# How to characterize good “models” of a category

Let ${\bf Cat}$ denote the category of small categories. Recall that for a category $\mathcal{C}$ and a functor $F\colon\mathcal{C}\to{\bf Cat}$, the Grothendieck construction of $F$, which I'll denote $\int F$, is a category and it comes equipped with a natural fibration $\int F\to\mathcal{C}$.

[For reference: The objects of $\int F$ are pairs $(c,x)$ where $c\in{\bf Ob}(\mathcal{C})$ and $x\in{\bf Ob}(F(\mathcal{C}))$, and a morphism $(c,x)\to(c',x')$ is a pair $(f,g)$ where $f\colon c\to c'$ in $\mathcal{C}$ and $g\colon F(f)(x)\to x'$ in the category $F(c')$.]

Now, given a category ${\mathcal D}$, I'll define a model of ${\mathcal D}$ to be a pair $({\mathcal C},F,e)$ where $\mathcal{C}$ is a category, $F\colon \mathcal{C}\to{\bf Cat}$ is a functor, and $e\colon\int F\to\mathcal{D}$ is a natural isomorphism. I will sometimes leave out $e$ if it is obvious. Allow me to leave morphisms of models undefined, as I'm not sure what I want here. [Supplying an appropriate definition of morphisms between models of $\mathcal{D}$ should be part of giving a good answer to this overflow question.]

Every category $\mathcal{D}$ has two canonical models which I'll denote by $(\mathcal{D},\{\ast\})$ and $(\{\ast\},\mathcal{D})$. The first is the functor $\mathcal{D}\to{\bf Cat}$ that sends every object of $\mathcal{C}$ to the terminal set, and the second is the functor $*\to{\bf Cat}$ that sends the terminal category to $\mathcal{D}$.

But there may be many models $F\colon\mathcal{C}\to{\bf Cat}$ of $\mathcal{D}$ that lie "in between" these two extreme cases, and some are "better than others" in the sense that the fibers of $\mathcal{D}=\int F\to\mathcal{C}$ are non-trivial yet "comprehensible" in some human sense.

Question: What can you say about the (yet-undefined) category of models of $\mathcal{D}$ that will clarify the above ideas?

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Maybe I should have included an example of a "nice" model. If $D$ is the Grothendieck construction of a faithful functor $F:C\to Set$, then we may consider $F$ a "nice" model of $D$. Another example might be to replace "$Set$" above with "$Poset$." –  David Spivak Aug 15 '10 at 21:05
In the definition of a model, you say that $e$ should be a natural isomorphism… but this doesn't quite seem to typecheck, since $e$ is a functor. Do you mean it should be an isomorphism or equivalence of categories, or something else? –  Peter LeFanu Lumsdaine Aug 16 '10 at 19:37