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Another good place to look are the notes of a course on stacks by Betrand Toen. Here's the link: http://www.math.univ-toulouse.fr/~toen/m2.html I think they pretty much do exactly what you are looking for. (Edit: new link http://ens.math.univ-montp2.fr/~toen/m2.html - Toen has moved)

Here's the quick summary: You will want to read section 1 of Cours 2 where the term geometric context is defined. It's basically a category with a Grothendieck topology with a fixed class of morphisms that you call geometric. The main example are commutative rings with the etale topology or the smooth topology. This induces coverings in the presheaf category in the standard way.

Then skip straight to Cours 5. Although you said that you are comfortable with descent this section is definitely worth a close look. It introduces a homotopy theory on the category of groupoids and shows that there always is a weakly equivalent groupoid such that your functor becomes strict. It then reformulates descent via homotopy limits. The upshot is a nice category of stacks, Definition 4.4.

Then jump straight to Cours 8, Definition 1.4. and you've got algebraic stacks. The only point where you will need schemes or algebraic spaces is for representable morphisms, but judging from the remark after the definition you can get around that as well.

2 added 1049 characters in body

Another good place to look are the notes of a course on stacks by Betrand Toen. Here's the link: http://www.math.univ-toulouse.fr/~toen/m2.html I think they pretty much do exactly what you are looking for.

Here's the quick summary: You will want to read section 1 of Cours 2 where the term geometric context is defined. It's basically a category with a Grothendieck topology with a fixed class of morphisms that you call geometric. The main example are commutative rings with the etale topology or the smooth topology. This induces coverings in the presheaf category in the standard way.

Then skip straight to Cours 5. Although you said that you are comfortable with descent this section is definitely worth a close look. It introduces a homotopy theory on the category of groupoids and shows that there always is a weakly equivalent groupoid such that your functor becomes strict. It then reformulates descent via homotopy limits. The upshot is a nice category of stacks, Definition 4.4.

Then jump straight to Cours 8, Definition 1.4. and you've got algebraic stacks. The only point where you will need schemes or algebraic spaces is for representable morphisms, but judging from the remark after the definition you can get around that as well.