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For simplicity, let us consider only a functor out of a small category $\mathcal{C}$ to $Set$, $$ f:\mathcal{C}\to Set, $$ The Grothendieck construction produces a category (category of elements) $El(f)$ whose objects are $\sqcup_{c\in \mathcal{C}} f(c)$. Grothendieck construction provides a universal way to compute the colimits.

My question:

Is there a co-form Grothendieck construction (or Grothendieck nstruction for fun)? The co-form of Grothendieck construction should give us a way to compute the limits.

By nonsense argument, this might be the (internal) Grothendieck construction for $f^{op}: \mathcal{C}^{op}\to Set^{op}$. But I hardly see what is this more concretely. First, the space of elements is dual to above form, perhaps $\Pi_{c\in \mathcal{C}} f(c)$.

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  • $\begingroup$ I'm not sure what "universal way to compute the colimits" you're referring to, but the only thing that comes to mind is the Grothendieck construction of a contravariant functor thought of as a functor $\mathcal{C}^{op}\to Set$. I wouldn't expect anything better than that because $Set$ and $Set^{op}$ are fundamentally different and this difference is very important in category theory. $\endgroup$ Apr 25, 2014 at 13:26
  • $\begingroup$ @EricWofsey The Grothendieck construction for contravariant functor is not I am seeking for. Grothendieck construction is essentially a colimit construction (more precisely, a oplax 2-categorical colimit). I am aware of $Set$ and $Set^{op}$ are very different. For a nice enough category $\mathcal{D}$, it is possible to speak of Grothendieck construction of functors $C\to D$. $\endgroup$
    – Ma Ming
    Apr 25, 2014 at 13:33
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    $\begingroup$ So you're just using "Grothendieck construction" to mean "explicit description of the appropriate 2-categorical (co)limit"? $\endgroup$ Apr 25, 2014 at 13:39
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    $\begingroup$ If I'm not mistaken, the (op)lax limit of a functor $\mathcal{C}\to Set\to Cat$ is just its ordinary 1-categorical limit, since all the natural transformations involved in defining a cone take place in a discrete category. So in fact I think your "Grothendieck nstruction" is in this case just a set, the ordinary limit of the functor $f$. If you let $f$ take values in non-discrete categories, things get more complicated and I don't immediately see a simple description. $\endgroup$ Apr 25, 2014 at 14:02
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    $\begingroup$ Wouldn't a construction be the co-form of a nstruction? $\endgroup$ Apr 26, 2014 at 4:51

2 Answers 2

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For any (pseudo)functor $f:C\to \mathrm{Cat}$, its Grothendieck construction $\mathrm{El}(f)$ is, as you have said, its oplax colimit. The lax limit of $f$ can also be computed as the category of sections of the Grothendieck construction. I.e. its objects are functors $s:C\to \mathrm{El}(f)$ for which the composite $C\xrightarrow{s} \mathrm{El}(f)\to C$ is the identity, and its morphisms are natural transformations lying over the identity of $C$.

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  • $\begingroup$ I didn't see your answer when I was writing mine --- I'll edit my answer to make a connection with yours. $\endgroup$ May 2, 2014 at 10:00
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Like Eric has said --- the construction trivializes on discrete (co)presheaves.

So, let me try to describe the general case. There are two difficulties: first, your question is about covariant functors, and in case of Grothendieck construction we are used to contravariant functors; second, I am going to describe the construction for more general lax functors, despite the fact that I always forget which functors are lax and which are op-lax :-)

The idea is that since the Gorthendieck construction is a generalisation of the Kleisli construction (i.e. a lax functor $1 \rightarrow \mathbf{Cat}$ is tantamount to a monad in $\mathbf{Cat}$, and the Grothendieck construction over such a lax functor gives the Kleisli resolution for the associated monad), the "dual Grothendieck construction" should be a generalisation of the Eilenberg-Moore construction. Specifically, the dual Grothendieck construction should yield a category consisting of collections of algebras and algebra homomorphisms.

Suppose $\Theta \colon \mathbb{C} \rightarrow \mathbf{Cat}$ is a lax functor. The dual Gothendieck construction of $\Theta$ is the category $\bigwedge \Theta$ defined as follows.

An object in $\bigwedge \Theta$ consists of the data:

  • for every object $A \in \mathbb{C}$ an object $\lambda_A \in \Theta(A)$,
  • for every morphism $f \colon A \rightarrow B \in \mathbb{C}$ a morphism $\lambda_f \colon \Theta(f)(\lambda_A) \rightarrow \lambda_B$

subject to the following laws:

  • (multiplication) if $A \overset{f}\rightarrow B \overset{g}\rightarrow C \in \mathbb{C}$, then: $\lambda_g \circ \Theta(g)(\lambda_f) = \lambda_{f \circ g} \circ \mu_{\lambda_A}$, where $\mu \colon \Theta(g) \circ \Theta(f) \rightarrow \Theta(g \circ f)$ is the transformation from the definition of lax functor; diagrammatically:

$$\require{AMScd}\begin{CD} \Theta(g)(\Theta(f)(\lambda_A)) @>{\mu_{\lambda_A}}>> \Theta(g \circ f)(\lambda_A)\\ @V{\Theta(g)(\lambda_f)}VV @VV{\lambda_{g \circ f}}V \\ \Theta(g)(\lambda_B) @>{\lambda_g}>> \lambda_C \end{CD}$$

  • (unit) if $A \in \mathbb{C}$, then: $\lambda_{\mathit{id}_A} \circ \eta_{\lambda_A} = \mathit{id}_{\lambda_A}$, where $\eta \colon \mathit{id} \rightarrow \Theta(\mathit{id})$ is another transformation from the definition of lax functor; diagrammatically:

$$\begin{CD} \lambda_A @>{\eta_{\lambda_A}}>> \Theta(\mathit{id}_A)(\lambda_A)\\ @| @VV{\lambda_{\mathit{id}_A}}V \\ \lambda_A @= \lambda_A \end{CD}$$

A morphism between defined in the above collections of algebras consists of a collection of morphisms between carriers, such that the structures of the algebras are preserved.

In case $\Theta$ is a discrete strict functor, all morphisms from the definition of an object of $\bigwedge \Theta$ collapse to identities.

[EDIT: connection with the other answer]

First, let me point that Mike's answer relies on an old observation made by Jean Benabou, that lax natural transformations between psuedofunctors $\Phi \rightarrow \Theta \colon \mathbb{C} \rightarrow \mathbf{Cat}$ are tantamount to functors between their Grothendieck constructions:

$$\begin{CD} \int \Phi @>>> \int \Theta \\ @V{\pi_\Phi}VV @VV{\pi_\Theta}V \\ \mathbb{C} @= \mathbb{C} \end{CD}$$

Everything else is almost tautological: $$\hom(1, \mathit{laxLim}(\Theta)) \approx \mathit{laxNat}(\delta_1, \Theta) \approx \hom(\pi_{\delta_1}, \pi_\Theta) \approx \hom(\mathit{id}, \pi_\Theta)$$ where the first isomorphism is the definition of lax limit $\mathit{laxLim}(\Theta)$, the second is Benabou's observation, and the third follows from the fact that the Grothendieck construction over terminal functor yields the identity on the base category (i.e. $\int \delta_1 = \int^{C \in \mathbb{C}} \mathbb{C}/C \times \delta_1(C) \approx \int^{C \in \mathbb{C}} \mathbb{C}/C \times 1 \approx \mathbb{C}/1 \approx \mathbb{C}$).

To link this explanation with my answer, notice that if $\Theta$ is a pseudofunctor, then the laws of unit and multiplication make $\lambda_{(-)}$ a functor to the Grothendieck construction $\int \Theta$ --- morphisms from $X$ to $Y$ over $f \colon A \rightarrow B$ in $\int \Theta$ consist of pairs $\langle f \colon A \rightarrow B, h \colon \Theta(f)(X) \rightarrow Y \rangle$; given another morphism $\langle g \colon B \rightarrow C, k \colon \Theta(g)(Y) \rightarrow Z \rangle$ their composition $k \circ h$ is defined as $k \circ \Theta(g)(h) \circ \mu^{-1}$, and identities in $\int \Theta$ are induced by $\eta^{-1}$.

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