Given a small category $S$, we may consider freely adjoining (small) colimits to $S$, giving the free cocompletion of $S$. Concretely, this is the category of presheaves on $S$. Dually, we may consider freely adjoining (small) limits to $S$, giving the free completion of $S$. Concretely, this is the category of copresheaves on $S$.
Alternatively, we may consider freely adjoining both (small) limits and colimits to $S$ simultaneously. (Note that this means there will be no distributivity of limits and colimits.) This is known as the free bicompletion of $S$.
Unlike the free completion or free cocompletion, it is not known whether a concrete description of the free bicompletion of a small category exists.
Abstractly, the free bicompletion construction is the coproduct of the two 2-monads associated to the free completion and the free cocompletion.
The Cauchy completion of a category is its free bicompletion under absolute limits and absolute colimits.
The free bicompletion of a category $C$ under an initial object and terminal object is given by adjoining two objects $\bot$ and $\top$ and defining:
The composition laws are uniquely determined. (This is equivalently the result of completing $C$ under a terminal object, and then cocompleting under an initial object.)
André Joyal. Free bicomplete categories. Mathematical Reports of the Academy of Sciences 17.5 (1995): 219-224. (link)
André Joyal. Free bicompletion of enriched categories. Mathematical Reports of the Academy of Sciences 17.5 (1995): 213-218. (link)
The above references do not contain proofs. For a sketch of the existence of free bicompletions of categories, see this MathOverflow question.
The free bicompletion of a category $C$ under nonempty products and nonempty coproducts is characterised as the category of nonempty $C$-valued contractible coherence spaces and $C$-maximal maps in:
A different approach for discrete categories is described in:
The free completion of a category $C$ under products, coproducts, and a zero object is characterised as the category of contractible $C$-coherence spaces in:
Hongde Hu and André Joyal, Coherence completions of categories and their enriched softness, Electronic Notes in Theoretical Computer Science 6 (1997): 174-190.
Hongde Hu and André Joyal, Coherence completions of categories, Theoretical Computer Science 227.1-2 (1999): 153-184.
Last revised on February 15, 2024 at 20:44:00. See the history of this page for a list of all contributions to it.