nLab
weak multilimit

Weak multilimits

Idea

A weak multilimit is a common generalization of multilimits and weak limits.

Definition

If F:DCF\colon D\to C is a diagram in a category CC, then a weak multilimit of FF is a (small) set LL of cones over FF such that any other cone over FF factors (not necessarily uniquely) through some (not necessarily unique) element of LL. If the factorization, and the cone factored through, are unique, then LL is a multilimit, whereas if LL is a singleton, then it is a weak limit.

The existence of weak multilimits is a “pure size condition” on CC, in the sense that if CC is a small category, then every small diagram in CC (that is, every functor F:DCF\colon D\to C where DD is also small) has a weak multilimit, namely the set of all cones over FF.

Of course, weak multilimits in C opC^{op} are called weak multicolimits in CC.

Examples

  • A weak multilimit of the empty diagram is a weak multi-terminal-object, also called a weakly terminal set: a small set TT of objects such that every object admits a morphism to some object in TT. The dual concept is a weakly initial set. These notions play a role in some statements of the adjoint functor theorem.

Revised on February 1, 2010 17:03:05 by Toby Bartels (173.60.119.197)