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:KCF:\mathbf K \to \mathbf C can be defined as the right and left adjoint, respectively, to the diagonal Δ ():C[K,C]\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C] (aka constant functor):

colimΔlim \colim \vdash \Delta \vdash \lim

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

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

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

C(X,limF)Nat(Δ X,F) \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:KCF : \mathbf K \to \mathbf C be a small diagram in C\mathbf C. Then recall that a 2-limit for FF is an object limF:C\lim F:\mathbf C equipped with a natural isomorphism

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

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

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

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

naturally in X:CX:\mathbf C.

Notice that by replacing 1:CCat1 : \mathbf C \to \mathrm{Cat} with an arbitrary weight WW we can easily give the definition of weighted quasilimit.

References

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

See also

Created on August 4, 2022 at 10:13:52. See the history of this page for a list of all contributions to it.