# Continuous functor

From Encyclopedia of Mathematics

A synonym of the concept "functor commuting with limits" . Let and be categories with limits. A one-place covariant functor is called continuous if for any diagram with an arbitrary small diagram scheme . In more detail, the equality above means the following: If is the limit of the diagram and , , are morphisms occurring in the definition of limit, then is the limit of the diagram .

A functor is continuous if and only if it commutes with products of arbitrary families of objects and kernels of pairs of morphisms. Every basic functor from to the category of sets is continuous.

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#### References

[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |

**How to Cite This Entry:**

Continuous functor. M.Sh. Tsalenko (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Continuous_functor&oldid=18119

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098