# Lax universality for lax limits

A lax limit is defined 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.

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 $h\Rightarrow\langle f,g\rangle$.

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.

Does such a structure exist in the literature?

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In other words, you want an object $P$ with projections $p\colon P\to A$ and $q\colon P\to B$ such that the induced functor $Hom(X,Y) \to Hom(X,A)\times Hom(X,B)$ has an adjoint? I haven't seen this specifically in the literature, but you may be interested in the related notion(s) of lax 2-adjunction: ncatlab.org/nlab/show/lax+2-adjunction –  Mike Shulman Nov 1 '10 at 20:33
That is indeed what I'm after. The particular case is smash product, but there's obvious generalization to an arbitrary (weighted) limit. Partly my concern is just terminological: I have a paper to write and I need a name for the gadget I have! But I'd also like to make sure it generalizes properly. –  Alan Jeffrey Nov 2 '10 at 1:30
This might be a good question for the categories mailing list. It seems like there might be a chance of its being in the literature somewhere, but if so, the people who would know may not be reading MO. –  Mike Shulman Nov 2 '10 at 5:11
BTW, is such a "lax product" uniquely characterized up to iso by its universal property? Offhand I don't see why it would be... and if not, I'd be a little uneasy about calling it any kind of "limit" rather than regarding it instead as some kind of algebraic structure on the category. –  Mike Shulman Nov 2 '10 at 5:13
The category theory mailing list may be the way to go. Dusko can be annoyed with me for asking the $n+1$th terminological question :-) "Lax products" aren't determined up to iso, but (quick scribbles on back of envelope) up to a "lax iso", that is for any candidates $Z$ and $Z'$ there are morphisms $f : Z \rightarrow Z'$ and $f' : Z' \rightarrow Z$ such that $1 \Rightarrow f;f'$ and $1 \Rightarrow f';f$. –  Alan Jeffrey Nov 2 '10 at 14:31

Adjunctions 'up to adjointness' have been considered before. Marta Bunge (Coherent extensions and relational algebras, Trans. AMS 197, 1974) called them 'lax adjunctions', John Gray (Formal category theory: Adjointness for 2-categories, LNM 391, 1974) called them 'quasi-adjunctions' (of some sort) and Barry Jay (Local adjunctions, 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 modifications (3-cells), and these two should be Yonedily equivalent.