Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
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 $\mathbf K$ 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):
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:
This means that, naturally in $X : \mathbf C$, there are adjunctions (instead of equivalences, like in a 2-adjunction) between hom-categories:
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
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
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.
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):
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.
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:
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]
Last revised on March 13, 2023 at 09:29:23. See the history of this page for a list of all contributions to it.