it is still useful since many stacks found in nature are of this form, such as the moduli of curves . Also, the differential-geometric analogue of such stacks are called orbifolds. The Etale condition implies the 2-functorsending a scheme to its groupoid of -torsors is representable as a stack over the Etale topology, but the Picard-stack of -torsors (equivalently the category of line bundles) is not representable. Stacks of this form are representable as stacks over the fppf-topology. Another reason for considering the fppf-topology versus the etale topology is over characteristic the Kummer sequenceis exact only as a sequence of fppf sheaves, but not as a sequence of etale sheaves. Defining algebraic stacks over other topologies Using other Grothendieck topologies on gives alternative theories of algebraic stacks which are either not general enough, or don't behave well with respect to exchanging properties from the base of a cover to the total space of a cover. It is useful to recall there is the following hierarchy of generalizationof big topologies on . Structure sheaf The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf on the site . This universal structure sheaf is defined asand the associated structure sheaf on a category fibered in groupoidsis defined aswhere comes from the map of Grothendieck topologies. In particular, this means is lies over , so , then . As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an -scheme for various topologies. For example, if is a category fibered in groupoids over , the structure sheaf for an open subscheme givesso this definition recovers the classic structure sheaf on a scheme. Moreover, for a quotient stack , the structure sheaf this just gives the -invariant sectionsfor in . Examples Classifying stacks Many classifying stacks for algebraic groups are algebraic stacks. In fact, for an algebraic group space over a scheme which is flat of finite presentation, the stack is algebraictheorem 6.1. See also Gerbe Chow group of a stack Cohomology of a stack Quotient stack Sheaf on an algebraic stack Toric stack Artin's criterion Pursuing Stacks Derived algebraic geometry References External links Artin's Axioms https://stacks.math.columbia.edu/tag/07SZ - Look at "Axioms" and "Algebraic stacks" Artin Algebraization and Quotient Stacks - Jarod Alper Papers Applications Mathoverflow threads Do algebraic stacks satisfy fpqc descent? Stacks in the fpqc topology fpqc covers of stacks