A category $\mathcal{C}$ is said to be a complete category if every small diagram has a limit, that is, a limiting cone exists over every small diagram (diagram such that collections of objects and morphisms are sets).

Of course, in a complete category, a product exists for any given set of objects. Also, a set of morphisms with common domain and codomain has an equalizer. Conversely, we have

in a category $\mathcal{C}$ , if the product exists for an arbitrary set of objects, and the equalizer exists for any pair of morphisms with common domain and codomain, then $\mathcal{C}$ is complete.

Examples

• Set is complete.
• Group is complete.
• Vector Space is complete
• R-module is complete for a given unital ring $R$ .
• Topological Space is complete.

A category $\mathcal{C}$ is said to be finitely complete if every finite diagram (sets of objects and morphisms are finite) has a limit.

A similar sufficient condition for a category $\mathcal{C}$ to be finitely complete is for $\mathcal{C}$ to possess a terminal object and that a pullback exists for every pair of morphisms with common codomain.

Examples

• Any complete category is clearly finitely complete.
• The subcategories of the above examples consisting of all objects with finite cardinality are finitely complete (but not complete).

Remark. The dual notion of a complete category is that of a cocomplete category, and the dual of a finitely complete category is called a finitely cocomplete category.