Continuous functor
From Encyclopedia of Mathematics
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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.
Comments
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
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