nLab
free coproduct completion

Contents

Contents

Idea

Given a category 𝒞\mathcal{C}, its free coproduct completion (or free sum completion) is the category PSh (𝒞)PSh_{\sqcup}(\mathcal{C}) (often denoted Fam(𝒞)Fam(\mathcal{C}), for families in 𝒞\mathcal{C}) obtained by freely adjoining coproducts of all objects of 𝒞\mathcal{C}.

(The following description is pretty immediate, but see also Hu & Tholen 1995, p. 281, 286.)

An explicit description of PSh (𝒞)PSh_{\sqcup}(\mathcal{C}) is:

  • Its objects are pairs consisting of

    1. an index set II \in Set

    2. an II-indexed set (X i𝒞) iI\big( X_i \in \mathcal{C} \big)_{i \in I} of objects of 𝒞\mathcal{C}.

  • Its morphisms (X i) iI(ϕ i) iI(Y j) jJ\big( X_i \big)_{i \in I} \xrightarrow{ \; ( \phi_i )_{i \in I} \; } \big( Y_j \big)_{j \in J} pairs consisting of

    • a function of index sets f:IJf \,\colon\, I \xrightarrow J.

    • an II-indexed set of morphisms ϕ i:X iY f(j)\phi_i \,\colon\, X_i \xrightarrow{\;} Y_{f(j)} in 𝒞\mathcal{C}.

  • The composition-law and identity morphisms are the evident ones.

Slightly more abstractly, this is equivalently the full subcategory

PSh (𝒞)PSh(𝒞) PSh_{\sqcup}(\mathcal{C}) \xhookrightarrow{\;} PSh(\mathcal{C})

of the category of presheaves over 𝒞\mathcal{C} on those which are coproducts of representables. The latter is the free cocompletion of 𝒞\mathcal{C} under all small colimits.

The Yoneda embedding hence factors through the free coproduct completion

y:𝒞PSh (𝒞)PSh(𝒞) y \;\colon\; \mathcal{C} \xhookrightarrow{\;} PSh_{\sqcup}(\mathcal{C}) \xhookrightarrow{\;} PSh(\mathcal{C})

Notice that the first inclusion here does not preserve coproducts (coproducts are freely adjoined irrespective of whether 𝒞\mathcal{C} already had some coproducts), but the second does. Both inclusions preserve those limits that exist.

Properties

Fairly immediate from the explicit definition above is:

Proposition

A category \mathcal{B} is equivalent to a free coproduct completion PSh (𝒞)PSh_{\sqcup}(\mathcal{C}) for a small category 𝒞\mathcal{C} if

  1. 𝒞\mathcal{C} \xhookrightarrow{\;} \mathcal{B} is a full subcategory of connected objects,

    i.e. X𝒞X \,\in\, \mathcal{C} \hookrightarrow \mathcal{B} means that the hom-functor (X,):Set\mathcal{B}(X,-) \,\colon\, \mathcal{B} \to Set preserves coproducts

    (which when 𝒞\mathcal{C} is extensive means equivalently that if XX is a coproduct, then one of the summands is initial, by this Prop.);

  2. each object of \mathcal{B} is a coproduct of objects in 𝒞\mathcal{C} \hookrightarrow \mathcal{B}.

(Carboni & Vitale 1998, Lem. 42)

Proposition

Any free coproduct completion is an extensive category.

(e.g. Carboni, Lack & Walters 1993)

Examples

The following examples follow as special cases of Prop. .

Example

The category Set is the free coproduct completion of the terminal category.

Example

( G G -sets are the free coproduct completion of G G -orbits)
Let GGrp(Set)G \,\in\, Grp(Set) be a discrete group. Write

Since every G-set XX decomposes as a disjoint union of transitive actions, namely of orbits of elements of XX, this inclusion exhibits GSetG Set as the free coproduct completion of G Orbt.

References

On limits in free coproduct completions:

In the context of regular and exact completions:

In the general context of extensive categories:

Last revised on October 13, 2021 at 05:31:10. See the history of this page for a list of all contributions to it.