170 VIII. HOMOTOPY COLIMIT AND LIMIT FUNCTORS AND HOMOTOPICAL ONES
while 49.1 and
51.4(ii)/
imply that
(ii) a sufficient condition for the homotopical cocompleteness (resp. complete-
ness) of X, i.e. the existence of homotopical colimit (resp. limit) systems
on X,
is that
(ii)' there exists a terminal (resp. initial) Kan extension along X^ (resp.
X^ ) which is left (resp. right) deformable (48.6)
Moreover it follows from 49.1 and 51.4(i) and (iii) that, if X is cocomplete (resp.
complete), then
(iii) there exist homotopical colimit (resp. limit) systems on X
iff
(iii)' there exist homotopy colimit (resp. limit) systems on X (49.2)
and
(iii)" if (iii) and (iii)' hold then the homotopy colimit (resp. limit) systems on
X are exactly the
colinrcat'-
(resp.
Ymr-cat'-)
presentations (b\A(\v)) of
the homotopical colimit (resp. limit) systems on X
so that
(iv) for cocomplete (resp. complete) homotopical categories the above notion of
homotopical cocompleteness (resp. completeness) coincides with the one
we considered in §49.
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