What is a stack?

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski for explaining the material.

Definition: A groupoid is a category where all the morphisms are invertible. Alternatively, a groupoid is a set of objects $A$, a set of morphisms $\Gamma$, and a collection of maps as described by the diagram below.

To describe stacks, we compare them with sheaves. Both start out with a space $X$ and a topology on it, so that we may consider open sets $U$.

In addition to these conditions, there is a triple intersection condition for stacks that does not have an analogous one in sheaves. It is given by:

for every $U_i,U_j,U_k$ and $s_i,s_j,s_k\in \widehat{\mathcal F}(U_i), \widehat{\mathcal F}(U_j), \widehat{\mathcal F}(U_k)$, respectively, such that there exist isomorphisms $\varphi_{ij}:s_i|_{U_i\cap U_j}\to s_j|_{U_i\cap U_j}$, $\varphi_{jk}:s_j|_{U_j\cap U_k}\to s_k|_{U_j\cap U_k}$, and $\varphi_{ik}:s_i|_{U_i\cap U_k}\to s_k|_{U_i\cap U_k}$, the diagram below commutes:

Example: A Hopf algebroid may be viewed as a functor into groupoids, so that with the appropriate topology, it becomes a stack. Indeed, by definition a Hopf algebroid is a pair of $k$-algebras $(A,\Gamma)$ such that $(\Spec(A),\Spec(\Gamma))$ is a groupoid object in affine schemes, or in other words, is a functor from affine schemes into groupoids.

References: nLab (article on groupoids)