nLab quasi-limit

Idea

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.

Definition

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

$$\colim \dashv \Delta \dashv \lim \,.$$

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:

$$\mathrm{qcolim} \dashv_{\mathrm{loc}} \Delta \dashv_{\mathrm{loc}} \mathrm{qlim}$$

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

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

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

Let $F : \mathbf K \to \mathbf C$ be a small diagram in $\mathbf C$. Then recall that a bilimit for $F$ is an object $\lim F:\mathbf C$ equipped with a natural family of equivalences

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

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 family of adjunctions

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

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.

Examples

• A quasi-terminal object in a 2-category $\mathbf C$ is an object $Q$ such that, for each other object $X: \mathbf C$, there exists a ‘quasi-universal map’ $t : X \to Q$ such that every other map $f : X \to Q$ admits a unique 2-cell $\phi_f : f \Rightarrow t$ to it. In $\mathbf C = \mathbf{Cat}$ 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 $A_1$ and $A_2$ in a 2-category $\mathbf C$ is an object $A_1 +_q A_2$ together with injections $i_j : A_j \to A_1 +_q A_2$ ($j=1,2$) which satisfies a relaxed version of the universal property of coproducts. Instead, we require that for any other object $X$ with maps $h_j : A_j \to X$ there is a map $h : A_1 +_q A_2 \to X$ and 2-cells $\lambda_j : h\circ i_j \Rightarrow h_j$ ($j=1,2$) which are terminal as such. This means that given any other $g : A_1 +_q A_2 \to X$ and 2-cells $\gamma_j : g \circ i_j \Rightarrow h_j$ ($j=1,2$), there is a 2-cell $\phi_{(g,\gamma_j)} : g \to h$ that factorizes $\gamma$ through $\lambda$: $(\tau \cdot i_j) \circ \lambda_j = \gamma_j$.

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.

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 391

  Springer (1974) [doi:10.1007/BFb0061280]

See also:

• “Very lax” 2 -dimensional co/limits, MathOverflow

• MacDonald, Stone, Soft adjunction between 2-categories, Journal of Pure and Applied Algebra, 60 (1989)

• Matteo Capucci, The Universal Property of External Choice (2022) [pdf]