Contents

category theory

# Contents

## Idea

For $R \,\colon\, \mathcal{D} \to \mathcal{C}$ a functor which preserves finite limits, the left adjoint profunctor

(1)$\array{ \mathcal{C} &\longrightarrow& Func(\mathcal{D}, Set)^{op} \\ c &\mapsto& \Big( d \mapsto \mathcal{C} \big( c ,\, R(d) \big) \Big) }$

factors through the pro-objects $Pro(\mathcal{D})$. This $\mathcal{C}\to Pro(\mathcal{D})$ is called the pro-left adjoint to $R$. Its extension $L\colon Pro(\mathcal{C})\to Pro(\mathcal{D})$ is a genuine left adjoint to $Pro(R)$, the proadjoint.

The analogous definition applies in the generality of $(\infty,1)$-category theory – if Set is replaced by ∞Grpd in (1) – to yield the notion of pro left-adjoint (∞,1)-functors, generalizing left adjoint (∞,1)-functors.

## Example

### Pro-étale homotopy type

Consider any (∞,1)-sheaf ∞-topos with its global sections geometric morphism

$\mathbf{H} \stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd \,.$

If here the constant ∞-stack inverse image $\Delta$ does have a further left adjoint (∞,1)-functor $\Pi$ (so that $\mathbf{H}$ is a locally ∞-connected (∞,1)-topos) then $\Pi$ may be understood as taking any object to its fundamental ∞-groupoid or geometric realization homotopy type.

In general $\Pi$ does not exist, but the pro-left adjoint $\Pi_{pro}$ of $\Delta$ may always be formed. This sends any object to what in the case of the $\infty$-topos over an étale site is called its étale homotopy type.

In general $\Pi_{pro}$ sends the terminal object to the shape of the $(\infty,1)$-topos of $\mathbf{H}$.

Discussion in the context of shape of an $(\infty,1)$-topos: