Limits - Category Theory - Cambridge III notes

4 Limits

Definition 4.1 (Diagram). Let $J$ be a category (almost always small, and often finite). By a diagram of shape $J$ in a category $C$ , we mean a functor $D : J \to C$ . The objects $D (j)$ , $j \in ob J$ are called vertices of $D$ , and morphisms $D (α)$ , $α \in mor J$ are called edges of $D$ .

For example, if $J$ is the category

a diagram of shape

J

is a commutative square in

C

If $J$ is instead

then a diagram of shape

J

is a not-necessarily-commutative square.

Definition 4.2 (Cone, limit). Let $D : J \to C$ be a diagram. A cone over $D$ consists of an object $A$ (its apex) together with morphisms $λ_{j} : A \to D (j)$ for each $j \in ob J$ (the legs of the cone) such that

commutes for each

α : j \to j^{'}

J

A morphism of cones $(A, (λ_{j} | j \in ob J)) \to (B, (μ_{j} | j \in ob J))$ is a morphism $f : A \to B$ such that $μ_{j} f = λ_{j}$ for all $j$ . We have a category $C o n e (D)$ of cones over $D$ ; a limit for $D$ is a terminal object of $C o n e (D)$ .

Dually, a colimit for $D$ is an initial cone under $D$ .

If $Δ : C \to [J, C]$ is the functor sending $A$ to the constant diagram with all vertices $A$ then a cone over $D$ is a natural transformation $Δ A \to D$ .

Also, $C o n e (D)$ is another name for $(Δ ↓ D)$ , defined as in Theorem 3.3 $^{o p}$ .

So by Theorem 3.3, $C$ has limits for all diagrams of shape $J$ if and only if $Δ$ has a right adjoint.

Example 4.3.

(a) Suppose $J = \emptyset$ . If $D : \emptyset \to C$ , then $C o n e (D) ≅ C$ , so a limit for $D$ is a terminal object.
(b) If $J = ∙ ∙$ , a diagram of shape $J$ is a pair $A, B$ , and a cone over it is a span
A limit for it is a categorical coproduct
Dually, a colimit for it is a coproduct
(c) If $J$ is a (small) discrete category, a (co)limit for $(A_{j} | j \in J)$ is a (co)product $\prod_{j \in J} A_{j}$ ( $\sum_{j \in J} A_{j}$ ).
(d) If $J$ is $∙ ⇉_{}^{} ∙$ , then a diagram of shape $J$ is a parallel pair $A ⇉_{g}^{f} B$ . A cone over it consists of
satisfying $f h = k = g h$ , or equivalently of $C \overset{h}{\to} A$ satisfying $f h = g h$ . So a limit for $A ⇉_{g}^{f} B$ is an equaliser for $(f, g)$ , as defined in Example 2.6(g).
(e) If $J$ is
then a diagram of shape $J$ is a cospan
A cone over it has $3$ legs, but if we omit the (redundant) middle one, it’s a span
completing the cospan to a commutative square.
A limit for
is called a pullback for $(f, g)$ . If $C$ has binary products and equalisers, we can construct pullbacks by forming the equaliser $A \times B ⇉_{g π_{2}}^{f π_{1}} C$ . Dually, colimits of shape $J^{o p}$ are called pushouts.
(f) If $M = {1, e}$ is the $2$ -element with $e^{2} = e$ , a diagram of shape $M$ is an object $A$ equipped with an idempotent $A \overset{e}{\to} A$ . A limit (respectively colimit) for $(A, e)$ is the monic (respectively epic) part of a splitting of $e$ .
Note that the functor $S e t \overset{F}{\to} [M, S e t]$ in Example 3.2(e) is $Δ$ , so this explains the coincidence of left and right adjoints.
(g) Suppose $J = ℕ$ is the ordered set of natural numbers. A diagram of shape $ℕ$ is a direct sequence
$A_{0} \to A_{1} \to A_{2} \to A_{3} \to \dots,$

and a colimit for it is called a direct limit $A_{\infty}$ .

Dually, we have inverse sequences
$\dots \to A_{2} \to A_{1} \to A_{0},$

and their limits are called inverse limits.

For example in topology, an infinite dimensional CW-complex $X$ is the direct limit of its $n$ -skeletons $X_{n}$ . In algebra, the ring of $p$ -adic integers is the limit of the inverse sequence
$\dots \to ℤ ∕ p^{3} ℤ \to ℤ ∕ p^{2} ℤ \to ℤ ∕ p ℤ \to {0}$

in $R n g$ .

Proposition 4.4. Assuming that:

$C$ a category

Then

(i) If $C$ has equalisers and all small products (including empty product), then $C$ has all small limits.
(ii) If $C$ has equalisers and all finite products (including empty product), then $C$ has all finite limits.
(iii) If $C$ has pullbacks and a terminal object, then $C$ has all finite limits.

Proof.

(i) & (ii) Let $D : J \to C$ be a diagram. Form the products $\begin{array}{l} P & = \prod_{j \in ob J} D (j) \\ Q & = \prod_{α \in mor J} D (cod α) \end{array}$
We have morphisms $P ⇉_{g}^{f} Q$ defined by $π_{α} f = π_{cod α}$ and $π_{α} g = D (α) π_{dom α}$ for all $α$ . Let $e \overset{e}{\to} P$ be an equaliser for $(f, g)$ . The morphisms $λ_{j} = π_{j} e : E \to P \to D (j)$ form a cone over $D$ , since for any $α : j \to j^{'}$ we have
$D (α) λ_{j} = D (α) π_{j} e = π_{α} g e = π_{α} f e = π_{j^{'}} e = λ j^{'} .$

It is a limit: given any cone $(A, (μ_{j} | j \in ob J))$ over $D$ , the $μ_{j}$ form a cone over the discrete diagram with vertices $D (j)$ , so they induce a unique $μ : A \to P$ . Then $f μ = g μ$ since the $μ_{j}$ s form a cone over $D$ , so $μ$ factors uniquely as $e ν$ , and $ν$ is the unique factorisation of $(μ_{j} | j \in ob J)$ through $(λ_{j} | j \in ob J)$ .
(iii) If $1$ is a terminal object of $C$ , then we can construct $A \times B$ as the pullback of
Then we can construct $\prod_{i = 1}^{n} A_{i}$ as $A_{1} \times (A_{2} \times (\dots \times A_{n}) \dots))$ .
To form an equaliser of $A ⇉_{g}^{f} B$ , consider the pullback of
Any cone
over this has $h = k = π_{1} (1_{A}, g) k = π_{1} (1_{A}, f) h$ . So a limit cone has the universal property of an equaliser for $(f, g)$ . □

Definition 4.5 (Limit preserving / reflecting / creating). Let $F : C \to D$ be a functor.

(a)
We say $F$ preserves limits of shape $J$ if, given $D : J \to C$ and a limit cone $(L, (λ_{j} | j \in ob J))$ for it, $(F L, (F λ_{j} | j \in ob J))$ is a limit for $F D : J \to D$ .
(b)
We say $F$ reflects limits of shape $J$ if given $D : J \to C$ , any cone over $D$ which maps to a limit cone in $D$ is a limit in $C$ .
(c)
We say $F$ creates limits of shape $J$ if, given $D : J \to C$ and a limit cone $(L, (λ_{j} | j \in ob J))$ over $F D$ , there exists a cone over $D$ whose image under $F$ is $≅ (L, (λ_{j}))$ , and any such cone is a limit in $C$ .

We say a category $C$ is complete if it has all small limits.

Corollary 4.6. In each of the statements of Proposition 4.4, we may replace ‘ $C$ has’ by either ‘ $D$ has and $G : D \to C$ preserves’ or ‘ $C$ has and $D \to C$ creates’.

Proof. Exercise. □

Example 4.7.

(a) The functor $G p \to S e t$ creates all small limits: given a family of groups ${G_{i} | i \in I}$ , there’s a unique structure on $\prod_{i \in I} G_{i}$ making the projections into homomorphisms, and it’s a product in $G p$ . Similarly for equalisers. But $G p \to S e t$ doesn’t preserve or reflect coproducts.
(b) The forgetful functor $T o p \to S e t$ preserves small limits and colimits, but doesn’t reflect them.
(c) The inclusion $A b G p \to G p$ reflects coproducts, but doesn’t preserve them.
A coproduct $A * B$ in $G p$ is nonabelian if both $A$ and $B$ are nontrivial. So the only cones in $A b G p$ thot could map to coproduct cones in $G p$ are those where either $A$ or $B$ is trivial. But if $A = {1}$ then $A \times B ≅ B$ in either category.
(d) If $D$ is a reflective subcategory of $C$ , the inclusion $D \to C$ creates any limits which exist.
Given $D : J \to D$ and a limit cone $(L, (x_{j} | j \in ob J))$ for it in $C$ , the morphisms $F L \overset{F x_{j}}{\to} F D (j) \overset{η_{D (j)}^{- 1}}{\to} D (j)$ (where $F$ is the left adjoint, and $η$ is the unit) form a cone over $D$ , so they induce a unique $u : F L \to L$ . Now $u η_{L} : L \to L$ is $1_{L}$ since it’s a factorisation of the limit through itself. So $η_{L} u η_{L} = η_{L}$ , i.e. $η_{L} u$ is a factorisation of $η_{L}$ through itself, so $η_{L} u = 1_{F L}$ . So the $η_{D (j)}^{- 1} (F λ_{j})$ form a limit cone in $C$ , and hence in $D$ .
(e) If $D$ has limits of shape $J$ , so does $[C, D]$ for any $C$ , and the forgetful functor $[C, D] \to D^{ob C}$ creates them (strictly).
Given $D : J \to [C, D]$ , we can regard it as a functor $J \times C \to D$ . For each $A \in ob C$ , $D (∙, A)$ is a diagram of shape $J$ in $D$ , so has a limit $(L A, (λ_{j, A} : L A \to D (j, A) | j \in ob J))$ . Given $f : A \to B$ in $C$ , the composites $L A \overset{λ_{j, A}}{\to} D (j, A) \overset{D (j, f)}{\to} D (j, B)$ form a cone over $D (∙, B)$ , so induce a unique $L f : L A \to L B$ . Functoriality of $L$ follows fro uniqueness, and this is the unique way of making $L$ into a functor which lifts the $λ_{j, ∙}$ to a cone in $[C, D]$ .

The fact that it’s a limit cone is straightforward.

Remark 4.8. In any category, $A \overset{f}{\to} B$ is monic if and only if

is a pullback. Hence, if

D

has pullbacks, then any monomorphism in

[C, D]

is pointwise monic, since its pullback along itself is contsructed pointwise.

Lemma 4.9. Assuming that:

$G : D \to C$ has a left adjoint

Then

G

preserves all limits which exist in

D

Proof 1. Suppose $(F ⊣ G)$ , and suppose $C$ and $D$ have limits of shape $J$ . Then the diagram

commutes, and all the functors in it have right adjoints, so

commutes up to isomorphism by Corollary 3.6. □

Proof 2. Suppose given $D : J \to D$ and a limit cone $(L, (λ_{j} | j \in ob J))$ over it. Give a cone $(A, (μ_{j} : A \to G D (j)))$ over $G D$ , the transposes $\bar{μ_{j}} : F A \to D (j)$ form a cone over $D$ by naturality of the adjunction, so induce a unique $\bar{μ} : F A \to L$ such that $λ_{j} \bar{μ} = \bar{μ_{j}}$ for all $j$ .

Then $μ : A \to G L$ is the unique morphism satisfying $(G λ_{j}) μ = μ_{j}$ for all $j$ . □

Lemma 4.10. Assuming that:

$J$ a diagram shape
$D$ has all limits of shape $J$
$G : D \to C$ preserves all limits of shape $J$

Then for each

A \in ob C

(A ↓ G)

has limits of shape

J

and the forgetful functor

(A ↓ G) \overset{U}{\to} D

creates them.

Proof. Suppose given $D : J \to (A ↓ G)$ ; write $D (j) = (U D (j), f_{j} : A \to G U D (j))$ and let $(L, (λ_{j} | j \in ob J))$ be a limit for $U D$ . Since the edges of $D$ are morphisms in $(A ↓ G)$ , the $f_{j}$ form a cone over $G U D$ , so there’s a unique $f : A \to G L$ satisfying $(G λ_{j}) f = f_{j}$ for all $j$ .

So $(L, f)$ is the unique lifting of $L$ to an object of $(A ↓ G)$ which makes the $λ_{j}$ into morphisms $(L, f) \to (U D (j), f_{j})$ in $(A ↓ G)$ . The fact that these morphisms form a limit cone is straightforward. □

Can we represent an initial object as a limit?

Lemma 4.11. Assuming that:

$C$ a category

Then specifying an initial object of

C

is equivalent to specifying a limit for

1_{C} : C \to C

Proof. First suppose $I$ is initial. The unique morphisms $I \to A$ , $A \in ob C$ , form a cone over $1_{C}$ , and it’s a limit cone since if $(A, (f_{B} : A \to B | B \in ob C))$ is any cone over $1_{C}$ , then $f_{I}$ is its unique factorisation through the one with apex $I$ .

Conversely, suppose given a limit $(I, (f_{A} : I \to A | A \in ob C))$ for $1_{C}$ . Then $I$ is weakly initial (i.e. it admits morphisms to every object of $C$ ); and if $g : I \to A$ then $g f_{I} = f_{A}$ . In particular, $f_{A} f_{I} = f_{A}$ for all $A$ , so $f_{I}$ is a factorisation of the limit cone through itself, so $f_{I} = 1_{I}$ and $I$ is initial. □

The ‘primitive’ Adjoint Functor Theorem follows from Lemma 4.10, Lemma 4.11 and Theorem 3.3. But it only applies to preorders (see Example Sheet).

Theorem 4.12 (General Adjoint Functor Theorem). Assuming that:

$D$ is complete and locally small

Then

G : D \to C

has a left adjoint if and only if

G

preserves small limits and satisfies the solution-set condition: for every

A \in ob C

, there’s a set

{(B_{i}, f_{i}) | i \in I}

of objects of

(A ↓ G)

which is collectively weakly initial.

Proof.

$\Rightarrow$ $G$ preserves limits by Lemma 4.9, and ${(F A, η_{A})}$ is a singleton solution-set for each $A$ .
$\Leftarrow$ By Lemma 4.10, the categories $(A ↓ G)$ are complete, and they’re locally small since $D$ is.
So we need to show: if $A$ is complete and locally small, and has a weakly initial set ${A_{i} | i \in I}$ , then it has an initial object. First form $P = \prod_{i \in I} A_{i}$ ; then $P$ is weakly initial. Now form the limit of the diagram with vertices $P$ and $P^{'}$ , with the morphisms $P \to P^{'}$ being all endomorphisms of $P$ .

Writing $I \overset{i}{\to} P$ for this, $I$ is still weakly initial. Suppose given $I ⇉_{g}^{f} B$ ; let $E \overset{e}{\to} I$ be their equaliser. There exists some $h : P \to E$ . Now $i e h : P \to P$ , but we also have $1_{P} : P \to P$ , so $i = 1_{P} i = i e h i$ . But $i$ is monic, so we get $e h i = 1_{I}$ , so $e$ is split epic, and hence $f = g$ . □

Example 4.13.

(a) Consider the forgetful functor $U : G p \to S e t$ . $G p$ has and $U$ preserves all small limits by Example 4.7(a), and $G p$ is locally small. Given $A$ , any $A \overset{f}{\to} U G$ factors through $A \to U G^{'}$ where $G^{'}$ is the subgroup generated by ${f (a) | a \in A}$ . Also $card G^{'} \leq \max {ℵ_{0}, card A}$ . Let $B$ be a set of this cardinality: considering all subsets $B^{'} \subseteq B$ , all group structures on $B^{'}$ and all functions $A \to B^{'}$ , we get a solution-set at $A$ .
(b) Let $C L a t$ be the category of complete lattices (posets with all joins and all meets). $U : C L a t \to S e t$ creates limits just like $U : G p \to S e t$ .
In 1965, A. Hales showed that there exist arbitrarily large complete lattices generated by 3 element subsets, so the solution-set condition fails for $A = {a, b, c}$ .

Now also that $C L a t$ doesn’t have a coproduct for 3 copies of ${0, a, 1}$ .

Definition 4.14 (Subobject). By a subobject of $A \in ob C$ , we mean a monomorphism $A^{'} ↣ A$ . We order subobjects by $(A^{'} ↣ A) \leq (A^{″} ↣ A)$ if there exists

We write

{Sub}_{C} (A)

for this preorder.

We say $C$ is well-powered if every ${Sub}_{C} (A)$ is equivalent to a small poset.

For example, $S e t$ is well-powered since the inclusions $A^{'} \subseteq A$ form a representative set of subobjects of $A$ . It is well-copowered since isomorphism classes of epimorphisms $A ↠ B$ correspond to equivalence relations on $A$ .

Lemma 4.15. Assuming that:

a pullback diagram
where $f$ is monic

Then

k

is monic.

Proof. Suppose given $D ⇉_{m}^{l} P$ with $k l = k m$ . Then $f h l = g k l = g k m = f h m$ , but $f$ is monic so $h l = h m$ . So $l$ and $m$ are both factorisations of

through the pullback, and hence

l = m

. □

Theorem 4.16 (Special Adjoint Functor Theorem). Assuming that:

$C$ and $D$ are locally small
$D$ is complete and well-powered
$D$ has a coseparating set of objects

Then

G : D \to C

has a left adjoint if and only if it preserves all small limits.

Proof.

$\Rightarrow$ is Lemma 4.9.
$\Leftarrow$ Let $A \in ob C$ . As in Theorem 4.12, $(A ↓ G)$ inherits completeness and locally smallness from $D$ : it also inherits well-poweredness since subobjects of $(B, f)$ in $(A ↓ G)$ are those $B^{'} \overset{m}{↣} B$ in $D$ such that $f$ factors through $G B^{'} \overset{G m}{↣} G B$ . (Note that the forgetful functor $(A ↓ G) \to D$ preserves monomorphisms by Remark 4.8). And if ${S_{i} | i \in I}$ is a coseparating set for $D$ , then ${(S_{i}, f) | i \in I, f \in C (A, G S_{i})}$ is a coseparating set for $(A ↓ G)$ .
So we need to show: if $A$ is complete, locally small and well-powered and has a coseparating set ${S_{i} | i \in I}$ , then $A$ has an initial object. First form $P = \prod_{i \in I} S_{i}$ ; now consider the limit of the diagram
whose edges are a representative set of subobjects of $P$ .

If $I$ is the apex of the limit cone, the legs $I \to P^{'}$ of the limit cone are all monic by the argument of Lemma 4.15, and in particular $I \to P$ is monic, and it’s a least subobject of $P$ .

If we had $I ⇉_{g}^{f} A$ , their equaliser $E \to I$ would be a subobject of $P$ contained in $I ↣ P$ , so $E \to I$ is an isomorphism, and hence $f = g$ .

Given any $A \in ob A$ form the product $Q = \prod_{(i, f)} S_{i}$ over all pairs $(i, f)$ with $f_{i} A \to S_{i}$ and the morphism $g : A \to Q$ with $π_{(i, f)} g = f$ for all $(i, f)$ . Since the $S_{i}$ are coseparating, $g$ is monic. We also have $h : P \to Q$ defined by $π_{(i, f)} h = π_{i}$ for all $(i, f)$ .

Form the pullback
then $l$ is monic by Lemma 4.15, so $I ↣ P$ factors as $I \to B \overset{l}{↣} P$ and hence we have $I \to B \overset{k}{\to} A$ . So $I$ is initial. □

Example 4.17. Consider the inclusion $K H a u s \overset{I}{\to} T o p$ . Tychonoff’s Theorem says $K H a u s$ is closed under (small) products in $T o p$ . It’s closed under equalisers, since equalisers of pairs in $K H a u s$ are closed inclusions.

So $K H a u s$ is complete, and $I$ preserves limits. $K H a u s$ and $T o p$ are locally small, and $K H a u s$ is well-powered since subobjects of $X$ is isomorphic to inclusions of closed subspaces. And $K H a u s$ has a coseparator $[0, 1]$ , by Uryson’s Lemma.

So by Theorem 4.16, $I$ has a left adjoint $β$ .

Remark 4.18.

(a) The construction in Theorem 4.16 is closely parallel to Čech’s original construction of $β$ .
Given a space, Čech constructs $P = \prod_{f : x \to [0, 1]} [0, 1]$ and the map $g : X \to P$ defined by $π_{f} g = f$ . Then he takes $β X$ to be the closure of the image of $g$ , i.e. the smallest subobject of $(P, g)$ in $(X ↓ I)$ .
(b) We could have constructed $β$ using Theorem 4.12: to get a solution-set for $I$ at an object $X$ of $T o p$ , note that any continuous $f : X \to I Y$ factors as $X \to I Y^{'} \to I Y$ where $Y^{'}$ is the closure of the image of $f$ , and then since $Y^{'}$ has a dense subspace of cardinality $\leq card X$ , we have $card Y^{'} \leq 2^{2^{card X}}$ .