bicategory $K(p)$ that we are after.

As recalled at limits and colimits by example in the section limits in presheaf categories, colimits (and hence pushouts) in the presheaf-category sSet $= Func(\Delta^{op}, Set)$ are computed for each object $[n] \in \Delta$ as ordinary colimits in Set.

For $n=0$ we see that $Q_0$ is the collection of objects of $C$ and one additional vertex $v$:

$Q_0 = N(C)_0 \coprod \{ v\} = p(Obj(E)) \coprod \{v \}$

For $n=1$ similarly we find that $Q_1$ consists of the 1-cells in in $C$ and in addition of one 1-cell $e : c \to v$ for each $e \in Obj(E)$ with $p(e) = c$ (this 1-cell is really the terminal 1-cell $e \to v$ in $E^{\triangleright}$ but with its source re-interpreted as being $p(e) = c$ according to the identification of $Q_0$ as above). In the fibrant replacement of $Q$ the composite of original 1-cells $c_1 \to c_2$ and the new 1-cells $e : c_2 \to v$ will be freely added, so that the general 1-morphism $c_1 \to v$ will consist of a 1-morphism $c_1 \to c_2$ in $C$ together with a lift of $c_2$ to $E$. This is just as in the comma category $c/p$.

For $n=2$ we have in $Q_2$ the 2-cells in $C$ as well as one 2-cell

$\array{ c_1 &&\to&& c_2 \\ & \searrow &{}^{(e_1 \to e_2)}\swArrow& \swarrow \\ && v }$

for each 1-cell $(e_1 \to e_2)$ in $N(E)$ with $p(e_1 \to e_2)$ = $(c_1 \to c_2)$.

In particular this means that if $e_2: c_2 \to v$ is a morphism in $Q$ and $c_1 \to c_2$ is a morphism in $C$, then the composite $c_1 \to c_2 \to v$ in $Q$ is homotopic to any compatible direct morphism $c_1 \to v$ in $Q$.

This means that forming the fibrant replacement of $Q$ in $sSet_{Joyal}$ will not throw in superfluous 1-morphisms on top of those we already discussed in the previous paragraph…

Now furthermore…

This formulation of the Grothendieck construction as an adjunction

$(L \dashv \int) : Fib(C) \stackrel{\leftarrow}{\to} [C^{op}, Cat]$

with the left adjoint given by hom-objects in a pushout object as above is the starting point for the vertical categorification described at (∞,1)-Grothendieck construction.

## Generalizations

### $n = 0$

The analog of the Grothendieck construction one categorical dimension down is the category of elements of a presheaf.

### $n = (\infty,0)$

The analog of the Grothendieck construction for ∞-groupoids is examined in detail in Heuts-Moerdijk 13.

The category of presheaves in groupoids is replaced by the model category of simplicial presheaves equipped with the projective model structure and the category of Grothendieck fibrations in groupoids is replaced by the model category of simplicial sets over the nerve of the source category, equipped with the contravariant model structure.

In this case there is not one, but two different functors that generalize the Grothendieck construction.

The first functor $h_!$ is a left adjoint, it implements the homotopy colimit using the diagonal of a bisimplicial set, and the second functor $r^*$ is a right adjoint, it uses the codiagonal (also known as the totalization) of a bisimplicial set. Both functors fit into adjunctions $h_!\dashv h^*$ and $r_!\dashv r^*$, where the other two adjoints can be seen as rectification functors: the right adjoint $h^*$ generalizes the cleavage construction, whereas the left adjoint $r_!$ generalizes the comma category construction above.

The two functors $h_!$ and $r^*$ become naturally weakly equivalent once we derive them, but they are not isomorphic. The functor $r^*$ restricted to the full subcategory of presheaves of groupoids recovers the nerve of the classical Grothendieck construction described above. The functor $h_!$ restricted to the same full subcategory does not even land in quasicategories, so it doesn’t give rise to a new construction in the classical case.

### $n = (\infty,1)$

The analog of the Grothendieck construction for (∞,1)-categories is described at Cartesian fibration and at universal fibration of (∞,1)-categories.

The correspondence between $(\infty,1)$-categorical cartesian fibrations $E \to C$ and (∞,1)-presheaves $C \to (\infty,1)Cat^{op}$ is modeled by the Quillen equivalence between the model structure on marked simplicial over-sets and the projective global model structure on simplicial presheaves.

For more details see

## Warning on terminology

The term ‘Grothendieck Construction’ is applied in the literature to at least two very different constructions (and as Grothendieck introduced so many new ideas and constructions to mathematics, perhaps there are others!). One concerns the construction of a fibered category from a pseudofunctor and will be treated in more detail in the entry on Grothendieck fibration. The other refers to constructing the Grothendieck group is in the context of K-theory from isomorphism classes of vector bundles on a space by the introduction of formal inverses, ‘virtual bundles’. This constructs an Abelian group from the semi-group of isomorphism classes.

## Examples

A representable functor $C(-,X) : C^{op} \to Set \hookrightarrow Cat$ maps under the Grothendieck construction to the slice category $C/X$. The corresponding fibrations $C/X \to C$ are also called representable fibered categories.

## References

Standard references are in

The left adjoint to the Grothendieck construction is discussed in §3.1.1 of

The analog for simplicial sets instead of groupoids is discussed in