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.
Recall limits and colimits of a given diagram can be defined as the right and left adjoint, respectively, to the diagonal (aka constant functor):
Suppose now is a 2-category and 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 and , 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 be a small diagram in . Then recall that a 2-limit for is an object equipped with a natural isomorphism
where is the diagonal functor.
A quasi-limit for is then an object equipped with a adjunctions
naturally in .
Notice that by replacing with an arbitrary weight we can easily give the definition of weighted quasilimit.
Quasi-limits and their duals have been defined in §I.7.9 of:
Springer 1974 (doi:10.1007/BFb0061280)
See also
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