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.)
The construction of free bicompletions of small categories under $\kappa$-small limits and colimits (for a regular cardinal $\kappa$), essentially using the theory of sketches, is given in:
Some of the general theory of free bicompletions is sketched (without proof) in the following two articles.
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)
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 May 13, 2024 at 13:32:24. See the history of this page for a list of all contributions to it.