Domenico Fiorenza
limit

Limit

Here I’ll try to reorganize the entry limit according to the structure discussed in the forum entry “objects” and tools to compute them. As this page will be complete I’ll move it into the Lab. Editing of this page is strongly encouraged :)

Contents

Idea

In category theory a limit of a diagram F:DCF\colon D \to C in a category CC is an object limFlim F of CC equipped with morphisms to the objects F(d)F(d) for all dDd \in D, such that everything in sight commutes. Moreover, the limit limFlim F is the universal object with this property, i.e. the “most optimized solution” to the problem of finding such an object.

One can think of the category of cones over F:DCF\colon D\to C as the collection of all displacements of arrows stemming from a fixed “source” CC (the base of the cone) suitably linked by morphisms.

This in turn can be regarded as a functor from the diagram [0]D[0]\star D to CC ‘extending’ FF (see the page over quasi-category); now the universal property makes limF\text{lim}\; F a terminal object in the category of these functors.

In the (,1)(\infty,1)-categorical setting, this suggests to define limits again in terms of terminal objects, but using the joins of quasi-categories instead of the join of categories?.

Definition

For KK and CC two quasi-categories and F:KCF : K \to C a morphism of quasi-categories (i.e. nothing more than a map of simplicial sets?. Then the limit of FF is, if it exists, the quasi-categorical terminal object in the over quasi-category C /FC_{/F}:

limF:=TerminalObj(C /F), \underset{\!\!\leftarrow}{\lim} \; F := TerminalObj(C_{/F}) \,,

where C /F=Hom F([0]K,C)C_{/F}=Hom_{F}([0] \star K, C ). (This is HTT Definition 1.2.13.4)

As an object of C /FC_{/F} the limit of FF can be regarded as a 0-simplex in the simplicial set nHom F([n]K,C)C [n]Kn\mapsto Hom_{F}([n] \star K, C )\subset C^{[n] \star K}: see HTT Notation 1.2.8.4 and the join of quasi-categories? page.

Generalization to weighted limits

Fosco: this generalization is not at all evident, since the definition of weighted limit seems to conflict (see the page weighted join) with the classical one.

There is an evident generalization to weighted limit?s: replace in the above the join [0]K[0] \star K with the weighted join [0] WK[0] \star_W K where WW is any functor W:DSetW \colon D \to Set – then called the weight. The WW-weighted limit of FF, lim WF\lim_W F, also written {W,F}\{W,F\}, is, if it exists, the quasicategorical terminal oject in Hom F([0] WK,C)Hom_{F}([0] \star_W K, C ).

To be continued…

Below this line is the entry limit as it was on the nLab on January 4, 2010.


Contents

Idea

In category theory? a limit of a diagram? F:DCF : D \to C in a category? CC is an object? limFlim F of CC equipped with morphisms to the objects F(d)F(d) for all dDd \in D, such that everything in sight commutes. Moreover, the limit limFlim F is the universal object with this property, i.e. the “most optimized solution” to the problem of finding such an object.

The limit construction has a wealth of applications throughout category theory and mathematics in general. In practice, it is possibly best thought of in the context of representable functor?s as a classifying space for maps into a diagram. So in some sense the limit object limFlim F “subsumes” the entire diagram F(D)F(D) into a single object, as far as morphisms into it are concerned. The corresponding universal object for morphisms out of the diagram is the colimit?.

Often, the general theory of limits (but not colimits!) works better if the source of FF is taken to be the opposite category? D opD^op (or equivalently, if FF is taken to be a contravariant functor?). This is what we do below. In any given situation, of course, you use whatever categories and functors you're interested in.

In some cases the category-theoretic notion of limit does reproduce notions of limit as known from analysis. See the examples below.

Global versus local

In correspondence to the local defintion of adjoint functor?s (as discussed there), there is a local definition of limits (in terms of cones), that defines a limit (if it exists) for each individual diagram, and there is a global definition, which defines the limit for all diagrams (in terms of an adjoint?).

If all limits over the given shape of diagrams exist in a category, then both definitions are equivalent.

See also the analogous discussion at homotopy limit?.

Terminology and notation

A limit is taken over a functor? F:D opCF : D^{op} \to C and since the functor comes equipped with the information about what its domain is, one can just write limF\lim F for its limit. But often it is helpful to indicate how the functor is evaluated on objects, in which case the limit is written lim dDF(d)\lim_{d \in D} F(d); this is used particularly when FF is given by a formula (as with other notation with bound variables.)

In some schools of mathematics, limits are called projective limits, while colimits are called inductive limits. Also seen are (repsectively) inverse limits and direct limits. Both these systems of terminology are alternatives to using ‘co-’ when distinguishing limits and colimits. The first system also appears in pro-object? and ind-object?.

Correspondingly, the symbols lim lim_{\leftarrow} and lim lim_{\rightarrow} are used instead of limlim and colimcolim. (Actually, the arrows should be directly underneath the ‘limlim’s, something like lim{\lim \atop \longleftarrow} and lim{\lim \atop \longrightarrow}. But the text should also be at normal size.)

Confusingly, many authors restrict the meanings of these alternative terms to (co)limits whose sources are directed set?s; see directed limit?. In fact, this is the original meaning; projective and inductive limits in this sense were studied in algebra before the general category-theoretic notion of (co)limit.

Local definition in terms of representable functors

There is a general abstract definition of limits in terms of representable functors, which we describe now. This reproduces the more concrete and maybe more familiar description in terms of universal cones, which is described further below.

Let in the following DD be a small category? and Set? the category of sets (possibly realized as the category USetU Set of UU-small sets with respect to a given Grothendieck universe?.)

Limit of a Set-valued functors

The limit of a Set-valued functor F:D opSetF : D^{op} \to Set is the hom-set?

limF:=Hom [D op,Set](pt,F)Set lim F := Hom_{[D^{op}, Set]}(pt, F) \in Set

in the functor category? [D op,Set][D^{op}, Set] (the presheaf? category), where

pt:D opSet pt : D^{op} \to Set
pt:d{*} pt : d \mapsto \{*\}

is the functor constant on the point?, i.e. the terminal? diagram.

The set limFlim F is equivalently called

The set limFlim F can be equivalently expressed as an equalizer? of a product?, explicitly:

limF{(x d) dD dDF(d)|(d iαd j)D:F(α)(x d j)=x d i} lim F \simeq \left\lbrace (x_d)_{d \in D} \in \prod_{d \in D} F(d) | \forall (d_i \stackrel{\alpha}{\to} d_j) \in D : F(\alpha)(x_{d_j}) = x_{d_i} \right\rbrace

In particular, the limit of a set-valued functor always exists.

Notice the important triviality that the covariant hom-functor? comutes with set-valued limits: for every set SS we have a bijection of sets

Hom Set(S,limF)limHom Set(S,F()), Hom_{Set}(S, lim F) \simeq \lim Hom_{Set}(S, F(-)) \,,

where Hom(S,F()):D opSetHom(S, F(-)) : D^{op} \to Set.

Limit of a functor with values in an arbitrary category

The above formula generalizes straightforwardly to

a notion of limit for functors F:D opCF : D^{op} \to C for CC an arbitrary category if we take the object “limFlim F” to be a presheaf? on CC. The true limFlim F is then, if it exists, the object of CC representing? this presheaf.

More precisely, using the the Yoneda embedding? Y:C[C op,Set]Y : C \to [C^{op}, Set] define for F:D opCF : D^{op} \to C the presheaf? lim^F[C op,Set]\hat \lim F \in [C^{op}, Set] by the analog of the above formula

(lim^F)(d)Hom [C op,Set](Y(d),limF):=limHom [C op,Set](Y(d),F()) (\hat \lim F)(d) \simeq Hom_{[C^{op}, Set]}(Y(d), lim F) := \lim Hom_{[C^{op}, Set]}(Y(d), F(-))

for all dDd \in D.

Here the lim\lim on the right is again that of Set?-valued functors defined before.

By the above this can also be written as

(lim^F)(c)=Hom [D op,Set](pt,Hom [C op,Set](c,F()) (\hat lim F)(c) = Hom_{[D^{op}, Set]}(pt , Hom_{[C^{op}, Set]}(c,F(-))

or, suppressing the subscripts for readability:

(lim^F)(c)=Hom(pt,Hom(c,F()). (\hat lim F)(c) = Hom(pt , Hom(c,F(-)) \,.

So also the presheaf?-valued limit always exist. If this presheaf is representble? by an object limFlim F of FF, then this is the limit of FF:

Hom(c,limF)Hom(pt,Hom(c,F())). Hom(c, \lim F) \simeq Hom(pt, Hom(c,F(-))) \,.

Generalization to weighted limits

In the above formulation, there is an evident generalization to weighted limit?s:

replace in the above the constant terminal functor pt:D opSetpt : D^{op} \to Set with any functor W:D opSetW : D^{op} \to Set – then called the weight –, then the WW-weighted limit of FF

lim WF \lim_W F

often written

{W,F} \{W,F\}

is, if it exists, the object representing the presheaf

dHom [D op,Set](W,Hom([C op,Set]))(c,F()), d \mapsto Hom_{[D^{op}, Set]}(W , Hom([C^{op}, Set]))(c,F(-)) \,,

i.e. such that

Hom(c,lim WF)Hom(W,Hom(c,F())) Hom(c, \lim_W F) \simeq Hom(W, Hom(c,F(-))) \,

naturally in cCc \in C.

Relation to continuous functors

The very definition of limit as above asserts that the covariant hom-functor? Hom(c,):CSetHom(c,-) : C \to Set commutes with forming limits. Indeed, the definition is equivalent to saying that the hom-functor? is a continuous functor?.

Definition in terms of universal cones

Unwrapping the above abstract definition of limits yields the following more hands-on description in terms of universal cones.

Unwrapping

Let F:D opCF : D^{op} \to C be a functor.

Notice that for every object cCc \in C an element

*Hom(pt,Hom(c,F())) * \to Hom(pt, Hom(c, F(-)))

is to be identified with a collection of morphisms

cF(d) c \to F(d)

for all dDd \in D, such that all triangles

c F(d i) F(f) F(d j) \array{ && c \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) }

commute. Such a collection of morphisms is called a cone over FF, for the obvious reason.

If the limit limFC\lim F \in C of FF exist, then it singles out a special cone given by the composite morphism

**Id limFHom C(limF,limF)Hom(pt,Hom(limF,F())), * \stackrel{* \mapsto Id_{\lim F}}{\to} Hom_C(\lim F, \lim F) \stackrel{\simeq}{\to} Hom(pt, Hom(\lim F, F(-))) \,,

where the first morphism picks the identity morphism? on limF\lim F and the second one is the defining bijection of a limit as above.

This cone

limF F(d i) F(f) F(d j) \array{ && \lim F \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) }

is called the universal cone over FF, because, again by the defining proprty of limit as above, everey other cone {cF(d)} dD\{c \to F(d)\}_{d \in D} as above is bijectively related to a morphism climFc \to \lim F

*{cF(d)} dDHom(pt,Hom(c,F()))Hom(c,limF). * \stackrel{\{c \to F(d)\}_{d \in D}}{\to} Hom(pt, Hom(c, F(-))) \stackrel{\simeq}{\to} Hom(c, \lim F) \,.

By inspection one finds that, indeed, the morphism climFc \to \lim F is the morphism which exhibits the factorization of the cone {cF(d)} dD\{c \to F(d)\}_{d \in D} through the universal limit cone

c F(d i) F(f) F(d j)= c limF F(d i) F(f) F(d j). \array{ && c \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } = \array{ && c \\ && \downarrow \\ && \lim F \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } \,.

An illustrative example is the following: a limit of the identity functor? Id c:CCId_c:C\to C is, if it exists, an initial object? of CC.

Global Definition in terms of adjoint of the constant diagram functor

Given categories DD and CC, limits over functors D opCD^{op} \to C may exist for some functors, but not for all. If it does exist for all functors, then the above local definition of limits is equivalent to the following global definition.

For DD a small category? and CC any category, the functor category? [D op,C][D^{op},C] is the category of DD-diagram?s in CC. Pullback along the functor D opptD^{op} \to pt to the terminal? category pt={}pt = \{\bullet\} induces a functor

const:C[D op,C] const : C \to [D^{op},C]

which sends every object of CC to the diagram functor constant on this object.

The left adjoint?

colim D:[D op,C]C colim_D : [D^{op},C] \to C

of this functor is, if it exists, the functor which sends every diagram to its colimit? and the right adjoint? is, if it exists, the functor

lim D:[D op,C]C lim_D : [D^{op},C] \to C

which sends every diagram to its limit. The Hom-isomorphisms of these adjunction?s state precisely the universal property of limit and colimit? given above.

Concretely this means that for all cCc \in C we have a bijection

Hom C(c,limF)Hom [D op,C](const X,F). Hom_C(c, \lim F) \simeq Hom_{[D^{op},C]}(const_X, F) \,.

Compare this with the discussion at Kan extension?.

Relation to Kan extension

From this perspective, a limit is a special case of a Kan extension?, as described there, namely a Kan extension to the point?.

Definition for (,1)(\infty,1)-categories

The definition of a limit as a terminal cone has a straightforward generalization to the context of (infinity,1)-categories?.

Definition

For KK and CC two quasi-categories? and F:KCF : K \to C a morphism of quasi-categories?, the limit over FF is, if it exists, the quasi-categorical terminal object? in the over quasi-categories C /FC_{/F}:

limF:=TerminalObj(C /F). lim F := TerminalObj(C_{/F}) \,.

For more details see limit in quasi-categories?.

Examples

Types of shapes of limit cones

Here are some important examples of limits, classified by the shape of the diagram:

Properties

Limits in Set are hom-sets

For F:D opSetF : D^{op} \to Set any functor and const *:D opSetconst_{*} : D^{op} \to Set the functor constant on the point?, the limit of FF is the hom-set?

limF[D op,Set](const *,F) lim F \simeq [D^{op}, Set](const_{*}, F)

in the functor category?, i.e. the set of natural transformation?s from the constant functor into FF.

Covariant Hom commutes with limits

For CC a locally small? category, for F:D opCF : D^{op} \to C a functor and writing C(c,F()):CD opSetC(c, F(-)) : CD^{op} \to Set, we have

C(c,limF)limC(c,F()). C(c, lim F) \simeq lim C(c, F(-)) \,.

Depending on how one introduces limits this holds by definition or is an easy consequence.

Proposition – limits in functor categories are computed pointwise

Let DD be a small category and let DD' be any category. Let CC be a category which admits limits of shape DD. Write [D,C][D',C] for the functor category?. Then * [D,C][D',C] admits DD-shaped limits; * these limits are computed objectwise (“pointwise”) in CC: for F:D op[D,C]F : D^{op} \to [D',C] a functor we have for all dDd' \in D' that (limF)(d)lim(F()(d))(lim F)(d') \simeq lim (F(-)(d')). Here the limit on the right is in CC.

Proposition – small limits commute with small limits

Let DD and DD' be small catgeories and let CC be a category which admits limits of shape DD as well as limits of shape DD'. Then these limits commute with each other, in that

for F:D op×D opCF : D^{op} \times {D'}^{op} \to C a functor , with corresponding induced functors F D:D op[D op,C]F_D : {D'}^{op} \to [D^{op},C] and F D:D op[D op,C]F_{D'} : {D}^{op} \to [{D'}^{op},C], then

limFlim D(lim DF D)lim D(lim DF D). lim F \simeq lim_{D} (lim_{D'} F_D ) \simeq lim_{D'} (lim_{D} F_{D'} ) \,.
Proposition – right adjoints commute with limits

Let R:CCR : C \to C' be a functor that is right adjoint? to some functor L:CCL : C' \to C. Let DD be a small category such that CC admits limits of shape DD. Then RR commutes with DD-shaped limits in CC in that

for F:D opCF : D^{op} \to C some diagram, we have

R(limF)lim(RF). R(lim F) \simeq lim (R \circ F) \,.
Proof

Using the adjunction isomorphism and the above fact that Hom commutes with limits, one obtains for every cCc' \in C'

C(c,R(limF)) C(L(c),limF) limC(L(c),F) limC(c,RF) C(c,lim(RF)).. \begin{aligned} C'(c', R (lim F)) & \simeq C(L(c'), lim F) \\ & \simeq lim C(L(c'), F) \\ & \simeq lim C'(c', R\circ F) \\ & \simeq C'(c', lim (R \circ F)) \,. \end{aligned} \,.

Since this holds naturally for every cc', the Yoneda lemma, corollary II? on uniquenes of representing objects implies that R(limF)lim(GF)R (lim F) \simeq lim (G \circ F).

Proposition – limits are equalizers of products

The limit of F:D opCF : D^{op} \to C is, if it exists, a subobject of the product of the F(d)F(d), namely the equalizer of

dObj(D)F(d) fMor(d)(F(f)p t(f)) fMor(D)F(s(f)) \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (F(f) \circ p_{t(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f))

and

dObj(D)F(d) fMor(d)(p s(f)) fMor(D)F(s(f)). \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (p_{s(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f)) \,.

In particular therefore, a category has all limits already if it has all products and equalizers.

See limits and colimits by example? for what this formula says for instance for the special case C=C = Set?.

Commutativity of limits with colimits

In general limits do not commute with colimits. But under a number of special conditions of interest they do. More on that at commutativity of limits and colimits?.

Revised on October 28, 2013 at 09:49:20 by Fosco Loregian