$\Pi_f$ for a morphism $f$ between simplicial sets

Question

This nLab article says presheaf categories (including $\mathsf{sSet}$, the category of simplicial sets) are locally cartesian closed. For presheaf categories, it can be proven by use of the category of elements.

How can I compute $\Pi_f \colon \mathsf{sSet}/X \to \mathsf{sSet}/Y$, which is the right adjoint functor of the pullback map, for $f \colon X \to Y$ in $\mathsf{sSet}$? Is there a simple description like the case of $\mathsf{Set}$, the category of sets?

Answer 1

Let $\mathcal{E} = [\mathbb{C}^{\mathrm{op}},\mathsf{Set}]$ be any presheaf topos, keeping in mind the example of $\mathcal{E} = \mathsf{sSet} = [\mathbf{\Delta}^{\mathrm{op}}, \mathsf{Set}]$.

First take $Y=1$, so that we can identify $f : X \to 1$ with $X$. In this case, the functor $\Pi_X : \mathcal{E}/X \to \mathcal{E}$ sends a map $p : A \to X$ to the object of sections $\Gamma_X(A)$. By doing some adjunctiony stuff, this can be characterised by $$(\Gamma_X(A))(c) \cong \{ \varphi : \mathsf{y}(c) \times X \to A\ |\ p \circ \varphi = \pi_2 : A \times X \to X \}$$ for all objects $c$ of $\mathbb{C}$, where $\mathsf{y} : \mathbb{C} \to \mathcal{E}$ is the Yoneda embedding.

• When $\mathcal{E}=\mathsf{Set}$, we can take $\mathbb{C}=\mathbf{1}$, so that this says $$\Gamma_X(A) \cong \{ \varphi : X \to A\ |\ p \circ \varphi = \mathrm{id}_X \}$$ You can check easily that this is equivalent to the classical 'choice function' definition.

• When $\mathcal{E}=\mathsf{sSet}$, this says precisely that $$\Gamma_X(A)_n \cong \{ \varphi : \Delta_n \times X \to A\ |\ p \circ \varphi = \pi_2 : A \times X \to X \}$$

Now note that $(\mathsf{sSet}/Y)/f \cong \mathsf{sSet}/X$, and $\mathsf{sSet}/Y$ is a presheaf topos, since it is equivalent to $\left[ \left( \int Y \right)^{\mathrm{op}}, \mathsf{Set}\right]$, where $\int Y$ is the category of elements of $Y$ (whose objects are pairs $(n,y)$, where $n \in \mathbb{N}$ and $y \in Y_n$).

Thus to find what $\Pi_f : \mathsf{sSet}/X \to \mathsf{sSet}/Y$ does, we can apply the above characterisation in the case $\mathcal{E} = \mathsf{sSet}/Y$, taking $\mathbb{C}=\int Y$, and transport along the isomorphisms $\left[\left(\int Y\right)^{\mathrm{op}},\mathsf{Set}\right] \cong \mathsf{sSet}/Y$ and $\mathcal{E}/f \cong \mathsf{sSet}/X$.

Explicitly: for each $n \in \mathbb{N}$ and $y \in Y_n$, let $$\Gamma_f(A)_{n,y} = \{ \varphi : \Delta_n \times_Y X \to A \text{ over } Y\ |\ p \circ \varphi = \pi_2 : A \times_Y X \to X \}$$ where $\Delta_n \times_Y X$ is the pullback of $y : \Delta_n \to Y$ (which corresponds by Yoneda with $y \in Y_n$) and $f$, and $A \times_Y X$ is the pullback of $p$ and $f$.

Define the simplicial set $\Gamma_f(A)$ by $$\Gamma_f(A)_n = \bigsqcup_{y \in Y} \Gamma_f(A)_{n,y}$$ The map $\Pi_f(p)$ is then given by the projection $\Gamma_f(A) \to Y$, computed componentwise.

Your actual question was, 'is there a simple description like in the case of $\mathsf{Set}$?'. Evidently, the answer is 'no'.