definition. topos [kostecki2011introduction, 7.1] [tt-004O]

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A topos or elementary topos is a category satisfying one of these equivalent conditions:

• it is a complete category with exponentials and subobject classifier
• it is a complete category with subobject classifier and its power object
• it is a cartesian closed category with equalizers and subobject classifier

Since the completeness of a category with subobject classifier implies its cocompleteness. Thus a topos not only has all finite limits, but also has all finite colimits.

This means that topos is such category which has, in particular,

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