pro-object in nLab

Context

Category theory

Limits and colimits

• Idea
• Definition
• As formal cofiltered limits
• Via spans
• Characterisations
• Examples
• Applications
• Etale homotopy theory.
• Shape theory.
• Related concepts
• References

Definition

(category of pro-objects)

The category of pro-objects of 𝒞\mathcal{C}, according to Grothendieck 1960, Section 2, is the full subcategory of the functor category Func(𝒞,Sets) opFunc(\mathcal{C}, Sets)^{op} (2)

(3)Pro(𝒞)↪Func(𝒞,Sets) op Pro(\mathcal{C}) \xhookrightarrow{\;\;} Func(\mathcal{C}, Sets)^{op}

on those functors which are cofiltered limits of representable functors under the opposite Yoneda embedding (2), hence of the form

(4)lim⟵i∈ℐy op(X i)≔lim⟵(ℐ→X •𝒞→y opFunc(𝒞,Sets) op), \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim} y^{op}(X_i) \;\; \coloneqq \;\; \underset{\longleftarrow}{\lim} \big( \mathcal{I} \xrightarrow{\;X_\bullet\;} \mathcal{C} \xrightarrow{\;y^{op}\;} Func(\mathcal{C}, Sets)^{op} \big) \,,

for ℐ\mathcal{I} a cofiltered category. These objects of Pro(𝒞)Pro(\mathcal{C}) are also called the pro-objects of 𝒞\mathcal{C}.

Remark

Under the identification of the objects of a category with those of its opposite category, the pro-objects (4) are equivalently filtered colimits of presheaves:

lim⟵i∈ℐy op(X i)≃lim⟶i∈ℐy(X i op)≔lim⟶(ℐ op→X • op𝒞 op→yFunc(𝒞,Sets)), \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim} y^{op}(X_i) \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} y(X^{op}_i) \;\; \coloneqq \;\; \underset{\longrightarrow}{\lim} \big( \mathcal{I}^{op} \xrightarrow{\;X^{op}_\bullet\;} \mathcal{C}^{op} \xrightarrow{\;y\;} Func(\mathcal{C}, Sets) \big) \,,

since the limit of a functor is the colimit of its opposite functor.

Lemma

The pro-objects regarded as functors (4) are objectwise on c∈𝒞c \in \mathcal{C} given by

lim⟶i∈ℐy op(X i):c↦lim⟶i∈ℐ𝒞(X i,c). \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} y^{op}(X_i) \;\; \colon \;\; c \;\mapsto\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} \mathcal{C}(X_i, c) \,.

Proof

This is the composite of the following sequence of natural bijections:

(lim⟵i∈ℐy op(X i))(c) ≃(lim⟶i∈ℐy(X i op))(c) ≃lim⟶i∈ℐ((y(X i op))(c)) ≃lim⟶i∈ℐFunc(𝒞,Sets)(y(c),y(X i)) ≃lim⟶i∈ℐ𝒞 op(c,X i) ≃lim⟶i∈ℐ𝒞(X i,c), \begin{aligned} \Big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim} y^{op}(X_i) \Big) (c) & \;\simeq\; \Big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} y(X^{op}_i) \Big) (c) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} \Big( \big(y(X^{op}_i)\big) (c) \Big) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} \; Func(\mathcal{C},Sets) \big( y(c), \, y(X_i) \big) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} \mathcal{C}^{op}(c, X_i) \\ & \;\simeq\; \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim} \mathcal{C}(X_i, c) \mathrlap{\,,} \end{aligned}

where

• the first step is Remark ;

• the second step uses that colimits of presheaves are computed objectwise;

• the third step is the Yoneda lemma;

• the fourth step is the Yoneda embedding (1);

• the fifth step is the definition of the opposite category.

Lemma

(hom-sets between pro-objects as limits of colimits of hom-sets in 𝒞\mathcal{C})
The hom-sets in Pro(𝒞)Pro(\mathcal{C}) (Def. ) are in natural bijection with limits of colimits of hom-sets in 𝒞\mathcal{C}, as follows:

Pro(𝒞)(lim⟵i∈ℐy op(X i),lim⟵j∈𝒥y op(Y j),)≃lim⟵j∈𝒥(lim⟶i∈ℐ𝒞(X i,Y j)) Pro(\mathcal{C}) \Big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} y^{op}(X_i), \, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} y^{op}(Y_j), \Big) \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} \Big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} \mathcal{C} \big( X_i, \, Y_j \big) \Big)

Proof

This is the composite of the following sequence of natural bijections:

Pro(𝒞)(lim⟵i∈ℐy op(X i),lim⟵j∈𝒥y op(Y j),) ≃Func(𝒞,Sets) op(lim⟵i∈ℐy op(X i),lim⟵j∈𝒥y op(Y j),) ≃Func(𝒞,Sets)(lim⟶j∈𝒥y(Y j op),lim⟶i∈ℐy(X i op),) ≃lim⟵j∈𝒥Func(𝒞,Sets)(y(Y j op),lim⟶i∈ℐy(X i op),) ≃lim⟵j∈𝒥((lim⟶i∈ℐy(X i op))(y(Y j op))) ≃lim⟵j∈𝒥(lim⟶i∈ℐ𝒞(X i,Y j)) \begin{aligned} Pro(\mathcal{C}) \Big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} y^{op}(X_i), \, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} y^{op}(Y_j), \Big) & \;\simeq\; Func(\mathcal{C},Sets)^{op} \Big( \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim} y^{op}(X_i), \, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} y^{op}(Y_j), \Big) \\ & \;\simeq\; Func(\mathcal{C},Sets) \Big( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{lim} y(Y^{op}_j), \, \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} y(X^{op}_i), \Big) \\ & \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} \; Func(\mathcal{C},Sets) \Big( y(Y^{op}_j), \, \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} y(X^{op}_i), \Big) \\ & \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} \Big( \big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} y(X^{op}_i) \big) \big( y(Y^{op}_j) \big) \Big) \\ & \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim} \Big( \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim} \mathcal{C} \big( X_i, \, Y_j \big) \Big) \end{aligned}

Here

• the first step is the fully faithfulness of the defining full subcategory-inclusion (3);

• the second step is the definition of the opposite category, using Remark ;

• the third step uses that hom-functors preserve limits, which means that 𝒦 op→𝒦(−,k)Sets\mathcal{K}^{op} \xrightarrow{\;\mathcal{K}(-,k)\;} Sets takes colimits in 𝒦\mathcal{K} to limits in Sets, here for 𝒦=Func(𝒞,Sets)\mathcal{K} = Func(\mathcal{C},Sets);

• the fourth step is the Yoneda lemma;

• the fifth step is Lemma .

Proposition

Let 𝒞\mathcal{C} be a category, and let 𝒜\mathcal{A} be a category with cofiltered limits. Suppose that we have a fully faithful functor i:𝒞→𝒜i: \mathcal{C} \rightarrow \mathcal{A} which lands in cocompact objects. Then lim 𝒜∘(i∘):Pro(𝒞)→𝒜lim_{\mathcal{A}} \circ (i \circ) : Pro(\mathcal{C}) \to \mathcal{A} is fully faithful, and hence defines an equivalence onto its image.

Proof

Let F:𝒟→𝒞F:\mathcal{D} \to \mathcal{C} and G:ℰ→𝒞G:\mathcal{E} \to \mathcal{C} be pro-objects. We then have a sequence of (natural) bijections:

Hom Pro(𝒞)(F,G)Hom_{Pro(\mathcal{C})}(F,G)

=lime:ℰcolimd:𝒟Hom 𝒞(Fd,Ge)= \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{C}}(F d, G e)

≅lime:ℰcolimd:𝒟Hom 𝒜(i(Fd),i(Ge))\cong \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{A}}(i (F d), i (G e))

≅lime:ℰHom 𝒜(limd:𝒟(i∘F)d,i(Ge))\cong \underset{e:\mathcal{E}}{lim} \, Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, i (G e))

≅Hom 𝒜(limd:𝒟(i∘F)d,lime:ℰ(i∘G)e)\cong Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, \underset{e:\mathcal{E}}{lim} (i \circ G) e)

=Hom 𝒜(lim(i∘F),lim(i∘G))= Hom_{\mathcal{A}}(lim (i \circ F), lim (i \circ G))

With these bijections being by definition of pro-object morphisms, fully faithfulness, cocompactness of i(Ge)i (G e), definition of a limit, and definition respectively.