Contents

category theory

# Contents

## Idea

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

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

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

• Its objects are pairs consisting of

1. an index set $I \in$ Set

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

• Its morphisms $\big( X_i \big)_{i \in I} \xrightarrow{ \; ( \phi_i )_{i \in I} \; } \big( Y_j \big)_{j \in J}$ pairs consisting of

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

• an $I$-indexed set of morphisms $\phi_i \,\colon\, X_i \xrightarrow{\;} Y_{f(j)}$ in $\mathcal{C}$.

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

Slightly more abstractly, this is equivalently the full subcategory

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

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

The Yoneda embedding hence factors through the free coproduct completion

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

Notice that the first inclusion here does not preserve coproducts (coproducts are freely adjoined irrespective of whether $\mathcal{C}$ 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 $PSh_{\sqcup}(\mathcal{C})$ for a small category $\mathcal{C}$ if

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

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

(which when $\mathcal{C}$ is extensive means equivalently that if $X$ 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 $\mathcal{C} \hookrightarrow \mathcal{B}$.

(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

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

Since every G-set $X$ decomposes as a disjoint union of transitive actions, namely of orbits of elements of $X$, this inclusion exhibits $G Set$ 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.