Contents

category theory

# Contents

## Idea

For $R \colon \mathcal{D} \to \mathcal{C}$ a functor which preserves finite limits then the left adjoint profunctor $\mathcal{C}\to Func(\mathcal{D}, \infty Grpd)^{op}$ factors through the pro-objects $Pro(\mathcal{D})$. This $\mathcal{C}\to Pro(\mathcal{D})$ is 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.

## 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 $\mathbf{H}$.

## References

• Marc Hoyois, A note on Étale homotopy, 2013 (pdf)
• E. Giuli, Pro-reflective subcategories, J. Pure Appl. Algebra 33 (1984) 19–29.

Last revised on June 14, 2019 at 09:04:43. See the history of this page for a list of all contributions to it.