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
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
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
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.
Fairly immediate from the explicit definition above is:
A category $\mathcal{B}$ is equivalent to a free coproduct completion $PSh_{\sqcup}(\mathcal{C})$ for a small category $\mathcal{C}$ if
$\mathcal{C} \xhookrightarrow{\;} \mathcal{B}$ 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.);
each object of $\mathcal{B}$ is a coproduct of objects in $\mathcal{C} \hookrightarrow \mathcal{B}$.
Any free coproduct completion is an extensive category.
The following examples follow as special cases of Prop. .
The category Set is the free coproduct completion of the terminal category.
($G$-sets are the free coproduct completion of $G$-orbits)
Let $G \,\in\, Grp(Set)$ be a discrete group. Write
$G Set$ for its category of group actions on sets,
$G Orbt \xhookrightarrow{\;} G Set$ for its orbit category, the full subcategory on the transitive actions, hence the sets of cosets $G/H$, for subgroups $H \subset G$.
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.
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.