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A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to consider sheaves that have values in categories.

Clearly, one did something not so smart here, because the category of (small) categories is in fact a 2-category, and one ignored this fact by having the sheaf take values in the category of categories.

This must have been the reason why people invented stacks - to have the right geometric/algebraic object that takes this fact into account. However, I am totally unclear how the axioms in the definition of a stack actually resemble this fact. So I am looking for a clarification of the exact difference of a sheaf with values in categories to a stack.

Please complete the following sentences, if possible:

A stack is a sheaf valued in categories plus the following extra structure...

/

A stack is a sheaf valued in categories except that one replaces the axiom X with the following weaker version...

/

A sheaf valued in categories is a stack with the following additional property...

/

etc.

Kofi
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    Perhaps your confusion stems from the formulation of stacks in terms of fibred categories instead of indexed categories. The definitions are otherwise very much analogous – the key point is that the category of descent data can be defined by a 2-limit. – Zhen Lin Aug 14 '14 at 08:05
  • Zhen Lin is right. In order to grasp the definition of a stack, it's best to first recast the definition of a sheaf in a language which is closer to the language employed by stacks: A sheaf on a topological space $X$ consists of a set $E$ together with a map $f : E \to \mathcal{O}(X)$ (where $\mathcal{O}(X)$ is the set of open subsets of $X$) and additional data ... such that ... The idea is that what would regularly be the value of the sheaf on an open subset $U$ is now the preimage $f^{-1}[{U}]$. – Ingo Blechschmidt Sep 02 '18 at 12:26

1 Answers1

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A category-valued sheaf transports a covering to a limit.

A stack transports a covering to a 2-limit.

As you say, the main difference is that a stack uses the 2-categorical structure of the 2-category of small categories.

Martin Brandenburg
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