Topos theory is the part of category theory that studies categories which are toposes. This includes in particular Grothendieck toposes, i.e. categories of sheaves.
There are always two ways to think of topos theory: as being
Introductions include
Ross Street, A survey of topos theory (notes for students, 1978) pdf
Tom Leinster, An informal introduction to topos theory (2010)
André Joyal, A crash course in topos theory -- The big picture, lecture series at Topos à l’IHES, November 2015, Paris
André Joyal, Geometric aspects of topos theory in relation with logical doctrines, talk at New Spaces for Mathematics and Physics, IHP Paris 2015 (video recording)
A standard textbook is
This later grew into the more detailed
A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in
A standard textbook on this case is
Introducing even category theory from the scratch while still managing to cover considerable ground, the following textbook is the royal road to topos theory for people with some background in first-order logic:
A gentle basic introduction is
A quick introduction of the basic facts of Grothendieck topos theory is chapter I, “Background in topos theory” in
A survey is in
Colin McLarty, The Uses and Abuses of the History of Topos Theory , Brit. J. Phil. Sci., 41 (1990) (JSTOR) PDF