nLab relation between preorders and (0,1)-categories



General idea

The notions of posets and prosets (in order theory), and of thin categories, truth-value-enriched categories and (0,1)-categories (in category theory and higher category theory) are all closely related and yet possibly subtly but crucially different (see also at category theory vs order theory):

In category theory-language:

Now, while every poset is in particular a proset, a proset need not be isomorphic (namely as a strict category) to a poset.

On the other hand, if we disregard strictness and assume the axiom of choice, then every proset is equivalent as a category to a poset: This is the statement that every category has a skeleton.

Finally, if prosets are regarded as actual categories this way (i.e. disregarding their strictness) then as such they are, equivalently:

  1. thin categories,

  2. truth value-enriched categories,

  3. (0,1)-categories.

Conversely, since the notion of skeletal category implies that of a strict category, one may say that posets are the skeletal truth value-/thin-/ ( 0 , 1 ) (0,1) -categories.

In doing so, one should just keep in mind that in various contexts (such as in various foundations of mathematics) strictness may matter and/or the axiom of choice may fail, in which case prosets are not equivalently posets and neither may be really equivalent to thin categories/(0,1)-categories.

Finally, beware that even these notions of categories may not always be equivalent: A context which uses the terminology “(0,1)-category” is less likely to even consider the notion of a strict category than a context using the terminology “thin category” or “truth value-enriched category”, which might even regard strict categories as the default notion.

In Type theory


Last revised on July 23, 2022 at 09:20:04. See the history of this page for a list of all contributions to it.