The terminal object is the limit of the empty functor F:∅→SetF: \emptyset \to Set. So a terminal object of SetSet is a set XX such that there is a unique function from any set to XX. This is given by any singleton set {a}\{a\}, where the unique function Y→{a}Y \to \{a\} from any set YY is the function that sends every element in YY to aa.

Product

Given two sets A,BA, B, the categorical product is the limit of the diagram (with no non-trivial maps)

A B. \array{ A & B }.

This is given by the usual product of sets, which can be constructed as the set of Kuratowski pairs

A×B={{{a},{a,b}}:a∈A,b∈B}. A \times B = \{ \{\{a\}, \{a, b\}\}: a \in A, b \in B\}.

We tend to write (a,b)(a, b) instead of {{a},{a,b}}\{\{a\}, \{a, b\}\}.

The projection maps π 1:A×B→A\pi_1: A \times B \to A and π 2:A×B→B\pi_2: A \times B \to B are given by

π 1(a,b)=a,π 2(a,b)=b. \pi_1(a, b) = a, \pi_2(a, b) = b.

To see this satisfies the universal property of products, given any pair of maps f:X→Af: X \to A and g:X→Bg: X \to B, we obtain a map (f,g):X→A×B(f, g): X \to A \times B given by

(f,g)(x)=(f(x),g(x)). (f, g)(x) = (f(x), g(x)).

More generally, given a (possibility infinite) collection of sets {A α} α∈I\{A_\alpha\}_{\alpha \in I}, the product of the discrete diagram consisting of these sets is the usual product ∏ α∈IA α\prod_{\alpha \in I} A_\alpha. This can be constructed as

∏ α∈IA α={f:I→⋃ α∈IA α∣f(α)∈A α for all α∈I}. \prod_{\alpha \in I} A_\alpha = \left\{f: I \to \bigcup_{\alpha \in I} A_\alpha \mid f(\alpha) \in A_\alpha\;\text{ for all }\;\alpha \in I\right\}.

Equalizer

Given a pair of functions f,g:X→Yf, g: X \to Y, the equalizer is the limit of the diagram

X ⇉fg Y. \array{ X & \underset{g}{\overset{f}{\rightrightarrows}} & Y }.

The limit is given by a map e:A→Xe: A \to X such that given any a:B→Xa: B \to X, it factors through ee if and only if f∘a=g∘af \circ a = g \circ a. In other words, aa factors through ee if and only if ima⊆{x∈X:f(x)=g(x)}\im a \subseteq \{x \in X: f(x) = g(x)\}. Thus the limit of the diagram is given by

A={x∈X:f(x)=g(x)}, A = \{x \in X: f(x) = g(x)\},

and the map e:A→Xe: A \to X is given by the inclusion.

Pullback

Given two maps f:A→Cf: A \to C and g:B→Cg: B \to C, the pullback is the limit of the diagram

A ↓ f B →g C. \array{ & & A\\ & & \downarrow^\mathrlap{f}\\ B & \underset{g}{\to} & C }.

This limit is given by

{(a,b)∈A×B:f(a)=g(b)}, \{(a, b) \in A \times B: f(a) = g(b)\},

with the maps to AA and BB given by the projections.

While the definition of a pullback is symmetric in ff and gg, it is usually convenient to think of this as pulling back ff along gg (or the other way round). This has more natural interpretations in certain special cases.

If g:B→Cg: B \to C is the inclusion of a subset (ie. is a monomorphism), then the pullback of ff along gg is given by

{a∈A:f(a)∈B}. \{a \in A: f(a) \in B\}.

So this is given by restricting ff to the elements that are mapped into BB.

Further, if f:A→Cf: A \to C is also the inclusion fo a subset, so that AA and BB are both subobjects of CC, then the above formula tells us that the pullback is simply the intersection of the two subsets.

Alternatively, we can view the map f:A→Cf: A \to C as a collection of sets indexed by elements of CC, where the set indexed by c∈Cc \in C is given by A c=f −1(c)A_c = f^{-1}(c). Under this interpretation, pulling ff back along gg gives a collection of sets indexed by elements of BB, where the set indexed by b∈Bb \in B is given b A g(b)A_{g(b)}.

General limits

Given a general Set-valued functor F:D→SetF : D \to Set, if the limit limFlim F exists, then by definition, for any set AA, a function f:A→limFf: A \to lim F is equivalent to a compatible family of maps f d:A→F(d)f_d: A \to F(d) for each d∈Obj(D)d \in Obj(D).

In particular, since an element of a set XX bijects with maps 1→X1 \to X from the singleton 1={∅}1 = \{\emptyset\}, we have

limF≅Set(1,limF)≅[D,Set](const 1,F), lim F \cong Set (1, lim F) \cong [D, Set](const_1, F),

where const 1const_1 is the functor that constantly takes the value 11. Thus the limit is given by the set of natural transformations from const 1const_1 to FF.

More concretely, a compatible family of maps 1→F(d)1 \to F(d) is given by an element s d∈F(d)s_d \in F(d) for each d∈Obj(d)d \in Obj(d), satisfying the appropriate compatibility conditions. Thus, the limit can be realized as a subset of the product ∏ d∈Obj(d)F(d)\prod_{d \in Obj(d)} F(d) of all objects:

limF={(s d) d∈∏ dF(d)|∀(d→fd′):F(f)(s d)=s d′}. lim F = \left\{ (s_d)_d \in \prod_d F(d) | \forall (d \stackrel{f}{\to} d') : F(f)(s_{d}) = s_{d'} \right\}.

Colimits

Initial object

The initial object in SetSet is a set XX such that there is a unique map from XX to any other set. This is given by the empty set ∅\emptyset.

Coproduct

(…)

Coequalizer

(…)

Pushout

(…)

General colimits

The colimit over a Set-valued functor F:D→SetF : D \to Set is a quotient set of the disjoint union ∐ d∈Obj(D)F(d)\coprod_{d \in Obj(D)} F(d):

colimF≃(∐ d∈DF(d))/ ∼, colim F \simeq \left(\coprod_{d\in D} F(d)\right)/_\sim \,,

where the equivalence relation ∼\sim is that which is generated by

((x∈F(d))∼(x′∈F(d′)))if(∃(f:d→d′)withF(f)(x)=x′). ((x \in F(d)) \sim (x' \in F(d')))\quad if \quad (\exists (f : d \to d') \quad with F(f)(x) = x') \,.

If DD is a filtered category then the resulting equivalence relation can be described as follows:

((x∈F(d))∼(x′∈F(d′)))iff(∃d″,(f:d→d″),(g:d′→d″)withF(f)(x)=F(g)(x′)). ((x \in F(d)) \sim (x' \in F(d')))\quad iff \quad (\exists d'', (f : d \to d''), (g: d' \to d'') \quad with F(f)(x) = F(g)(x')) \,.

(If DD is not filtered, then this description doesn’t yield an equivalence relation.)

Limits and colimits of topological spaces

We discuss limits and colimits in the category Top of topological spaces.

examples of universal constructions of topological spaces:

AAAA\phantom{AAAA}limits	AAAA\phantom{AAAA}colimits
\, point space\,	\, empty space \,
\, product topological space \,	\, disjoint union topological space \,
\, topological subspace \,	\, quotient topological space \,
\, fiber space \,	\, space attachment \,
\, mapping cocylinder, mapping cocone \,	\, mapping cylinder, mapping cone, mapping telescope \,
	\, cell complex, CW-complex \,

Definition

Let {X i=(S i,τ i)∈Top} i∈I\{X_i = (S_i,\tau_i) \in Top\}_{i \in I} be a class of topological spaces, and let S∈SetS \in Set be a bare set. Then

For {S→f iS i} i∈I\{S \stackrel{f_i}{\to} S_i \}_{i \in I} a set of functions out of SS, the initial topology τ initial({f i} i∈I)\tau_{initial}(\{f_i\}_{i \in I}) is the topology on SS with the minimum collection of open subsets such that all f i:(S,τ initial({f i} i∈I))→X if_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i are continuous.
For {S i→f iS} i∈I\{S_i \stackrel{f_i}{\to} S\}_{i \in I} a set of functions into SS, the final topology τ final({f i} i∈I)\tau_{final}(\{f_i\}_{i \in I}) is the topology on SS with the maximum collection of open subsets such that all f i:X i→(S,τ final({f i} i∈I))f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I})) are continuous.

Example

For XX a single topological space, and ι S:S↪U(X)\iota_S \colon S \hookrightarrow U(X) a subset of its underlying set, then the initial topology τ intial(ι S)\tau_{intial}(\iota_S), def. , is the subspace topology, making

ι S:(S,τ initial(ι S))↪X \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X

a topological subspace inclusion.

Example

Conversely, for p S:U(X)⟶Sp_S \colon U(X) \longrightarrow S an epimorphism, then the final topology τ final(p S)\tau_{final}(p_S) on SS is the quotient topology.

Proposition

Let II be a small category and let X •:I⟶TopX_\bullet \colon I \longrightarrow Top be an II-diagram in Top (a functor from II to TopTop), with components denoted X i=(S i,τ i)X_i = (S_i, \tau_i), where S i∈SetS_i \in Set and τ i\tau_i a topology on S iS_i. Then:

The limit of X •X_\bullet exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions p ip_i which are the limiting cone components:

lim⟵ i∈IS i p i↙ ↘ p j S i ⟶ S j. \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.

Hence

lim⟵ i∈IX i≃(lim⟵ i∈IS i,τ initial({p i} i∈I)) \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)
The colimit of X •X_\bullet exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. for the component maps ι i\iota_i of the colimiting cocone

S i ⟶ S j ι i↘ ↙ ι j lim⟶ i∈IS i. \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.

Hence

lim⟶ i∈IX i≃(lim⟶ i∈IS i,τ final({ι i} i∈I)) \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)

(e.g. Bourbaki 71, section I.4)

Proof

The required universal property of (lim⟵ i∈IS i,τ initial({p i} i∈I))\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) is immediate: for

(S,τ) f i↙ ↘ f j X i ⟶ X i \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_i }

any cone over the diagram, then by construction there is a unique function of underlying sets S⟶lim⟵ i∈IS iS \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.

The case of the colimit is formally dual.

Example

The limit over the empty diagram in TopTop is the point *\ast with its unique topology.

Example

For {X i} i∈I\{X_i\}_{i \in I} a set of topological spaces, their coproduct ⊔i∈IX i∈Top\underset{i \in I}{\sqcup} X_i \in Top is their disjoint union.

In particular:

Example

The equalizer of two continuous functions f,g:X⟶⟶Yf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the equalizer of the underlying functions of sets

eq(f,g)↪S X⟶g⟶fS Y eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y

(hence the largets subset of S XS_X on which both functions coincide) and equipped with the subspace topology, example .

Example

The coequalizer of two continuous functions f,g:X⟶⟶Yf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the coequalizer of the underlying functions of sets

S X⟶g⟶fS Y⟶coeq(f,g) S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g)

(hence the quotient set by the equivalence relation generated by f(x)∼g(x)f(x) \sim g(x) for all x∈Xx \in X) and equipped with the quotient topology, example .

Example

For

A ⟶g Y f↓ X \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X }

two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted

A ⟶g Y f↓ ↓ g *f X ⟶ X⊔ AY.. \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.

(Here g *fg_\ast f is also called the pushout of ff, or the cobase change of ff along gg.) If gg is an inclusion, one also write X∪ fYX \cup_f Y and calls this the attaching space.

By example the pushout/attaching space is the quotient topological space

X⊔ AY≃(X⊔Y)/∼ X \sqcup_A Y \simeq (X\sqcup Y)/\sim

of the disjoint union of XX and YY subject to the equivalence relation which identifies a point in XX with a point in YY if they have the same pre-image in AA.

(graphics from Aguilar-Gitler-Prieto 02)

Example

As an important special case of example , let

i n:S n−1⟶D n i_n \colon S^{n-1}\longrightarrow D^n

be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space ℝ n\mathbb{R}^n).

Then the colimit in Top under the diagram, i.e. the pushout of i ni_n along itself,

{D n⟵i nS n−1⟶i nD n}, \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,,

is the n-sphere S nS^n:

S n−1 ⟶i n D n i n↓ (po) ↓ D n ⟶ S n. \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.

(graphics from Ueno-Shiga-Morita 95)

Limits and colimits in a preordered set

(..)

Limits and colimits in functor categories

The main point is that limits of functors are computed objectwise. See there for more

Colimit of a representable functor

The colimit of a representable functor (with values in Set) is the point, i.e. the terminal object in Set.

One can readily see this from a universal element-style argument, by direct inspection of cocones.

𝒞(D,C) ⟵(D→fC) * 𝒞(C,C) ↘ ↙ {C→idC} \array{ \mathcal{C}(D,C) && \overset{ (D \overset{f}{\to}C)^\ast }{\longleftarrow} && \mathcal{C}(C,C) \\ & \searrow && \swarrow \\ && \left\{ C \overset{id}{\to}C \right\} }

However, this style of reasoning does not easily generalize to higher category theory. The following gives a more abstract argument that is short and generalizes. We state it in (∞,1)-category theory just for definiteness of notation:

Proposition

(homotopy colimit over a representable functor is contractible)

Let 𝒞\mathcal{C} be a small (∞,1)-category and consider an ∞-groupoid-valued (∞,1)-functor F:𝒞 op→Groupoids ∞F \colon \mathcal{C}^{op} \to Groupoids_\infty which is representable, i.e. in the image of the (∞,1)-Yoneda embedding F≃yCF \simeq y C:

𝒞 ⟶AAyAA Func(𝒞 op,Groupoids ∞) C ↦ yC≔𝒞(−,C) \array{ \mathcal{C} & \overset{ \phantom{AA} y \phantom{AA} }{\longrightarrow} & Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \\ C &\mapsto& y C \coloneqq \mathcal{C}(-,C) }

Then the (∞,1)-colimit over this (∞,1)-functor is contractible, i.e. is the point, the terminal object in ∞Groupoids:

lim⟶𝒞(yC)≃* \underset{ \underset{\mathcal{C}}{\longrightarrow} }{\lim} (y C) \;\simeq\; \ast

Proof

The terminal ∞\infty-groupoid *\ast is characterized by the fact that for each S∈Groupoids ∞S \in Groupoids_\infty we have Groupoids ∞(*,S)≃SGroupoids_\infty(\ast, S) \simeq S. Therefore it is sufficient to show that lim⟶(yC)\underset{\longrightarrow}{\lim}\big(y C\big) has the same property:

Groupoids ∞(lim⟶(yC),S) ≃Func(𝒞 op,Groupoids ∞)(yC,constS) ≃(constS)(C) ≃S \begin{aligned} Groupoids_\infty \left( \underset{\longrightarrow}{\lim} \big( y C \big) , S \right) & \simeq\; Func( \mathcal{C}^{op} ,\, Groupoids_\infty ) \left( y C , const S \right) \\ & \simeq\; (const S)(C) \\ & \simeq\; S \end{aligned}

Here the first step is the (∞,1)-adjunction

Func(𝒞 op,Groupoids ∞)AA⊥AA⟵const⟶lim⟶Groupoids ∞ Func \big( \mathcal{C}^{op}, Groupoids_\infty \big) \underoverset { \underset{const}{\longleftarrow} } { \overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow} } {\phantom{AA}\bot\phantom{AA}} \Groupoids_\infty

and the second step is the (∞,1)-Yoneda lemma

Examples of limits

In the following examples, DD is a small category, CC is any category and the limit is taken over a functor F:D op→CF : D^{op} \to C.

Simple diagrams

the limit of the empty diagram D=∅D = \emptyset in CC is, if it exists the terminal object;
if DD is a discrete category, i.e. a category with only identity morphisms, then a diagram F:D→CF : D \to C is just a collection c ic_i of objects of CC. Its limit is the product ∏ ic i\prod_i c_i of these.
if D={a→→b}D = \{a \stackrel{\to}{\to} b\} then limFlim F is the equalizer of the two morphisms F(b)→F(a)F(b) \to F(a).
if DD has an terminal object II (so that II is an initial object in D opD^{op}), then the limit of any F:D op→CF : D^{op} \to C is F(I)F(I).

Filtered limits

if DD is a poset, then the limit over D opD^{op} is the supremum over the F(d)F(d) with respect to (F(d)≤F(d′))⇔(F(d)←F(≤)F(d′))(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\leftarrow} F(d'));
the generalization of this is where the term “limit” for categorical limit (probably) originates from: for DD a filtered category, hence D opD^{op} a cofiltered category, one may think of (d→fd′)↦(F(d)←F(f)F(d′)(d \stackrel{f}{\to} d') \mapsto (F(d) \stackrel{F(f)}{\leftarrow} F(d') as witnessing that F(d′)F(d') is “larger than” F(d)F(d) in some sense, and limFlim F is then the “largest” of all these objects, the limiting object. This interpretation is perhaps more evident for filtered colimits, where the codomain category CC is thought of as being the opposite C=E opC = E^{op}. See the motivation at ind-object.

In terms of other operations

If products and equalizers exist in CC, then limit of F:D op→CF : D^{op} \to C can be exhibited as a subobject of the product of the F(d)F(d), namely the equalizer of

∏ d∈Obj(D)F(d)→⟨F(f)∘p F(t(f))⟩ f∈Mor(D)∏ f∈Mor(D)F(s(f)) \prod_{d \in Obj(D)} F(d) \stackrel{\langle F(f) \circ p_{F(t(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f))

and

∏ d∈Obj(D)F(d)→⟨p F(s(f))⟩ f∈Mor(D)∏ f∈Mor(D)F(s(f)). \prod_{d \in Obj(D)} F(d) \stackrel{\langle p_{F(s(f))} \rangle_{f \in Mor(D)} }{\to} \prod_{f \in Mor(D)} F(s(f)) \,.

See the explicit formula for the limit in Set in terms of a subset of a product set.

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

Limits in presheaf categories

Consider limits of functors F:D op→PSh(C)F : D^{op} \to PSh(C) with values in the category of presheaves over a category CC.

Proposition

(limits of presheaves are computed objectwise)

Limits of presheaves are computed objectwise:

limF:c↦limF(−)(c) lim F : c \mapsto lim F(-)(c)

Here on the right the limit is over the functor F(−)(c):D op→SetF(-)(c) : D^{op} \to Set.

Similarly for colimits

Similarly colimits of presheaves are computed objectwise.

Proposition

The Yoneda embedding Y:C→PSh(C)Y : C \to PSh(C) commutes with small limits:

Let F:D op→CF : D^{op} \to C, then we have

Y(limF)≃lim(Y∘F) Y(lim F) \simeq lim (Y\circ F)

if limFlim F exists.

Warning The Yoneda embedding does not in general preserve colimits.

Limits in under-categories

Limits in under categories are a special case of limits in comma categories. These are explained elsewhere. It may still be useful to spell out some details for the special case of under-categories. This is what the following does.

Proposition

Limits in an under category are computed as limits in the underlying category.

Precisely: let CC be a category, t∈Ct \in C an object, and t/Ct/C the corresponding under category, and p:t/C→Cp : t/C \to C the obvious projection.

Let F:D→t/CF : D \to t/C be any functor. Then, if it exists, the limit over p∘Fp \circ F in CC is the image under pp of the limit over FF:

p(limF)≃lim(pF) p(\lim F) \simeq \lim (p F)

and limF\lim F is uniquely characterized by lim(pF)\lim (p F).

Proof

Over a morphism γ:d→d′\gamma : d \to d' in DD the limiting cone over pFp F (which exists by assumption) looks like

limpF ↙ ↘ pF(d) →pF(γ) pF(d′) \array{ && \lim p F \\ & \swarrow && \searrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

By the universal property of the limit this has a unique lift to a cone in the under category t/Ct/C over FF:

t ↙ ↓ ↘ limpF ↓ ↙ ↘ ↓ pF(d) →pF(γ) pF(d′) \array{ && t \\ & \swarrow &\downarrow & \searrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

It therefore remains to show that this is indeed a limiting cone over FF. Again, this is immediate from the universal property of the limit in CC. For let t→Qt \to Q be another cone over FF in t/Ct/C, then QQ is another cone over pFp F in CC and we get in CC a universal morphism Q→limpFQ \to \lim p F

t ↙ ↓ ↘ Q ↓ ↙ ↓ ↘ ↓ limpF ↓ ↙ ↘ ↓ pF(d) →pF(γ) pF(d′) \array{ && t \\ & \swarrow & \downarrow & \searrow \\ && Q \\ \downarrow & \swarrow &\downarrow & \searrow & \downarrow \\ && \lim p F \\ \downarrow & \swarrow && \searrow & \downarrow \\ p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d') }

A glance at the diagram above shows that the composite t→Q→limpFt \to Q \to \lim p F constitutes a morphism of cones in CC into the limiting cone over pFp F. Hence it must equal our morphism t→limpFt \to \lim p F, by the universal property of limpF\lim p F, and hence the above diagram does commute as indicated.

This shows that the morphism Q→limpFQ \to \lim p F which was the unique one giving a cone morphism on CC does lift to a cone morphism in t/Ct/C, which is then necessarily unique, too. This demonstrates the required universal property of t→limpFt \to \lim p F and thus identifies it with limF\lim F.

Remark: One often says “pp reflects limits” to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if U:A→CU: A \to C is monadic (i.e., has a left adjoint FF such that the canonical comparison functor A→(UF)-AlgA \to (U F)\text{-}Alg is an equivalence), then UU both reflects and preserves limits. In the present case, the projection p:A=t/C→Cp: A = t/C \to C is monadic, is essentially the category of algebras for the monad T(−)=t+(−)T(-) = t + (-), at least if CC admits binary coproducts. (Added later: the proof is even simpler: if U:A→CU: A \to C is the underlying functor for the category of algebras of an endofunctor on CC (as opposed to algebras of a monad), then UU reflects and preserves limits; then apply this to the endofunctor TT above.)

Further resources

Pedagogical vidoes that explain limits and colimits are at

The Catsters, General Limits and Colimits

Last revised on April 20, 2023 at 17:58:37. See the history of this page for a list of all contributions to it.