Lax universality for lax limits - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:55:02Z http://mathoverflow.net/feeds/question/44463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44463/lax-universality-for-lax-limits Lax universality for lax limits Alan Jeffrey 2010-11-01T17:21:05Z 2010-11-01T20:57:51Z <p>A lax limit is <a href="http://ncatlab.org/nlab/show/2-limit" rel="nofollow">defined</a> to be a 2-limit, except that the cone is only required to commute up to specified transformations, not up to isomorphism. In particular, the limit is defined up to isomorphism, and on examples such as product where there are no 2-cells in the cone, lax limits and 2-limits coincide.</p> <p>I'm working with an example, whose properties are similar to smash product on partial orders with bottom. This gives rise to a lax version of a product, where the equalities $\langle f,g\rangle;p=f$ and $\langle f,g\rangle;q=g$ are replaced by 2-cells, and the uniqueness property becomes a universality property: for any other candidate $h$, we have a unique 2-cell $h\Rightarrow\langle f,g\rangle$.</p> <p>Generalizing, it seems that there should be an "even laxer" notion of limit, where the adjunction used to define the limit is specified itself by an adjunction rather than by an equivalence of categories.</p> <p>Does such a structure exist in the literature?</p> http://mathoverflow.net/questions/44463/lax-universality-for-lax-limits/44487#44487 Answer by Finn Lawler for Lax universality for lax limits Finn Lawler 2010-11-01T20:57:51Z 2010-11-01T20:57:51Z <p>Adjunctions 'up to adjointness' have been considered before. Marta Bunge (<em>Coherent extensions and relational algebras</em>, Trans. AMS 197, 1974) called them 'lax adjunctions', John Gray (<em>Formal category theory: Adjointness for 2-categories</em>, LNM 391, 1974) called them 'quasi-adjunctions' (of some sort) and Barry Jay (<em>Local adjunctions</em>, JPAA 53, 1988) called them 'local adjunctions'. The formulations differ somewhat, but what they have in common is a family of adjunctions $$D(F A, X) \leftrightarrows C(A, G X)$$ between hom categories. Another way to generalize, of course, is to turn the triangle equalities into coherent <a href="http://ncatlab.org/nlab/show/modification" rel="nofollow">modifications</a> (3-cells), and these two should be Yonedily equivalent.</p> <p>Often the functors F and G here are allowed to be lax, which complicates things because they can't in general be whiskered onto lax transformations. The unit and counit are often only lax too. There is a Yoneda lemma for lax transformations, written down <a href="http://ncatlab.org/nlab/show/lax+natural+transformation" rel="nofollow">here</a> (by me, so any mistakes are my fault), but I for one haven't yet managed to massage it into a form that would be useful for these local/lax adjunctions.</p>