nLab
weighted limit

Context

Category theory

Enriched category theory

Limits and colimits

Contents

Idea

The notion of weighted limit (also called indexed 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 \colon 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 \colon 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 this example 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(-)) \colon 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 \colon K \to Set;

  2. to generalize everything straightforwardly from the Set-enriched context to arbitrary VV-enriched contexts (see below).

The idea is that the weight W:KVW \colon 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 the end 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}

Motivation from enriched category theory

Let VV be a monoidal category.

Imagine you’re tasked to write down the definition of limit in a category CC enriched over VV. You would start saying there is a diagram F:KCF \colon K \to C and a limit is a universal cone over it, i.e. it’s the universal choice of an object cc together with an arrow f k:cF(k)f_k \colon c \to F(k) for each object kk of KK.

Here’s where you stop and ask yourself: what is ‘an arrow’ in CC? CC has no hom-sets — it has hom-objects — hence what’s ‘an element’ of C(c,F(k))C(c, F(k)) in VV?

There are two ways to specify an element of an object XX in a monoidal category (V,I,)(V, I, \otimes):

  1. Give an arrow IXI \to X (think of sets, where elements of XX are indeed the same thing as arrows {*}X\{*\} \to X. These are called global elements of XX, and are more often than not a misbehaved notion of element, since often II is ‘too big’ to thoroughly probe XX (on the other hand, notice the underlying category of an enriched category is defined by taking global elements of the hom-objects)
  2. Give any arrow into XX. These are called generalized elements, and the existence of the Yoneda embedding assures us they completely capture the categorical structure of VV.

Hence you now say: a cone over FF is a choice of a generalized element f kf_k of C(c,F(k))C(c, F(k)), for every kk in KK. This means specifying an arrow W kC(c,F(k))W_k \to C(c, F(k)) in VV, for each kk. It’s now quite natural to ask for the functoriality of this choice in kk, hence we end up defining a ‘generalized cone’ over FF as an element

[K,V](W(),C(c,F())) [K, V](W(-), C(c, F(-)))

Hence WW is simply a uniform way to specify the sides of a cone. A confirmation that this is indeed the right definition of limit in the enriched settings come from the fact that ‘conical completeness’ (a conical limit now is one where W=ΔIW = \Delta I, hence we pick only global element) is an inadequate notion, see for example Section 3.9 in Kelly’s book (aptly named The inadequacy of conical limits).

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

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 shape of pullback diagrams 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-enriched 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, postcomposed 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

  • Max Kelly, section 3.1, p. 37 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press 1982, 245 pp.; remake: TAC reprints 10 (tac:tr10, pdf)

In

is given an account of lectures by Mike Shulman on the subject. The definition appears there as definition 3.1, p. 4 (in a form a bit more general than the one above).

The analogous notion of weighted (infinity,1)-limit is discussed in

  • Martina Rovelli, Weighted limits in an (∞,1)-category, 2019, arxiv:1902.00805

Last revised on September 18, 2021 at 05:18:25. See the history of this page for a list of all contributions to it.