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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
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up vote 2 down vote accepted

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.

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 here (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.

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Thanks for the links! Unfortunately, none of these naming schemes lend themselves to a name of the form adjective limit/colimit/product/pullback/etc. Perhaps "locally lax"? Or "univerally lax" to stress that it's the universality of the limit that's being made lax? I note that this is one of the places where the 1-category terminology and the 2-category terminology are in mild conflict: a weak limit weakens the universality of the limit, whereas a lax limit relaxes the cone. –  Alan Jeffrey Nov 2 '10 at 1:53
Re: weak vs lax: yes, that's one reason why "weak" is an unfortunate choice of word to mean "up to iso/equivalence" generally. But there's not much we can do about that, I think. –  Mike Shulman Nov 2 '10 at 5:10
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