nLab thread

A thread is an element of a cofiltered limit of topological spaces (usually studied in the generality of projective spectra?) or (in a more rarely used terminology) of a cofiltered limit of sets.

Thus let F:D opSetF: D^{op}\to Set be a functor where DD is a small filtered category (for example, a directed set). Then limF=lim dDF(d)lim F = lim_{d\in D} F(d) consists of families (s d) dD(s_d)_{d\in D} where s dF(d)s_d \in F(d) and for every morphism δ:de\delta:d\to e in DD, F(δ)(s e)=s dF(\delta)(s_e) = s_d. Such families are called threads.

If F:D opTopF: D^{op}\to Top is a functor where DD is a small filtered category then limFlim F has the same underlying set (of threads) as the composition UFU\circ F where U:TopSetU:Top\to Set is the forgetful functor; the topology of limFlim F is the subspace topology on lim(UF)lim (U\circ F) understood as a subset of the Cartesian product dF(d)\prod_d F(d) equipped with the (product)Tihonov's topology.

Last revised on May 11, 2012 at 04:38:18. See the history of this page for a list of all contributions to it.