Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int_{\mathcal{C}}} P$$ Now there are (at least) two ways to see this (from what I've managed to ascertain.) You might derive the equivalence as it were 2-categorically using the universal property of $\widehat{\mathcal{C}}$ which is the equivalence $\textbf{Cat}/\widehat{\mathcal{C}} = \textbf{Func }(\widehat{\mathcal{C}}) \cong \textbf{DFib}(\mathcal{C})$ where $A$ is any small category and $\textbf{DFib}$ are the discrete fibrations over $\mathcal{C}$. You do this by showing that any functor $A \rightarrow \widehat{\mathcal{C}}/P$ corresponds to a functor $A \rightarrow \widehat{\mathcal{C}}$ and a natural transformation $A \rightarrow 1 \overset{P}{\rightarrow} \mathcal{C}$ (where $1$ is terminal in $\textbf{Cat}$) and hence that the corresponding morphism of fibrations corresponds to a discrete fibration of ${\int_{\mathcal{C}}}P$ and hence that

$$\textbf{Func }(\widehat{\mathcal{C}}/P) \cong \textbf{DFib}(\int_{\mathcal{C}}P) \cong \textbf{Func }(\widehat{\int_{\mathcal{C}}P}) $$

which by universality gives the desired equivalence. Now I have to admit I'm a bit sketchy on the details of this, as it uses many non-obvious properties of fibrations. A more detailed version of this approach was given as a partial answer to this question.

I am more interested in the more elementary approach, i.e. establishing an explicit equivalence between the two categories, so that I can understand, morally, why this equivalence ought to hold. The only natural functor I've been able to find between the two is given firstly by factorizing $$\int_{\mathcal{C}}P \overset{\pi_P}{\rightarrow} \mathcal{C} \overset{y}{\rightarrow} \widehat{\mathcal{C}}$$ through the Yoneda embedding to $\widehat{\int_{\mathcal{C}}P}$ and the canonical colimit functor $L : \widehat{\int_{\mathcal{C}}P} \rightarrow \widehat{\mathcal{C}}$ given since $y \circ \pi_P$ is a functor to a cocomplete category, and then getting to $\widehat{\mathcal{C}}/P$ by pullback $p$ along $P \rightarrow 1$. So the end result is a functor $$ p \circ L: \widehat{\int_{\mathcal{C}}P} \rightarrow \widehat{\mathcal{C}}/P$$ Does this functor provide the desired equivalence? There seems to be significant loss of information at $p$, but on the other hand $L$ does have the property that it makes the two respective Yonneda embeddings commute (i.e. $L \circ y = y \circ \pi_P$), so morally it seems that the information lost at $p$ should be the same amount of information lost by the projection $\pi_P : \int_{\mathcal{C}}P \rightarrow \mathcal{C}$ (which ought to make $p \circ L$ full and faithful; and I think essential surjectivity is straightforward.)

But I also get the sense I am chatting nonsense. If this functor does nothing of the sort then the question, I suppose, is what is the most elementary functor that provides this equivalence?

Any helpful comments will be greatly appreciated both by myself and my bloodshot eyes.


You can verify this equivalence elementarily (without the language of fibrations etc.):

Assume first that $C$ is the category with only one morphism (i.e. the terminal category), so that a presheaf on it is just a set. Then the statement is as follows: If $P$ is a set, then a set $F$ together with a map $F \to P$ is the same as to give a family of sets indexed by $P$. But this is obvious, right? Given $F \to P$, we may look at its fibers $F_s$, where $s$ runs through the elements $s \in P$. Since $F$ is the disjoint union of the $F_s$, it is also clear how to write down the inverse.

For general $C$ it works in the same way, $C$ is just a sort of parametrization.

If $F \in \widehat{C} / P$, i.e. $F$ is a presheaf on $C$ together with a morphism $F \to P$, then define a presheaf $G$ on $\int_C P$ as follows: If $(X,s)$ is an element of $P$, i.e $s \in P(X)$, then let $G(X,s)$ be the fiber of $s$ with respect to $F(X) \to P(X)$.

Conversely, if $G$ is a presheaf on $\int_C P$, then define a presheaf $F$ on $C$ as follows: For $X \in C$ let $F(X) = \coprod_{s \in P(X)} G(X,s)$. We have a natural projection $F(X) \to P(X)$, which gives rise to a morphism $F \to P$.

Now it is straight forward to check that these assignments actually define functors which are pseudo-inverse to each other.


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