It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category $y/\ulcorner F \urcorner$ in the category $\mathrm{CAT}$ of

$$\mathcal{C} \xrightarrow{y} \widehat{\mathcal{C}} \xleftarrow{\ulcorner F \urcorner} 1.$$

Question: does this generalise to presheaves $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Gpd}$ of groupoids (or even categories)? Mere comma categories yield discrete fibrations, hence won't give the expected answer. So the question is whether some other construction with a similar flavour could do.

Probably, if the construction does generalise, then it will also work for pseudo-functors to $\mathrm{Gpd}$ or $\mathrm{Cat}$, but I'm really interested in strict functors.

Note: I'm half-aware of another universal property of the Grothendieck construction as an oplax colimit. Is it related?

  • 1
    $\begingroup$ It's not really a "lax pullback" but rather a comma category. $\endgroup$
    – Zhen Lin
    Jan 22, 2014 at 10:22
  • $\begingroup$ I thought this was the same in $\mathrm{CAT}$, what's the subtlety i'm missing? $\endgroup$ Jan 22, 2014 at 11:44
  • $\begingroup$ The lax pullback is symmetric with respect to the given two functors, the comma category is not. $\endgroup$
    – Zhen Lin
    Jan 22, 2014 at 12:04
  • 2
    $\begingroup$ There is a general definition of "lax limit", and lax pullbacks are the evident special case. $\endgroup$
    – Zhen Lin
    Jan 22, 2014 at 13:26
  • 1
    $\begingroup$ I see. For the record, a definition of lax limit may be found in Kelly's "Elementary Observations on 2-Categorical Limits". Edited the question to avoid the ambiguity, thanks for pointing this out. $\endgroup$ Jan 22, 2014 at 16:31

3 Answers 3


Possibly the comma 2-category gives what you want.

If you think of $C^{op}\rightarrow Gpd$ as a functor of 2-categories (e.g. pass to the nerve for some model of $(\infty,2)$ categories) and then take the comma object $$ \matrix{(r\downarrow F)&\rightarrow& C \\ \downarrow & & \downarrow _{r} \\ 1 & \rightarrow^F & Gpd^{C^{op}}} $$

(where $r$ is the Yoneda imbedding.)

then an object is then a pair $(c\in C,\varphi:rc\rightarrow F)$, and a morphism is a pair $(\Psi:c\rightarrow c',M:\varphi\rightarrow \varphi'\circ r\Psi)$.

There is a canonical map from the Grothendiek construction into this comma category, and I suspect it is a homotopy equivalence, so that when you take the corresponding usual category you get the same thing.

  • $\begingroup$ In which general sense is this a comma object? I only know what a comma object is in a 2-category, but you seem to be using a more general notion here, right? $\endgroup$ May 27, 2014 at 7:29
  • $\begingroup$ it's pretty much the same idea. The square is final in the category of squares with the same bottom right corner. In this case it's the comma object in the 3-category of 2-categories. $M$ here is a non trivial 3-morphism. $\endgroup$
    – Adam Gal
    May 27, 2014 at 11:09
  • $\begingroup$ It's not quite the comma object in any 3-category of 2-categories, because the transformation inhabiting the square is only lax natural, and there is no 3-category or tricategory containing lax natural transformations (they only satisfy interchange laxly). It's probably some kind of comma object in an appropriate kind of "lax-Gray-category", but I don't know if anyone has worked that out formally. $\endgroup$ Mar 24, 2021 at 22:36

Another way of phrasing the isomorphism of categories $\int^{\mathcal{C}}\mathcal{F}\congよ\downarrow[\mathcal{F}]$ is by saying that $\int^{\mathcal{C}}\mathcal{F}$ is the full subcategory of $\mathsf{PSh}(\mathcal{C})_{/\mathcal{F}}$ spanned by the representable presheaves. A quick way to prove this is as follows:

  • First, note that $よ\downarrow[\mathcal{F}]$ is the category of elements of the functor $\mathsf{Nat}(h_{(-)},\mathcal{F})\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Sets}$.
  • Then, the Yoneda lemma gives a natural isomorphism $\mathsf{Nat}(h_{(-)},\mathcal{F})\cong\mathcal{F}$. This is an isomorphism in $\mathsf{PSh}(\mathcal{C})$.
  • As functors preserve isomorphisms, it follows that $\int^{\mathcal{C}}\colon\mathsf{PSh}(\mathcal{C}^{\mathsf{op}})\longrightarrow\mathsf{Cats}$ sends the isomorphism of presheaves $\mathcal{F}\cong\mathsf{Nat}(h_{(-)},\mathcal{F})$ to an isomorphism of categories $\int^{\mathcal{C}}\mathcal{F}\cong\underbrace{\int^{\mathcal{C}}\mathsf{Nat}(h_{(-)},\mathcal{F})}_{\congよ\downarrow[\mathcal{F}]}$.

To give such a description for the Grothendieck construction, we first pass to bicategories. There, given a pseudopresheaf $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ from a bicategory $\mathcal{C}$ to the $2$-category $\mathsf{Cats}_{\mathsf{2}}$ of small categories, functors, and natural transformations, we can repeat the above strategy:

  • This time, we consider the locally full sub-bicategory of $\mathsf{PseudoPSh}(\mathcal{C})_{/\mathcal{F}}$ spanned by the representable pseudopresheaves. This is the bicategory $よ(\mathcal{C})_{/\mathcal{F}}$ where List item enter image description here enter image description here enter image description here

and where vertical and horizontal composition of $2$-morphisms is defined as in $\mathsf{PseudoPSh}(\mathcal{C})$ (we also have to specify the associators and unitors, but let's not since this description is already quite long).

  • Now, note that $よ(\mathcal{C})_{/\mathcal{F}}$ is the bicategory of elements (as defined in arXiv:1212.6283) of the pseudopresheaf $\mathsf{PseudoNat}(\mathsf{h}_{(-)},\mathcal{F})\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$.
  • Just as functors preserve isomorphisms, functors of tricategories preserve biequivalences. We can then apply Proposition 3.3.6 of arXiv:1212.6283 to the equivalence $\mathsf{PseudoNat}(\mathsf{h}_(-),\mathcal{F})\cong\mathcal{F}$ provided by the bicategorical Yoneda lemma to obtain a biequivalence $\int^{\mathcal{C}}\mathcal{F}\cong\underbrace{\int^{\mathcal{C}}\mathsf{PseudoNat}(\mathsf{h}_{(-)},\mathcal{F})}_{\cong よ(\mathcal{C})_{/\mathcal{F}}}$.

Finally we connect the above back to the Grothendieck construction. To this end, observe that―just as the category of elements of a presheaf $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Sets}$ agrees with the Grothendieck construction of $\mathcal{F}_{\mathsf{disc}}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}$―the Grothendieck construction of a pseudofunctor $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ agrees with the bicategory of elements of $\mathcal{F}\colon\mathcal{C}^{\mathsf{op}}_\mathsf{bi}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$, where $\mathcal{C}^{\mathsf{op}}_\mathsf{bi}$ is the discrete bicategory associated to $\mathcal{C}$ (this is $\mathcal{C}$ but with discrete $\mathsf{Hom}$-categories, i.e. for each $A,B\in\mathrm{Obj}(\mathcal{C})$, we have $\mathsf{Hom}_{\mathcal{C}^{\mathsf{op}}_\mathsf{bi}}(A,B)\overset{\mathrm{def}}{=}\mathrm{Hom}_{\mathcal{C}}(A,B)_{\mathsf{disc}}$).

All this is to say the following: the Grothendieck construction of a pseudofunctor $\mathcal{F}\colon\mathcal{C}^\mathsf{op}\longrightarrow\mathsf{Cats}_{\mathsf{2}}$ is equivalent to the category $よ_\mathsf{disc}(\mathcal{C})_{/\mathcal{F}}$ where enter image description here enter image description here

(Sorry for the long answer!)

  • $\begingroup$ That's the way to go! I just found your comment while looking for textbooks or similar which would say it just this way, but in vain. It would be good to boost the nLab entry at ncatlab.org/nlab/show/Grothendieck+construction with this kind of perspective. $\endgroup$ Dec 27, 2021 at 16:56

The answer seems to be "no": Consider a strong-/pseudofuntor $F:C^\mathrm{op}\to \mathrm{Cat}$.

The object "sets" are isomorphic: they are given by pairs $(x,a_x)$ with $x\in C$ and $a_x\in F(x)$. But arrows in the Grothendieck construction / category of elements are given by pairs

$$(f:x\to y,\varphi: a_x\to f^*a_y)$$

where $f^*:=F(f)$.

On the other hand: Arrows in the respective comma-category are given by triples

$$(f:x\to y, \Phi: F\to F, [\varphi]: \Phi_x(a_x) \to f^*a_y).$$

Composition is similar but the endo-transformation $\Phi$ does not need to be the identity.

These categories are in general not equivalent: Consider the case where $C=1$ for suitable $F$.

  • 1
    $\begingroup$ This is precisely what i meant by: "Mere comma categories yield discrete fibrations, hence won't give the expected answer." My question was thus about what should replace comma categories to yield the correct construction. $\endgroup$ Jan 24, 2014 at 10:17
  • $\begingroup$ Ah sry; misread the question then. $\endgroup$ Jan 24, 2014 at 13:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.