nLab quasi-limit



2-Category theory

Limits and colimits



In 2-category theory, the notion called quasi-limits [Gray (1997)] or soft limits [MacDonald & Stone (1989)] is a weakening of the notion of bilimit so as to make them satisfy a ‘quasi-universal’ property. This means that instead of being singled out up to unique isomorphism, quasi-limits are identified only up to unique adjunction.


Recall that limits and colimits of shape K\mathbf K are computed by the right and left adjoint, respectively, to the diagonal functor Δ ():C[K,C]\Delta_{(-)} : \mathbf C \to [\mathbf K, \mathbf C] (the one which constant functors):

colimΔlim. \colim \dashv \Delta \dashv \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 locΔ locqlim \mathrm{qcolim} \dashv_{\mathrm{loc}} \Delta \dashv_{\mathrm{loc}} \mathrm{qlim}

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

C(X,qlimF)Nat(Δ X,F) \mathbf C(X, \mathrm{qlim} 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 bilimit for FF is an object limF:C\lim F:\mathbf C equipped with a natural family of equivalences

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

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 family of adjunctions

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

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.


  • A quasi-terminal object in a 2-category C\mathbf C is an object QQ such that, for each other object X:CX: \mathbf C, there exists a ‘quasi-universal map’ t:XQt : X \to Q such that every other map f:XQf : X \to Q admits a unique 2-cell ϕ f:ft\phi_f : f \Rightarrow t to it. In C=Cat\mathbf C = \mathbf{Cat} quasi-terminal objects are categories with a terminal object.

From this example alone one can see quasi-limits are, in general, not unique up to isomorphism, like limits and other universal constructions. The universality of the objects is replaced by universality of their quasi-universal maps. A consequence of this fact is that we can’t say the quasi-limit but a quasi-limit.

The following is an example of quasi-colimit from (Gray ‘74, p.199):

  • A quasi-coproduct of A 1A_1 and A 2A_2 in a 2-category C\mathbf C is an object A 1+ qA 2A_1 +_q A_2 together with injections i j:A jA 1+ qA 2i_j : A_j \to A_1 +_q A_2 (j=1,2j=1,2) which satisfies a relaxed version of the universal property of coproducts. Instead, we require that for any other object XX with maps h j:A jXh_j : A_j \to X there is a map h:A 1+ qA 2Xh : A_1 +_q A_2 \to X and 2-cells λ j:hi jh j\lambda_j : h\circ i_j \Rightarrow h_j (j=1,2j=1,2) which are terminal as such. This means that given any other g:A 1+ qA 2Xg : A_1 +_q A_2 \to X and 2-cells γ j:gi jh j\gamma_j : g \circ i_j \Rightarrow h_j (j=1,2j=1,2), there is a 2-cell ϕ (g,γ j):gh\phi_{(g,\gamma_j)} : g \to h that factorizes γ\gamma through λ\lambda: (τi j)λ j=γ j(\tau \cdot i_j) \circ \lambda_j = \gamma_j.

In Capucci (2022) it is shown that this (with 2-cells reversed though) is the universal property of a certain example of monoidal product common in game theory, called external choice.

  • A quasi-product is dual to a quasi-coproduct in the sense of reversing 1-cells. Reversing 2-cells is also possible, but there doesn’t seem to be an agreed terminological scheme to distinguish between the two kinds of quasi-limits.

Notably, the product of sets, which in the 2-category of relations loses its usual universal property, becomes a quasi-product there. Notice both this and the above example of quasi-coproduct are unique up to iso.


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

See also:

Last revised on March 13, 2023 at 09:29:23. See the history of this page for a list of all contributions to it.