Subject: categories: bounded = accessible

From: Jiri Adamek <[email-masked]>
Date: Wed, 29 Mar 2000 10:43:24 +0200 (MET DST)

In the community investigating coalgebras of functors F: K-> K (i.e.,
comma-categories Id/F ) the following condition in case  K = Set is 
quite often used:

Definition: A set functor is called bounded if there exists a cardinal
k  such that for every element  a  of every coalgebra there exists
a subcoalgebra of cardinality less than  k  containing  a . ("Bounded
at  k " is used  for the given  k.)

So I wondered how this is related to accessibility, and here is the answer:

Proposition: A set functor is bounded iff it is accessible.

In fact, if  k  is an infinite, regular cardinal, then k-accessible = 
bounded at  k . For, suppose that  F  is bounded at  k , then we
prove that in the usual category of elements of  F  for every object  
(A,a) there exists a morhpism  m: (B,b) -> (A,a) with  card B  < k .
It then follows easily that  F  is k-accessible (see e.g. 2.17 in the book
of Rosicky and myself). We can assume  A  nonempty, and choose  x  in  A.
Let p:A -> FA be the coalgebra given by the constant map to  a . There
exists a subcoalgebra m: (B,q) -> (A,p) for some  q: B -> FB with
card B < k  such that  x = m(y) for an element  y  of  B . Then                 b = q(y) clearly fulfills a = Fm(b), as requiered.
Conversely, suppose that  F  is k-accessible. Given a coalgebra  
p: A -> FA , consider the collection  m_i: A_i -> A  of all subsets
of cardinality less than  k  . Since  F  preserves k-directed unions,
for each  i  there exists an i' such that m_i  is contained in  m_i'
and  the image of p.m_i  is contained in that of Fm_i' . After
iterating i -> i' -> (i')' ... k times we find a  j  such that
m_i  is contained in  m_j , and  the image of  p.m_j  is contained in that
of Fm_j. The latter means that  p  can be restricted to  p_j: A_j -> FA_j
forming a subcoalgebra of the coalgebra above. Since every element of
FA  is contained in the image of some  Fm_i , this proves that  F  is
bounded at  k .

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