Domenico Fiorenza
weighted join

Idea

Idea

An initial object? in a category DD is an object \emptyset with exactly one morphism from emtyset\emtyset to any object XX of DD. An obvious generalization is that of a weighted initial object?, i.e. an object W\emptyset_W with exactly W XW_X morphisms to any object XX; the ‘function’ WW which assigns to an object XX the number W XW_X of morphisms from W\emptyset_W to XX is the weight.

This is nicely formalized in terms of functors: let W:DSetW: D\to Set be a functor. Then a weighted initial object with weight WW is, if it exists, an object emtyset W\emtyset_W representing the functor WW, i.e., such that

Hom( W,X)W(X), Hom(\emptyset_W,X)\cong W(X),

naturally in XX.

The trivial weight W(X)={pt}W(X)=\{pt\} corresponds to the usual notion of initial object.

Mike Shulman: This is an interesting notion, but it is not a special case of the general notion of weighted limit. The weight for a weighted limit of a diagram F:DCF\colon D\to C is a functor W:DSetW\colon D\to Set, not CSetC\to Set. Since DD is the empty category for initial objects, no nontrivial weights are possible in that case. Of course, what you call a “WW-weighted initial object” is the same a representation of WW.

Domenico Fiorenza: I see the choice CC for the category was a bad one.. I had written everithing in terms of CC here since when there’s just one category in sight it’s customary to call it CC, but if one is reading at this entry with weighted limits in mind, this should rather be DD. Now I’ll edit the page using DD instead; I hope this will make clear what I had in mind.

Mike Shulman: Changing the name of the category doesn’t change the meaning of the definition, does it? I don’t understand how this is supposed to make it clearer.

Domenico Fiorenza: My fault: when you said that the weight for a weighted limit of a diagram F:DCF\colon D\to C is a functor W:DSetW\colon D\to Set, not CSetC\to Set, I guessed that in the way I had written things I had made you get confused, since in the previous version of this page WW was a functor from CC to SetSet. Thus I tried to clarify thing changing the notation accordingly.

So let me write here what I am suggesting the weighted limit lim WFlim_WF could be, for a functor F:DCF:D\to C, to see if I’m able to make the aim of this entry clearer: lim WFlim_WF should be a terminal object in the category of functors from {} WD\{\top\}\star_WD to CC exteding FF; here {} WD\{\top\}\star_WD is the category obtatined by adding a WW-weighted initial object to DD.

Mike Shulman: Ah, yes! It is quite right that a WW-weighted cone (or cylinder) over a diagram F:DCF\colon D\to C is the same as a functor from the cograph of WW (considered as a profunctor D1D ⇸ 1) to CC which restricts to FF along the inclusion of DD. In fact, this is the basis for a natural generalization of weighted limits, where we consider profunctors DKD ⇸ K for arbitrary KK, rather than merely K=1K=1; there are some remarks about this generalization at proarrow equipment.

I would suggest, however, that you not use the phrase “WW-weighted initial object” since that construction isn’t actually itself a weighted (co)limit; it’s just something you can use to help define the weighted limit.

If DD is any quasi-category, W:DSetW:D\to Set is a functor, and {}\{\top\} denotes the terminal quasi-category, the weighted join {} WD\{\top\}\star_W D is the quasi-category obtained from DD by ‘adding’ a WW-weighted initial object to DD.

More in general, if KK is another quasi-category, the weighted join K WDK\star_W D is the quasi-category intuitively obtained by making W(X)W(X) arrows stem from any object of KK to any object XX of DD.

(details on the formal construction of K WDK\star_W D to be added)

Mike Shulman: It seems as though possibly you would want to start from a functor W:K op×DSpacesW\colon K^{op}\times D\to Spaces, rather than simply DSetD\to Set. In that case, perhaps your “weighted join” is the same as the cograph of a profunctor (in the quasicategory context)?

Fosco: I think Mike Shulman is right, as I was motivated to open this question on MO in order to clarify the role of profunctors in the definition of limits. It seems pretty obvious to me that the generalization which leads to consider representing object limF\lim \; F for the functor M opSets M^\text{op}\to \Sets sending mMm\in M to Cones(m,F)Sets C(*,M(m,F))\Cones (m,F)\cong \Sets^C(*, M(m,F)) can be easily carried on to consider

  • representing object lim WF\lim^W \; F for a general functor W:CSetsW\colon C\to \Sets and the functor mSets C(W,M(m,F))m\mapsto \Sets^C(W, M(m,F));
  • representing object lim WF\lim^W \; F for a general functor W:CVW\colon C\to V and the functor m[C,V](W,M(m,F))m\mapsto [C,V](W, M(m,F)), for VV a suitable Benabou cosmos?.

This led me to wonder if, in the same vein that regards the good old limit limF\lim \; F as the terminal object in the category Fun F([0]C,D)\text{Fun}^F([0]\star C,D), where [0]C[0]\star C is the collage along the terminal profunctor, then the limit of FF weighted by W:CSetsW\colon C\to \Sets can be seen as the terminal object in the category Fun F([0] WC,D)\text{Fun}^F([0]\star_W C,D), where [0] WC[0]\star_W C is merely the cograph of WW (I’m certain you see the pattern!).

Revised on October 31, 2013 at 07:05:34 by Domenico Fiorenza