Michael Shulman
posetal object

An object AA in a 2-category KK is posetal if the category K(X,A)K(X,A) is a preorder (equivalent to a poset) for all objects XX of KK. Posetal objects are also called (0,1)-truncated objects since K(X,A)K(X,A) is a (0,1)-category (a poset).

More explicitly, AA is posetal iff any parallel 2-cells α,β:fg:XA\alpha,\beta:f \;\rightrightarrows\; g: X\;\rightrightarrows\; A are equal. If KK has finite limits, this is equivalent to saying that A 2A pprA^{\mathbf{2}} \to A^{ppr} is an equivalence, where 2\mathbf{2} is the “walking arrow” and pprppr is the “walking parallel pair of arrows.”

We write pos(K)pos(K) for the full sub-2-category of posetal objects; it is a (1,2)-category and is closed under limits in KK.

Revised on January 30, 2009 20:10:20 by Mike Shulman (75.3.140.11)