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, the left adjoint profunctor

(1)π’ž ⟢ Func(π’Ÿ,Set) op c ↦ (dβ†¦π’ž(c,R(d))) \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(π’Ÿ)Pro(\mathcal{D}). This π’žβ†’Pro(π’Ÿ)\mathcal{C}\to Pro(\mathcal{D}) is called 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.

The analogous definition applies in the generality of ( ∞ , 1 ) (\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-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 Π pro\Pi_{pro} sends the terminal object to the shape of the ( ∞ , 1 ) (\infty,1) -topos of H\mathbf{H}.

References

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

See also:

  • E. Giuli, Pro-reflective subcategories, J. Pure Appl. Algebra 33 (1984) 19–29.

Last revised on October 3, 2021 at 04:36:44. See the history of this page for a list of all contributions to it.