nLab
pro-object in an (infinity,1)-category

Contents

Context

(,1)(\infty,1)-Category theory

Limits and colimits

Contents

Definition

For small (∞,1)-categories

For CC a small (∞,1)-category and κ\kappa a regular cardinal, the (,1)(\infty,1)-category of κ\kappa-pro-objects in CC is the opposite (∞,1)-category of ind-objects in the opposite of CC:

Pro κ(C):=(Ind κ(C op)) op. Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op} \,.

For κ=ω\kappa = \omega we write just Pro(C)Pro(C).

By the properties listed there, if CC has all κ\kappa-small (∞,1)-limits then this is equivalent to

Lex κ(C,Grpd) opFunc(C,Grpd) op \cdots \simeq Lex_\kappa(C, \infty Grpd)^{op} \subset Func(C,\infty Grpd)^{op}

the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve these limits.

For large (∞,1)-categories

Generalizing this definition:

Definition

If 𝒞\mathcal{C} is a possibly non-small but accessible ( , 1 ) (\infty,1) -category with finite ( , 1 ) (\infty,1) -limits, we write :

Pro(𝒞)AccLexFunc(C,Grpd) op Pro(\mathcal{C}) \;\coloneqq\; AccLexFunc(C ,\, \infty Grpd)^{op}

for the opposite \infty -category of \infty -functors 𝒞\mathcal{C} \to ∞Grpd which are

  1. left exact in that they preserve finite ( , 1 ) (\infty,1) -limits;

  2. accessible.

(Lurie 2009, Def. 7.1.6.1)
Remark

Yet more generally, if CC is just locally small, then one can take Pro(𝒞)Pro(\mathcal{C}) to be the \infty-category of small functors whose Grothendieck construction is cofiltered?. Equivalently, Pro(mathca;C)Pro(\mathca;{C}) consists of the functors which are “small cofiltered limits of representables”.

Remark

For 𝒞\mathcal{C} a possibly large but accessible ( , 1 ) (\infty,1) -category which is tensored over ∞Grpd, in that there is a natural equivalence

Grpd(S,𝒞(X,Y))Grpd(SX,Y) \infty Grpd \big( S ,\, \mathcal{C}(X,Y) \big) \;\; \simeq \;\; \infty Grpd \big( S \cdot X ,\, Y \big)

then it is still a full sub- ( , 1 ) (\infty,1) -category of its pro-objects, in the sense of Def. , via the usual ( , 1 ) (\infty,1) -Yoneda embedding:

(1)𝒞 Pro(𝒞) c 𝒞(c,) \array{ \mathcal{C} &\xhookrightarrow{\phantom{---}}& Pro(\mathcal{C}) \\ c &\mapsto& \mathcal{C}(c,-) }

This is because the above tensoring means that 𝒞(c,)\mathcal{C}(c,-) is a right\,adjoint ( , 1 ) (\infty,1) -functor and these preserve limits and are accessible (by this Prop.)

References

The large version is mentioned in:

See also:

Last revised on October 4, 2021 at 04:59:34. See the history of this page for a list of all contributions to it.