nLab
weak multilimit

Idea

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

Definition

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

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

Of course, weak multilimits in $C^{\mathrm{op}}$ are called weak multicolimits in $C$.

Examples

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