nLab
weighted limit

Context

Category theory

Enriched category theory

Limits and colimits

Contents

Idea

The notion of weighted limit is naturally understood from the point of view on limits as described at representable functor.

Weighted limits make sense and are considered in the general context of VV-enriched category theory, but restrict attention to V=V= Set for the moment, in order to motivate the concept.

Let KK denote the small category which indexes diagrams over which we want to consider limits and eventually weighted limits. Notice that for

F:KSet F : K \to Set

a Set-valued functor on KK, the limit of FF is canonically identified simply with the set of cones with tip the singleton set pt={}pt = \{\bullet\}:

limF=[K,Set](Δpt,F). lim F = [K,Set](\Delta pt, F) \,.

This means, more generally, that for

F:KC F : K \to C

a functor with values in an arbitrary category CC, the object-wise limit of the functor FF under the Yoneda embedding

C(,F()):KFCYSet C op C(-,F(-)) : K \stackrel{F}{\to} C \stackrel{Y}{\to} Set^{C^{op}}

which appears in the discussion in example 1 at representable functor can be expressed by the right side of

limC(,F())=[K,Set](Δpt,C(,F())). lim C(-,F(-)) = [K,Set](\Delta pt, C(-,F(-))) \,.

(Recall that this is the limit over the diagram C(,F()):KSet C opC(-,F(-)) : K \to Set^{C^{op}} which, if representable defines the desired limit of FF.)

The idea of weighted limits is to

  1. allow in the formula above the particular functor Δpt\Delta pt to be replaced by any other functor W:KSetW : K \to Set;

  2. to generalize everything straightforwardly from the Set-enriched context to arbitrary VV-enriched contexts.

The idea is that the weight W:KVW : K \to V encodes the way in which one generalizes the concept of a cone over a diagram FF (that is, something with just a tip from which morphisms are emanating down to FF) to a more intricate structure over the diagram FF. For instance in the application to homotopy limits discussed below with VV set to SimpSet the weight is such that it ensures that not only 1-morphisms are emanating from the tip, but that any triangle formed by these is filled by a 2-cell, every tetrahedron by a 3-cell, etc.

Definition

Let VV be a closed symmetric monoidal category. All categories in the following are VV-enriched categories, all functors are VV-functors.

A weighted limit over a functor

F:KC F : K \to C

with respect to a weight or indexing type functor

W:KV W : K \to V

is, if it exists, the object lim WFClim^W F \in C which represents the functor (in cCc \in C)

[K,V](W,C(c,F())):C opV, [K,V](W, C(c,F(-))) : C^{op} \to V \,,

i.e. such that for all objects cCc \in C there is an isomorphism

C(c,lim WF)[K,V](W(),C(c,F())) C(c, lim^W F) \simeq [K,V](W(-), C(c,F(-)))

natural in cc.

(Here [K,V][K,V] is the VV-enriched functor category, as usual.)

In particular, if C=VC = V itself, then we get the direct formula

lim WF[K,V](W,F). lim^W F \simeq [K,V](W,F) \,.

This follows from the above by by the coend manipulation

[K,V](W(),C(c,F())) := kKV(W(k),V(c,F(k))) kKV(c,V(W(k),F(k)) V(c, kKV(W(k),F(k)) =:V(c,[K,V](W,F)). \begin{aligned} [K,V](W(-),C(c,F(-))) &:= \int_{k \in K} V(W(k),V(c,F(k))) \\ & \simeq \int_{k \in K} V(c,V(W(k),F(k)) \\ & \simeq V(c, \int_{k \in K} V(W(k),F(k)) \\ & =: V(c, [K,V](W,F)) \,. \end{aligned}

Examples

Homotopy limits

For VV some category of higher structures, the local definition of homotopy limit over a diagram F:KCF : K \to C replaces the ordinary notion of cone over FF by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.

One can convince oneself that for the choice of SimpSet for VV this is realized in terms of the weighted limit lim WFlim^W F with the weight WW taken to be

W:KSimpSet W : K \to \Simp\Set
W:kN(K/k), W : k \mapsto N(K/k) \,,

where K/kK/k denotes the over category of KK over kk and N(K/k)N(K/k) denotes its nerve.

This leads to the classical definition of homotopy limits in SimpSet\Simp\Set-enriched categories due to

  • A.K. Bousfield and D.M. Kan, Homotopy limits, completions, and localizations

See for instance also

  • Nicola Gambino, Weighted limits in simplicial homotopy theory (pdf or pdf)

In some nice cases the weight N(K/)N(K/-) can be replaced by a simpler weight; an example is discussed at Bousfield-Kan map.

Homotopy pullback

For instance in the case that K={rts}K = \{r \to t \leftarrow s\} is the pullback diagram we have

W(r)={r} W(r) = \{r\}
W(s)={s} W(s) = \{s\}
W(t)=N({rts}) W(t) = N( \{r \to t \leftarrow s\} )

and W(rt):{r}{rts}W(r \to t) : \{r\} \to \{r \to t \leftarrow s\} injects the vertex rr into {rts}\{r \to t \leftarrow s\} and similarly for W(st)W(s \to t).

This implies that for F:KCF : K \to C a pullback diagram in the SimpSet-eriched category CC, a WW-weighted cone over FF with tip some object cCc \in C, i.e. a natural transformation

WC(c,F()) W \Rightarrow C(c, F(-))

is

  • over rr a “morphism” from the tip cc to F(r)F(r) (i.e. a vertex in the Hom-simplicial set C(c,F(r))C(c,F(r)));

  • similarly over ss;

  • over tt three “morphisms” from cc to F(t)F(t) together with 2-cells between them (i.e. a 2-horn in the Hom-simplicial set C(c,F(t))C(c,F(t)))

  • such that the two outer morphisms over tt are identified with the morphisms over rr and ss, respectively, postcompoised with the morphisms F(rt)F(r \to t) and F(st)F(s \to t), respectively.

So in total such a WW-weighted cone looks like

c F(r) F(rt) F(t) F(st) F(s) \array{ &&& c \\ & \swarrow &\Rightarrow& \downarrow &\Leftarrow& \searrow \\ F(r) && \stackrel{F(r \to t)}{\to} & F(t) & \stackrel{F(s \to t)}{\leftarrow} && F(s) }

as one would expect for a “homotopy cone”.

References for homotopy limits in terms of weighted limits

Details of this are discussed for instance in the book

  • Hirschhorn, Model categories and their localization

To compare with the above discussion notice that

  • The functor

    W:=N(K/) W := N(K/-)

    is discussed there in definition 14.7.8 on p. 269.

  • the VV-enriched hom-category [K,V][K,V] which on VV-functors S,TS,T is the end [K,V](S,T)= kKV(S(k),T(k))[K,V](S,T) = \int_{k \in K} V(S(k), T(k)) appears as hom K(S,T)hom^K(S,T) in definition 18.3.1 (see bottom of the page).

  • for VV set to SimpSet the above definition of homotopy limit appears in example 18.3.6 (2).

Related pages

References

A standard reference is

In

Revised on April 14, 2012 11:01:10 by Noam Zeilberger (85.53.12.113)