nLab
strict 2-limit

Context

2-Category theory

Limits and colimits

Contents

Idea

In general, a 2-limit is the sort of limit appropriate in a (weak) 2-category. See 2-limit for details. However, when we happen to be in a strict 2-category we also have another notion at our disposal. Since strict 2-categories are just categories enriched over Cat, we can apply the usual notions of weighted limits in enriched categories verbatim. (Historically, these were called 2-limits while the up-to-isomorphism limits were called bilimits.)

Because enriched category theory doesn’t know anything about the 2-categorical nature of Cat, the resulting limits can have cones that commute strictly and have universal properties expressed by isomorphisms of categories; thus they can be evil. However, such strict limits often turn out to be technically useful even if we are fundamentally only interested in the non-srict notions, since in many strict 2-categories we can use tools of enriched category theory to construct strict limits, and then by considering suitably non-evil strict limits we can construct (non-strict) limits. This is reminiscent of the use of strict structures in homotopy theory as a tool to get at weak ones, and in fact a precise comparison can be made (see below).

Classification

By a limit we will mean the fully 2-categorical notion described at 2-limit, in which cones commute up to isomorphism and the universal property is expressed by an equivalence of categories.

It just occured to me that ‘strict initial object’ conflicts with this. But unlike ‘weak limit’, that doesn’t generalise very far.

Heh, you’re right. I suppose we could try calling strict initial objects stable initial objects, which would make more sense anyway since they are really the 0-ary version of a stable coproduct. But there’s probably not likely to be any real confusion created by the two uses of strict.

  • A strict 2-limit (or just strict limit) in a strict 2-category is just a Cat-enriched (weighted) limit. This means that its cones must commute strictly (although weakness can be built in via the weighting, see below), and its universal property is expressed by an isomorphism of categories. Note that a strict limit is not necessarily a limit, because it may be evil. (cf. red herring principle.)

  • A pseudo limit (or strict pseudo limit if it is necessary to emphasize the strictness) is a limit whose cones commute up to coherent 2-cell isomorphism, but whose universal property can still be expressed by an isomorphism of categories. For any weight WW, there is another weight WW' (a cofibrant replacement of WW) such that pseudo WW-weighted limits are equivalent to strict WW'-weighted ones. The idea is that WW' includes explicitly all the extra isomorphisms in a pseudo WW-cone. Since any isomorphism of categories is a fortiori an equivalence of categories, any pseudo limit is also a limit.

  • A strict lax limit is a limit whose cones commute only up to a coherent transformation in one direction, but again whose universal property is expressed by an isomorphism. Likewise we have strict oplax limits where the transformation goes in the other direction. Strict lax and oplax limits can also be rephrased as strict (non-lax) limits for a different weight. As in the pseudo case, any strict (op)lax limit is also an (op)lax limit.

More generally, any non-evil strict limit (one which doesn’t demand equality of objects) will also be a limit. Two formal versions of this statement involve flexible limits and the more restrictive PIE-limits. In particular, any strict flexible limit is also a limit. Since pseudo limits are PIE-limits, it follows that any strict 2-category which admits (strict) PIE-limits also admits all limits, even if it fails to admit some (evil) strict limits. The category of algebras and pseudo morphisms for any 2-monad, such as MonCat?, is a good example of a 2-category having strict PIE-limits but not all strict limits.

Pseudo limits and homotopy limits

If there is a model category structure on the 1-category underlying the given strict 2-category CC, then in addition to whatever 2-categorical notions of limit exist in CC, there is the notion of homotopy limits in CC. If CC is a model 2-category with the “trivial” or “natural” model structure constructed in (Lack 2006), then these two notions coincide (Gambino 2007). For example, this is the case in Cat and Grpd, so the examples listed at homotopy limit are also examples of pseudo limits. In general, homotopy limits in a model 2-category give (non-strict) limits in its “homotopy 2-category.”

Examples

Any ordinary 1-limit can be made into a strict 2-limit simply by boosting up its ordinary universal property (a bijection of sets) to an isomorphism of hom-categories. Thus we have strict products, strict pullbacks, strict equalizers, and so on. Of these, strict products (including terminal objects) are non-evil (and thus are also limits), while others such as pullbacks and equalizers tend to be evil.

  • For example, a strict terminal object is an object 1 such that K(X,1)K(X,1) is isomorphic to the terminal category, for any object XX.

  • Likewise, a strict product of AA and BB is an object A×BA\times B with projections p:A×BAp:A\times B\to A and q:A×BBq:A\times B\to B such that (1) given any f:XAf:X\to A and g:XBg:X\to B, there exists a unique h:XA×Bh:X\to A\times B such thath ph=fp h = f and qh=gq h = g (equal, not isomorphic), and (2) given any h,k:XA×Bh,k:X\to A\times B and α:phpk\alpha: p h \to p k and β:qhqk\beta:q h \to q k, there exists a unique γ:hk\gamma:h\to k such that pγ=αp\gamma = \alpha and qγ=βq\gamma =\beta.

As mentioned above, adding pseudo in front of an ordinary limit has a precise meaning: it means that all the triangles in the limit cone now commute up to specified isomorphism, and the universal property is still expressed by an isomorphism of categories. In particular, there is still a specified projection to each object in the diagram. For example:

  • The pseudo pullback of a cospan AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a universal object PP equipped with projections p:PAp:P\to A, q:PBq:P\to B, and r:PCr:P\to C and 2-cell isomorphisms fprf p \cong r and gqrg q \cong r.

  • The pseudo equalizer of a pair of arrows f,g:ABf,g:A\rightrightarrows B is a universal object EE equipped with morphisms h:EAh:E\to A and k:EBk:E\to B and 2-cell isomorphisms fhkf h \cong k and ghkg h \cong k.

These are to be distinguished from:

  • The iso-comma object of AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a universal object PP equipped with projections p:PAp:P\to A and q:PBq:P\to B and a 2-cell isomorphism fpgqf p \cong g q.

  • The iso-inserter of f,g:ABf,g:A\rightrightarrows B is a universal object EE equipped with a morphism e:EAe:E\to A and a 2-cell isomorphism fegef e \cong g e.

The pseudo pullback, pseudo equalizer, iso-comma object, and iso-inserter are all strict Cat-weighted limits; their universal property is expressed by an isomorphism of categories. Usually the pseudo pullback and iso-comma object are not isomorphic, and likewise the pseudo equalizer and iso-inserter are not isomorphic. However, both the pseudo pullback and iso-comma object are non-evil and represent a pullback; therefore they are equivalent when they both exist. Likewise, the pseudo equalizer and iso-inserter both represent an equalizer, and are equivalent when they both exist.

If one is mostly interested in (non-strict) limits, then there is little harm in using “pseudo pullback” to mean “iso-comma object” or “pullback,” as is common in the literature. However, with lax limits the situation is more serious. Speaking precisely, in the lax version of a limit, the triangles in the limiting cone are made to commute up to a specified transformation in one direction, but there are still specified projections to each object in the diagram. For example:

  • The strict lax limit of an arrow f:ABf:A\to B is a universal object LL equipped with projections p:LAp:L\to A and q:LBq:L\to B and a 2-cell fpqf p \to q.

  • The strict lax pullback of a cospan AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a universal object PP equipped with projections p:PAp:P\to A, q:PBq:P\to B, r:PCr:P\to C, and 2-cells fprf p \to r and gqrg q \to r.

In particular, the strict lax pullback is quite different from the following more common limit.

  • The comma object of a cospan AfCgBA \overset{f}{\to} C \overset{g}{\leftarrow} B is a generalization of the comma category in CatCat; it is a universal object (f/g)(f/g) equipped with projections p:(f/g)Ap:(f/g)\to A and q:(f/g)Bq:(f/g)\to B and a 2-cell fpgqf p \to g q.

Even in their non-strict forms, the lax pullback and comma object are distinct. Usually the comma object is the more important one, but calling it a “lax pullback” should be avoided.

Here are some more important examples of 2-limits, all of which come in strict and weak forms and are non-evil.

  • The inserter of a pair of parallel arrows f,g:ABf,g:A \;\rightrightarrows\; B is a universal object II equipped with a map i:IAi:I\to A and a 2-cell figif i \to g i.

  • The equifier of a pair of parallel 2-cells α,β:fg:AB\alpha,\beta: f\to g: A\to B is a universal object EE equipped with a map e:EAe:E\to A such that αe=βe\alpha e = \beta e.

  • The inverter of a 2-cell α:fg:AB\alpha:f\to g:A\to B is a universal object VV with a map v:VAv:V\to A such that αv\alpha v is invertible.

  • The power of an object AA by a category CC is a universal object A CA^C equipped with a functor CK(A C,A)C\to K(A^C,A).

References

Revised on April 14, 2014 20:49:34 by David Roberts (129.127.4.122)