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One application of pseudo 2-limits (bilimits) in algebraic geometry is already found in the definition itself of stacks with values in a 2-category admitting bilimits (i.e. a discrete 1-cell-contravariant pseudo functor defined on a site). Of course there is the general notion of descent, etc. and all of its applications. There are also 2-colimit generalisations of aspects of Grothendieck's Galois theory (from SGA1) and aspects of SGA2.

The question is then whether there are "interesting" situations (not necessarily from algebraic geometry) where one must use lax 2-limits and cannot get away with just using pseudo 2-limits? (Or 2-colimits?)

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  • $\begingroup$ Isn't the comma category some kind of lax 2-limit? $\endgroup$
    – Harry Gindi
    Commented Jan 5, 2011 at 13:31
  • $\begingroup$ Yes it is; it's a weighted lax 2-limit. $\endgroup$
    – Nic Palmero
    Commented Jan 5, 2011 at 14:59
  • $\begingroup$ After your remark about the comma category (which contains the notion of slice category), it occurred to me that there is a natural application of lax 2-colimits in the bicategory of topoi, viz., the famous "Medaille en Or" exercise from SGA4.1, Exp. IV, exercice I.4.10, one major application of which is to relate the sheaves on the crystalline site to the Zariski localisations of suitable thickenings (cf. Berthelot's thesis, chp. III). Cisinski and Baez also discussed the relation of the petit/gros topos in 2009 on the n-category cafe also. $\endgroup$
    – Nic Palmero
    Commented Jan 5, 2011 at 16:40
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    $\begingroup$ No, a comma category is not a lax 2-limit. It is a weighted (pseudo) 2-limit. See nlab.mathforge.org/nlab/show/2-limit#lax. $\endgroup$
    – Mike Shulman
    Commented Jan 5, 2011 at 20:36
  • $\begingroup$ Of course every pseudo functor is a lax 2-functor, so your claiming that it cannot be a lax 2-limit (without changing its weighting)? $\endgroup$
    – Nic Palmero
    Commented Jan 6, 2011 at 2:47

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First I should point out that (as you may know) in most of the 2-categorical literature, "bilimit" and "pseudo (2-)limit" are not used synonymously. The former has a cone which commutes up to specified isomorphism, and has a universal property expressed by an equivalence of categories, while the latter also has a cone which commutes up to specified isomorphism, but has a universal property expressed by an isomorphism of categories.

The nLab uses "2-limit" to mean "bilimit," and so will I from now on.

Now, at least if one allows weighted 2-limits, then there is no situation in which one "must" use lax 2-limits and cannot get away with ordinary ones, since every weighted lax 2-limit can be identified with a non-lax weighted 2-limit which uses a modified weight.

However, lax 2-limits do occur naturally in some places, such as the Eilenberg-Moore object of a monad in a 2-category. Note that weighted 2-limits which are neither unweighted ("conical") 2-limits or lax versions of conical 2-limits do also occur frequently, such as the comma categories mentioned in the comments.

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