Density Theorem via Weighted Colimits: Confusing Variance

Question

Here is a question concerning Emily Riehl's Categorical Homotopy Theory about weighted colimits. There is some $(-)^{\mathrm{op}}$-story confusing me. Let me begin by recalling the definition of (unenriched) weighted colimits and briefly describing the relevant results.

• For functors $F: \mathscr{C} \to \mathscr{M}, W : \mathscr{C}^{\mathrm{op}} \to \mathbf{Set}$ the colimit of $F$ weighted by $W$ is $\operatorname{colim}^W F \in \mathscr{M}$ representing the functor $\mathbf{Set}^{\mathscr{C}^{\mathrm{op}}}(W, \mathscr{M}(F-,m))$.
• One shows $\operatorname{colim}^W F \cong \int^{c \in \mathscr{C}} Wc \cdot Fc$ where $\cdot : \mathbf{Set} \times \mathscr{M} \to \mathscr{M}$ denotes the copower/tensor [Riehl, p. 84].
• One shows $\operatorname{colim}^W F \cong \operatorname{colim}_{\mathbf{el}(W)} F \Sigma$ where $\Sigma: \mathbf{el}(W) \to \mathscr{C}$ is the category of elements [Riehl, 7.2.4].

I'm confused about Example 7.2.7 in two spots:

1. Since $\mathbf{el}(\mathscr{C}(-,c)) \simeq \mathscr{C}_{/c}$ one obtains $Fc \cong \operatorname{colim}^{\mathscr{C}(-,c)}F \cong \int^{x \in \mathscr{C}} \mathscr{C}(x,c) \cdot Fx$. It is now stated that in case $F: \mathscr{C} \to \mathbf{Set}$ this formula is symmetric so that $$ Fc \cong \int^{x \in \mathscr{C}} Fx \cdot \mathscr{C}(x,c) \cong \operatorname{colim}^F \mathscr{C}(-,c).$$ I'm confused: coends take functors with domain $\mathscr{C}^{\mathrm{op}} \times \mathscr{C}$ but this is not symmetric, one has to remember the variance. So what is meant here?
2. In the same example, Emily concludes $$F \cong \operatorname{colim}_{\mathbf{el}(F)} \mathscr{C}(c,-).$$ Now even the variance of the representable functor has changed. What is going on here? How did $\mathscr{C}(-,c)$ become $\mathscr{C}(c,-)$?

Answer 1

Point 2. is a matter of unfortunate notation on Emily's side. We have $$ Fc\cong \mathrm{colim}^F\,\mathscr{C}(-,c), $$ where the colimit is taken over the first entry (so the ''$-$'' sign here denotes the input over which you take the colimit), and we need to consider $F$ as a functor $(\mathscr{C}^\mathrm{op})^\mathrm{op}\to\mathbf{Set}$. Using the contravariant Grothendieck construction to build $\mathbf{el}(F)\to\mathscr{C}^\mathrm{op}$, your third bullet then shows that $$ Fc\cong\mathrm{colim}(\mathbf{el}(F)\to\mathscr{C}^\mathrm{op}\xrightarrow{\mathscr{C}(-,c)}\mathbf{Set}). $$ Now we can write this as $Fc\cong\mathrm{colim}_{(x\in\mathscr{C},a\in F(x))\in\mathbf{el}(F)}\,\mathscr{C}(x,c)$. This isomorphism is natural in $c$, and therefore we get $F(+)\cong\mathrm{colim}_{(x,a)\in\mathbf{el}(F)}\mathscr{C}(x,+)$. Shortening the notation, we could write this as $F\cong\mathrm{colim}_{\mathbf{el}(F)}\,\mathscr{C}(x,-)$.

However, as you note, the colimit is now taken over $x$, and unlike before the symbol ''$-$'' has nothing to do with the colimit. Moreover, Emily writes $c$ instead of $x$, which is confusing since just before it $c$ played a different role. The $c$ Emily writes at this moment has nothing to do with the $c$ before that.