nLab free bicompletion

Contents

Context

Category theory

Limits and colimits

Contents

Idea

Given a small category SS, we may consider freely adjoining (small) colimits to SS, giving the free cocompletion of SS. Concretely, this is the category of presheaves on SS. Dually, we may consider freely adjoining (small) limits to SS, giving the free completion of SS. Concretely, this is the category of copresheaves on SS.

Alternatively, we may consider freely adjoining both (small) limits and colimits to SS simultaneously. (Note that this means there will be no distributivity of limits and colimits.) This is known as the free bicompletion of SS.

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 CC under an initial object and terminal object is given by adjoining two objects \bot and \top and defining:

    C(,)=1C(,c)=1C(c,)=1C(,)=1C(,)=1 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 CC under a terminal object, and then cocompleting under an initial object.)

References

  • 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 CC under nonempty products and nonempty coproducts is characterised as the category of nonempty CC-valued contractible coherence spaces and CC-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 CC under products, coproducts, and a zero object is characterised as the category of contractible CC-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.