nLab free bicompletion

Idea

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.

Examples

• 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:

  $$C(\bot, \bot) = 1 \qquad C(\bot, c) = 1 \qquad C(c, \top) = 1 \qquad C(\bot, \top) = 1 \qquad C(\top, \top) = 1 \qquad$$

  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.)

Related concepts

• free cocompletion, free completion

• Cauchy completion

References

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:

• H. Kaphengst and H. Reichel, Finite limit-colimit completions of small categories, Algebraische Modelle, Kategorien und Gruppoide, pp. 21–33.

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:

• Hongde Hu, Contractible coherence spaces and maximal maps, Electronic Notes in Theoretical Computer Science 20 (1999): 309-319. (link)

A different approach for discrete categories is described in:

• Dominic J. D. Hughes?, A canonical graphical syntax for non-empty finite products and sums, Technical report (2002).

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.