Notes on weak units of group-like 1- and 2-stacks. (English) Zbl 1408.18011
Summary: The weak units of strict monoidal 1- and 2-categories are defined respectively in [J. Kock, Math. Proc. Camb. Philos. Soc. 144, No. 1, 53–76 (2008; Zbl 1147.18004)] and [A. Joyal and J. Kock, Doc. Math. 18, 71–110 (2013; Zbl 1319.18001)]. In this paper, we define them for group-like 1- and 2-stacks. We show that they form a contractible Picard 1- and 2-stack, respectively. We give their cohomological description which provides for these stacks a representation by complexes of sheaves of groups. Later, we extend the discussion to the monoidal case. We consider the (2-)substack of cancelable objects of a monoidal 1-(2-)stack. We observe that this (2-)substack is trivially group-like, its weak units are the same as the weak units of the monoidal 1-(2-)stack, and therefore we can recover the contractibility results in [Kock, loc. cit.] and [Joyal and Kock, loc. cit.] by analysing it.
MSC:
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 14A20 | Generalizations (algebraic spaces, stacks) |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
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