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:
a preordered set (or proset, for short) is
a partially ordered set (or poset, for short) is
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:
Conversely, since the notion of skeletal category implies that of a strict category, one may say that posets are the skeletal truth value-/thin-/-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.
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Last revised on July 23, 2022 at 09:20:04. See the history of this page for a list of all contributions to it.