nLab quasi-limit

Idea

Quasilimits (sometimes called soft limits) are a relaxation of 2-limits so as to make them satisfy a ‘quasi-universal’ property, i.e. instead of being singled out up to unique isomorphism, they are identified only up to unique adjunction. They’ve been defined by Gray in his seminal 1974 book on 2-category theory.

Definition

Recall limits and colimits of a given diagram $F:\mathbf K \to \mathbf C$ can be defined as the right and left adjoint, respectively, to the diagonal $\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C]$ (aka constant functor):

\colim \vdash \Delta \vdash \lim

Suppose now $\mathbf C$ is a 2-category and $\mathbf K$ is a small 2-category. Gray defines quasi(co)limits by relaxing the previous adjunction to be only a local adjunction:

\mathrm{qcolim} \vdash_{\mathrm{lax}} \Delta \vdash_{\mathrm{lax}} \mathrm{qlim}

This means that, naturally in $X : \mathbf C$ and $F : \mathbf K \to \mathbf C$ , there are adjunctions (instead of equivalences, like in a 2-adjunction) between hom-categories:

\mathbf C(X, \lim F) \rightleftarrows \mathrm{Nat}(\Delta_X, F)

Alternatively, one can give a definition in terms of a weakened representability property.

Let $F : \mathbf K \to \mathbf C$ be a small diagram in $\mathbf C$ . Then recall that a 2-limit for $F$ is an object $\lim F:\mathbf C$ equipped with a natural isomorphism

\phi_X : \mathbf C(X, \lim F) \cong \mathrm{Nat}(1, \mathbf C(X, F(-)))

where $\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C]$ is the diagonal functor.

A quasi-limit for $F$ is then an object $\mathrm{qlim} F:\mathbf C$ equipped with a adjunctions

\psi_X : \mathbf C(X, \mathrm{qlim}F) \rightleftarrows \mathrm{Nat}(1, \mathbf C(X, F(-)))

naturally in $X:\mathbf C$ .

Notice that by replacing $1 : \mathbf C \to \mathrm{Cat}$ with an arbitrary weight $W$ we can easily give the definition of weighted quasilimit.

References

Quasi-limits and their duals have been defined in §I.7.9 of:

John Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, Vol. 391.
Springer 1974 (doi:10.1007/BFb0061280)