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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?

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    $\begingroup$ It's not really a "lax pullback" but rather a comma category. $\endgroup$
    – Zhen Lin
    Jan 22 '14 at 10:22
  • $\begingroup$ I thought this was the same in $\mathrm{CAT}$, what's the subtlety i'm missing? $\endgroup$ Jan 22 '14 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 '14 at 12:04
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    $\begingroup$ There is a general definition of "lax limit", and lax pullbacks are the evident special case. $\endgroup$
    – Zhen Lin
    Jan 22 '14 at 13:26
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    $\begingroup$ 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 '14 at 16:31
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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.

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  • $\begingroup$ 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 '14 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 '14 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 at 22:36
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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 $F$.

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    $\begingroup$ 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 '14 at 10:17
  • $\begingroup$ Ah sry; misread the question then. $\endgroup$ Jan 24 '14 at 13:32
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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!)

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