nLab
pro-left adjoint

Contents

Contents

Idea

For R:π’Ÿβ†’π’žR \colon \mathcal{D} \to \mathcal{C} a functor which preserves finite limits then the left adjoint profunctor π’žβ†’Func(π’Ÿ,∞Grpd) op\mathcal{C}\to Func(\mathcal{D}, \infty Grpd)^{op} factors through the pro-objects Pro(π’Ÿ)Pro(\mathcal{D}). This π’žβ†’Pro(π’Ÿ)\mathcal{C}\to Pro(\mathcal{D}) is the pro-left adjoint to RR. Its extension L:Pro(π’ž)β†’Pro(π’Ÿ)L\colon Pro(\mathcal{C})\to Pro(\mathcal{D}) is a genuine left adjoint to Pro(R)Pro(R), the proadjoint.

Example

Pro-Γ©tale homotopy type

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

HβŸΆΞ“βŸ΅Ξ”βˆžGrpd. \mathbf{H} \stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd \,.

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

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

In general Ξ  pro\Pi_{pro} sends the terminal object to the shape of H\mathbf{H}.

References

  • Marc Hoyois, A note on Γ‰tale 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.