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

Definition

For small (∞,1)-categories

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

$$Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op} \,.$$

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

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

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

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

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

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

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

2. accessible.

(Lurie 2009, Def. 7.1.6.1)

Remark

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

Remark

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

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

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

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

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

Related concepts

• ind-object / ind-object in an (∞,1)-category

• pro-object

• shape of an $\infty$-topos

References

The large version is mentioned in:

• Jacob Lurie, around def. 7.1.6.1 of: Higher Topos Theory, 2009

See also:

• Ilan Barnea, Yonatan Harpaz, Geoffroy Horel, Pro-categories in homotopy theory (arXiv:1507.01564).