# "Very lax" $2$-dimensional co/limits

In the setting of $$1$$-categories, there are two (unweighted) variant notions of limits, namely limits and colimits. For bicategories, there are sixteen of them:

Each of these notions has an associated "strength", given by whether we require an equivalence of categories like $$\mathsf{Hom}_{\mathcal{D}}(X,\mathsf{bilim}^{\mathsf{lax}}(D)) \overset{\mathrm{eq}}{\cong} \mathsf{LaxNat}(\Delta_{X},D)$$ or an isomorphism of categories like

$$\mathsf{Hom}_{\mathcal{D}}(X,\mathsf{lim}^{\mathsf{lax}}(D)) \cong \mathsf{LaxNat}(\Delta_{X},D).$$

Bicategorical adjunctions, on the other hand, are usually discussed using three different "strengths": equivalences, isomorphisms, and adjunctions. This leads to the following question:

Question. Have the "very lax" notions of $$2$$-dimensional co/limits corresponding to weakening the equivalences of categories in the above table to be merely adjunctions been studied before? Are there any interesting/natural examples of them?

• As Sori Lee let me know by email, I initially forgot to include the 13th and 14th rows in the table above corresponding to 2-bilimits and 2-bicolimits, and have now updated the table to reflect this. (Thanks for letting me know about this, Sori!) Oct 5, 2023 at 19:57

## 1 Answer

I believe these kinds of (co)limits are the ones defined in Gray's Formal Category Theory: Adjointness for $$2$$-categories I.7.9.1 (as quasi-(co)limits); some examples and computations of them are given further on in I.7.10, I.7.11 and I.7.12, and recover several classical constructions such as (co)comma objects or Eilenberg–Moore (or Kleisli) categories, as well as the usual description, when the target $$2$$-category is $$\mathfrak{Cat}$$, in terms of sections of an associated opfibration. Note that Gray defines them in the $$3$$-dimensional—or rather, (lax) Gray-categorical—setting, using his notion of quasi-adjunction, a.k.a. soft adjunction or lax adjunction or local adjunction ("adjunction up to adjunctions"), this last description (established in MacDonald–Stone's Soft adjunction between $$2$$-categories) being what recovers the definition you are asking for.

However, I have not read this book in enough detail to be able to answer confidently on what exactly Gray's somewhat idiosyncratic terminology corresponds to (I am only writing this as an answer because it is slightly too long for a comment), though various flavours of laxity, not all of which have stood the test of time, are allowed—see the discussion at the end of Definition I.7.1 for a list; I think the relevant kind for your question (and in general use) should be "strict weak quasi-adjunction" or "transcendental quasi-adjunction".

• Thank you! This is precisely what I was looking for! Mar 14, 2021 at 22:35
• Incidentally, I also noticed the following point since asking this question: since every equivalence can be refined into an adjoint equivalence, it follows that e.g. every lax bilimit is also a 'lax soft limit', and similarly for the other variants. This means that the soft notions are a weakening of the uniqueness of the "bi" notions, though the latter still provide a kind of "'non-unique but canonical' choice" for the former. So perhaps from this point of view they are not as interesting as it may seem at first... Mar 14, 2021 at 22:38