Definition. topos [kostecki2011introduction, 7.1] [tt-004O]
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    A topos, or elementary topos, is a category with all finite limits, exponentials, and a subobject classifier. Equivalently, exponentials may be replaced by power objects. See [maclane1992sheaves, sec. IV.1, pp. 161--163].

    Every elementary topos also has all finite colimits; see [maclane1992sheaves, sec. IV.5, pp. 180--184]. Arbitrary small limits and colimits are not part of the elementary definition. They do exist in a Grothendieck topos of sheaves on a site; see [maclane1992sheaves, sec. III.6, pp. 134--135].

    Thus an elementary topos has, in particular,

    1. terminal object
    2. equalizers
    3. pullbacks
    4. all other finite limits
    5. exponential objects
    6. subobject classifier
    7. all finite colimits