Commentationes Mathematicae Universitatis Carolinae
Horst Herrlich
Reflective Mac Neille completions of fibre-small categories need not be
fibre-small
Commentationes Mathematicae Universitatis Carolinae, Vol. 19 (1978), No. 1, 147--149
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OOMMENTATIONES MATHEMATICA1 UNI¥1RSITATIS CABOLTNAK
19,1 (1978)
REFI£CTIYE MAC NEILLE COMPLETIONS OP FIBRE-SMALL CATEGORIES
NEED NOT BE FIBRE-SMALL
Horst HERRLICH, Bremen
Abstract; See Title
Key words; Initial completion, universal initial completion, Mac Neille completion, semi-topological functor,
topologically-algebraic functor, fibre-smallness, strong
fibre-smallness.
AMS; 18J330, 18A35
Ref. 2.; 2.726.23
Mac Neille completions have been defined in C2]. Categories having reflective Mac Neille completions, have been
characterized by Wischnewsky and Tholen C81, Hoffmann [5lf
Aiajaek [13 and by Herrlich and Strecker C33 as those (A,U),
for which U is semi-topological. Categories, having fibresmall Mac Neille completions, have been characterized by
Adamek C11 and by Herrlich and Strecker C 41 as those (A,U),
which are strongly fibre-small. The title statement provides
a negative answer to a problem posed by Adajiek til, p« 22.
The example is as follows.
Let ( H , 6 ) be a large complete lattice. Let X be the
following category:
Objects: X Q , B^ , C^ , D^ for all oc € It
Morphisms:
- 147
f.. j» D
D .
*•
and all
r
<&<(*)
identities,
Composition is uniquely determined by the fact that morphism
classes horn (XfI) contain at most one element.
Let 4 be the subcategory of X, obtained by removing 3L.,
i d x , a l l r«-, t P*
f
Q*» * * »
and
a11 h
with
«cfB
and
fl > °^ •
let U : A — * X be the embedding functor. Then U is not only semi-topological, but even topologicaliy-algebraic in the sense
of Y.H. Hong C73 and S.S. Hong £63, i.e. any U-source has some (generating, initial)-factorization
•A^rjj^ «
x—в^üå
üm
i
m
i
as indicated by the following table:
X
-^UA^
g
(1)
X = B^ anđ 4 ŕ i ) i t I | n {{r^ïvih^ìßx*})
= ф
(2)
X « B ^ a n d - f f i l i ^ I I rч Цr^lvih^ìßxsoí)
фф
(3)
X = CÄ and 4 f i Ü € I I m ÍІІÛQJУÍ
(4)
X = C^ anđ I f ^ ţ i c l l r% UiUQjУІҲ^ßìß^mi)
(5)
X * X Q ,f
шuўi^йSílg^ ш í f j i * Ш
g
(6)
X * D^, f « в u p ^ c Л І đ ^ * 4 f i l i t Щ
ă
я
"-w
г
id
\cß\ß>ФІ)фф
«ф
c«
ą*
Hence, by C3l f (A f U) has not only a reflective Mac Neille
148 -
*
r
fiГ
completion but even a reflective universal initial completion. Since the g« : X Q — * UD« are pairwise non-equivalent
semi-final U-morphismsf (A,U) is not strongly fibre-small.
Hence its Mac Neille completion is not fibre-small.
R e f e r e n c e s
[11
J. A M M B K ; Fibre completions of concrete categories,
preprint.
[21
H. HERRLICH: Initial completions, Math. Z. 1^0, 101110 (1976).
[31
H. HERRLICH, G.E. STHECKER: Semi-universal maps and universal initial completions, preprint.
C41 H. HERRLICH, G.E. SflECKER; Small-fibred initial completions, preprint.
[5] R.E. HOFFMANN: Note on semi-topological functors, preprint .
[61
S.S. HONG: Categories in which every mono-source is
initial, Kyungpook Math. J. lg, 133-139 (1975).
[71
I.H. HONG; Studies on categories of universal topological algebras, fhesis.McMaster University 1974.
[81
W. THOUSN: Semi-topological functors I., preprint.
F.S. Mathematik
Universitlt Bremen
28 Bremen
Fed. Rep. Germany
(Oblatum 21.12. 1977)
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