nLab (-1)-category



Higher category theory

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1-categorical presentations


As a degenerate case of the general notion of n-category, (1)(-1)-categories may be understood as truth value. Compare the concept of 0-category (a set) and (−2)-category (which is trivial). The point of (1)(-1)-categories (a kind of negative thinking) is that they complete some patterns in the periodic table of nn-categories. (They also shed light on the theory of homotopy groups and n-stuff.)

For example, there should be a 00-category of (1)(-1)-categories; this is the set of truth values, classically

(1)Cat:={true,false}. (-1)Cat := \{true, false\} \,.

Similarly, (2)(-2)-categories form a (1)(-1)-category (specifically, the true one).

If we equip the category of (1)(-1)-categories with the monoidal structure of conjunction (the logical AND operation), then a category enriched over this is a poset; an enriched groupoid is a set. Notice that this doesn't fit the proper patterns of the periodic table; we see that (1)(-1)-categories work better as either 00-posets or as (1)(-1)-groupoids. Nevertheless, there is no better alternative for the term ‘(1)(-1)-category’.

For an introduction to (1)(-1)-categories and (2)(-2)-categories see page 11 and page 34 of

(1)(-1)-categories and (2)(-2)-categories were discovered (or invented) by James Dolan and Toby Bartels. To witness these concepts in the process of being discovered, read the discussion here:

Last revised on May 29, 2023 at 10:39:52. See the history of this page for a list of all contributions to it.