41.02.02 · category-theory / limits-colimits

Constructing Limits: Products, Equalizers, Preservation, and Filtered Colimits

shipped3 tiersLean: none

Anchor (Master): Mac Lane 1998 *Categories for the Working Mathematician* 2e (Springer GTM 5) Ch. V, IX in full; Adámek-Rosický 1994 *Locally Presentable and Accessible Categories* (Cambridge LMS 189) Ch. 1 (finitely presentable objects, filtered colimits, accessible categories); Borceux 1994 *Handbook of Categorical Algebra 1* (Cambridge) §2.13, Ch. 6 (flat functors, filtered colimits, ind-objects); Gabriel-Ulmer 1971 *Lokal präsentierbare Kategorien* (Springer LNM 221)

Intuition Beginner

Once you know that a limit is the best object hanging consistently over a pattern of objects and arrows, the next question is practical: how do you actually build one? It would be tedious to invent a fresh construction for every shape of pattern. Two simple ingredients suffice. If a category lets you form products (the best object over a bunch of objects with no arrows) and equalizers (the best object on which two parallel arrows agree), then you can build the limit of any pattern at all. You bundle every object of the pattern into one big product, then carve out the part where the pattern's own arrows are respected. That carving is an equalizer.

A second question is how limits behave when you carry them from one category to another by a translation. Some translations keep the best object best: they preserve limits. Others are even better behaved and build the limit upstairs out of the limit downstairs: they create limits. Knowing which translations do this lets you compute a complicated limit by handing it to a friendlier category.

The last theme is a special kind of pattern, a filtered one, where any two pieces can always be merged further along. Filtered patterns behave like growing unions, and their colimits are gentle: they get along with finite limits, so you can swap the order of the two operations.

Visual Beginner

Think of building a limit in two stages. First, lay all the pattern's objects side by side and take their product — one apex with a leg to each. This apex is too generous: it ignores the pattern's internal arrows. Second, keep only the part of the apex where those arrows are honored. That filtering step is the equalizer.

stage	what you form	what it does
1. product	one apex over all objects at once	provides a leg to every object, ignores internal arrows
2. equalizer	the sub-apex respecting the arrows	keeps only the consistent families
result	the limit of the whole pattern	best cone, built from two stock pieces

A filtered pattern looks different. Picture a growing tower of rooms where any two rooms share a larger room further up. The colimit is the whole building you reach in the limit of growth — and because the tower keeps merging, a finite check made inside the building can already be made in some single large-enough room.

The picture shows the whole strategy: complicated limits are never built from scratch. You always assemble them from products and equalizers, and you compute them in whichever category makes those two pieces easy.

Worked example Beginner

Build the limit of a small pattern in the category of sets, using only a product and an equalizer. Take two sets $A = {1, 2, 3}$ and $B = {1, 2}$ with a single arrow between them: the function $f : A \to B$ that sends $1, 2 \mapsto 1$ and $3 \mapsto 2$ . The pattern is " $A$ with $f$ landing in $B$ ".

Step 1. Form the product apex $A \times B$ , all ordered pairs. It has $3 \times 2 = 6$ elements. Its two legs read off the first and second slots.

Step 2. The pattern's arrow $f$ imposes a condition: a consistent family $(a, b)$ must have $f (a) = b$ . So compare two functions from $A \times B$ to $B$ : one reads off the second slot, $b$ ; the other applies $f$ to the first slot, $f (a)$ . The equalizer keeps the pairs where these agree.

Step 3. List the survivors: $(1, 1)$ has $f (1) = 1$ , agree; $(2, 1)$ has $f (2) = 1$ , agree; $(3, 2)$ has $f (3) = 2$ , agree. The pairs $(1, 2), (2, 2), (3, 1)$ all fail. So the limit has three elements: ${(1, 1), (2, 1), (3, 2)}$ .

What this tells us: the limit is a copy of $A$ , since each element of $A$ pairs with exactly one forced value in $B$ . We never guessed this; the product-then-equalizer recipe produced it. Building over the six-element product and carving by the arrow's condition gave the right three-element answer.

Check your understanding Beginner

Formal definition Intermediate+

Fix a category $C$ . A product of a family $(X_{i})_{i \in I}$ is a limit of the discrete diagram on $I$ : an object $\prod_{i} X_{i}$ with projections $pr_{i}$ universal among cones ^{[Mac Lane 1998]}. An equalizer of a parallel pair $f, g : A ⇉ B$ is a limit of the parallel-pair shape: a map $e : E \to A$ with $f \circ e = g \circ e$ , universal among such maps. A category has products if every set-indexed family has one, and has equalizers if every parallel pair has one.

Definition (preservation, reflection, creation of limits). Let $U : C \to D$ be a functor and $F : J \to C$ a diagram. $U$ preserves the limit of $F$ if, whenever $(L, π)$ is a limit cone over $F$ , the image $(U L, U π)$ is a limit cone over $U F$ . $U$ reflects limits if a cone $(L, π)$ over $F$ is a limit whenever its image $(U L, U π)$ is a limit over $U F$ . $U$ creates the limit of $F$ if, given a limit cone $(M, σ)$ over $U F$ in $D$ , there is a cone over $F$ in $C$ — unique up to unique isomorphism — whose image is $(M, σ)$ , and every such cone is a limit. A functor preserving all small limits is continuous; dually, preserving all small colimits is cocontinuous.

Definition (filtered category). A category $J$ is filtered if it is nonempty and: (i) for any two objects $j, k$ there is an object $ℓ$ with morphisms $j \to ℓ$ and $k \to ℓ$ ; (ii) for any two parallel morphisms $u, v : j ⇉ k$ there is a morphism $w : k \to ℓ$ with $w \circ u = w \circ v$ . A filtered colimit is a colimit of a diagram $F : J \to C$ with $J$ filtered. A directed colimit is the special case where $J$ is a directed poset (every pair has an upper bound); the names direct limit and $lim$ are standard. Cofiltered and the dual cofiltered limit (inverse limit, $lim$ ) reverse all arrows.

Definition (finitely presentable and finitely generated objects). In a category $C$ with filtered colimits, an object $X$ is finitely presentable (or compact) if the representable functor $C (X, -) : C \to Set$ preserves filtered colimits: every morphism $X \to lim_{j} F (j)$ factors through some $F (j)$ , and two factorisations agree after a further map in $J$ . $X$ is finitely generated if $C (X, -)$ preserves filtered colimits of diagrams with monomorphic transition maps. A sifted category is one for which colimits commute with finite products in $Set$ ; filtered categories are sifted, and so is the simplex-like category indexing reflexive coequalizers.

Counterexamples to common slips Intermediate+

Preservation is not automatic. The inclusion $Ab ↪ Grp$ does not preserve the coproduct: the coproduct in $Ab$ is the direct sum, in $Grp$ the free product, and these differ. Forgetful functors often preserve limits while destroying colimits.
Filtered is strictly stronger than connected, and matters for the commutation. A pushout shape (a span $∙ \leftarrow ∙ \to ∙$ ) is connected but not filtered, and pushout colimits do not commute with finite limits in $Set$ . The filtered condition (ii) on parallel arrows is exactly what is needed; dropping it breaks the interchange.
Finitely generated is weaker than finitely presentable. For modules, finitely presentable means finitely generated with a finitely generated module of relations; over a non-Noetherian ring a finitely generated module can fail to be finitely presentable. The categorical hom-preservation definitions separate the two notions cleanly.

Key theorem with proof Intermediate+

The signature result is the completeness theorem: products and equalizers suffice to construct every limit, with the limit exhibited as a single equalizer of two maps between products.

Theorem (completeness from products and equalizers). Let $C$ have all small products and all equalizers, and let $F : J \to C$ be a diagram with $J$ small. Then $lim F$ exists and is the equalizer $E$ of the parallel pair $$ s,t\colon \prod_{j\in\mathrm{Ob},\mathcal{J}} F(j) ;\rightrightarrows; \prod_{(u\colon j\to k)\in\mathrm{Mor},\mathcal{J}} F(k), $$ where the component of $s$ at $u : j \to k$ is $F (u) \circ pr_{j}$ and the component of $t$ at $u$ is $pr_{k}$ . Dually, coproducts and coequalizers give cocompleteness ^{[Mac Lane 1998]}.

Proof. Write $P = \prod_{j} F (j)$ and $Q = \prod_{u} F (k)$ , the second product indexed by all morphisms of $J$ . A morphism $h : C \to P$ is exactly a family $(λ_{j})_{j}$ with $λ_{j} = pr_{j} \circ h : C \to F (j)$ , by the universal property of $P$ . Composing $h$ with $s$ gives the family whose $u$ -component is $F (u) \circ λ_{j}$ ; composing with $t$ gives the family whose $u$ -component is $λ_{k}$ . Thus $s \circ h = t \circ h$ holds if and only if $F (u) \circ λ_{j} = λ_{k}$ for every $u : j \to k$ , which is precisely the cone condition on $(λ_{j})$ . So maps $C \to P$ that equalize $s, t$ correspond bijectively, and naturally in $C$ , to cones over $F$ with apex $C$ .

Now let $e : E \to P$ be the equalizer of $s, t$ . By the universal property of $E$ , a map $C \to E$ is the same as a map $C \to P$ equalizing $s, t$ , hence the same as a cone over $F$ with apex $C$ . Setting $π_{j} = pr_{j} \circ e : E \to F (j)$ makes $(E, π)$ a cone (it equalizes $s, t$ , so satisfies the cone condition), and the bijection just established says every cone factors through $(E, π)$ uniquely. Therefore $(E, π)$ is terminal in $Cone (F)$ , i.e. $E = lim F$ . The dual statement is the same argument read in $C^{op}$ , turning products into coproducts and equalizers into coequalizers. $□$

Bridge. This construction builds toward the entire machinery of preservation and creation, and it appears again in 41.03.01, where the limit functor is exhibited as right adjoint to the diagonal and the equalizer-of-products formula becomes the explicit description of that adjoint's value. The foundational reason the reduction works is that a cone is a morphism into a product subject to equalizing conditions, so the two-stage shape — bundle, then carve — is forced by the definition of a cone rather than chosen; this is exactly the structure that makes a functor preserving products and equalizers automatically continuous, since it preserves each stage separately. The construction is dual to the colimit case throughout, coproducts-and-coequalizers giving cocompleteness by the same proof in the opposite category. Putting these together, the bridge is that "complete" and "has products and equalizers" name the same categories, which is what lets the next sections reduce questions about all limits to questions about these two stock shapes — and lets right adjoints, shown to preserve both, preserve every limit at a stroke.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the forgetful functor $U : Grp \to Set$ creates all small limits.

Hint

Given a limit of underlying sets, equip it with the unique group structure making the projections homomorphisms.

Answer

Let $F : J \to Grp$ and let $(M, σ)$ be the limit in $Set$ of $U F$ , realised as compatible families $M = {(x_{j}) ∣ F (u) (x_{j}) = x_{k}} \subseteq \prod_{j} ∣ F (j) ∣$ . Define multiplication componentwise: $(x_{j}) (y_{j}) = (x_{j} y_{j})$ , which again lands in $M$ because each $F (u)$ is a homomorphism, and similarly for the identity and inverses. This is the unique group structure on $M$ making every projection $σ_{j}$ a homomorphism, and with it $(M, σ)$ is a cone in $Grp$ that is terminal, since any compatible family of homomorphisms into the $F (j)$ assembles to a homomorphism into $M$ . So $U$ lifts the limit uniquely and the lift is a limit: $U$ creates limits.

Exercise 4 (medium, symbolic).

In $Set$ , exhibit the filtered colimit of the system $Z /2 \to Z /4 \to Z /8 \to \dots$ of cyclic groups under the doubling-compatible inclusions $Z / 2^{n} ↪ Z / 2^{n + 1}$ , $1 \mapsto 2$ , and identify the resulting group.

Hint

A directed colimit of these inclusions is the union; track where $2^{- n}$ -torsion lands.

Answer

The transition maps embed $Z / 2^{n}$ as the $2^{n}$ -torsion subgroup of the next stage. The directed (hence filtered) colimit is the union of all the $2^{n}$ -torsion, which is the Prüfer group $Z [1/2] / Z = Z (2^{\infty})$ , the group of dyadic rationals modulo $1$ . Each element of the colimit comes from a single stage, and equality of two elements is detected at a common later stage — the defining behaviour of a filtered colimit, where every finite amount of data already appears in one object of the system.

Exercise 5 (medium, symbolic).

Show that in $Set$ a filtered colimit of injections is again an injection of each stage into the colimit, and use this to argue that finite sets are finitely presentable.

Hint

A map out of a finite set is a finite tuple of elements; each lands in some stage, and finitely many can be merged.

Answer

Let $F : J \to Set$ be filtered with colimit $lim F$ , computed as $⨆_{j} F (j)$ modulo identifying $x \in F (j)$ with $F (u) (x)$ for $u : j \to k$ . A map $f : S \to lim F$ from a finite set $S = {s_{1}, \dots, s_{n}}$ assigns each $s_{i}$ a class represented in some $F (j_{i})$ ; by filteredness (i) the finitely many $j_{i}$ map to a common $ℓ$ , so all representatives live in $F (ℓ)$ , and $f$ factors through $F (ℓ)$ . If two maps $S \to F (ℓ)$ become equal in the colimit, each of the $n$ equalities is witnessed by a later map, and by filteredness (i) and (ii) one $ℓ \to ℓ^{'}$ witnesses all of them. Hence $Set (S, -)$ preserves filtered colimits, so finite sets are finitely presentable.

Exercise 6 (medium, symbolic).

Give a finite limit and a filtered colimit in $Set$ whose order of evaluation can be exchanged, and verify the canonical comparison map is a bijection on a small instance.

Hint

Take the equalizer (a finite limit) of two natural maps between two filtered colimits, and compute both orders.

Answer

Take the constant filtered system $F (j) = N$ along $N$ as a directed poset, with identity transitions, so $lim F = N$ ; and the parallel pair $id, succ : N ⇉ N$ . The equalizer of $id, succ$ is empty (no $n$ with $n = n + 1$ ), and the filtered colimit of the equalizers (each empty) is empty; conversely the equalizer of the colimit maps $id, succ : N ⇉ N$ is again empty. Both orders give $\emptyset$ , and the canonical map $lim (eq) \to eq (lim)$ is the identity on $\emptyset$ , a bijection — illustrating that filtered colimits commute with the finite limit "equalizer" in $Set$ .

Exercise 7 (hard, symbolic).

Prove that filtered colimits commute with finite products in $Set$ : for a filtered $J$ and functors $F, G : J \to Set$ , the canonical map $lim_{j} (F (j) \times G (j)) \to (lim_{j} F (j)) \times (lim_{j} G (j))$ is a bijection.

Hint

Surjectivity merges the two indices via filteredness (i); injectivity merges two witnessing later maps via (i) and (ii).

Answer

Surjectivity. An element of the target is a pair $([x], [y])$ with $x \in F (j)$ , $y \in G (k)$ . By filteredness (i) choose $ℓ$ with $j \to ℓ$ , $k \to ℓ$ ; transport $x, y$ to $F (ℓ), G (ℓ)$ . Then $(x_{ℓ}, y_{ℓ}) \in F (ℓ) \times G (ℓ)$ maps to $([x], [y])$ , so the map is onto.

Injectivity. Suppose $(x, y), (x^{'}, y^{'}) \in F (j) \times G (j)$ (after moving both to a common $j$ by (i)) have equal images, so $[x] = [x^{'}]$ and $[y] = [y^{'}]$ in the respective colimits. Each equality is witnessed by a later map: some $u : j \to m$ with $F (u) (x) = F (u) (x^{'})$ and some $v : j \to n$ with $G (v) (y) = G (v) (y^{'})$ . By (i) take $j \to p$ through which both $m, n$ map, then by (ii) coequalize the two resulting composites $j ⇉ p$ by a single $p \to q$ ; over $q$ both coordinates agree, so $(x, y)$ and $(x^{'}, y^{'})$ have the same class in $lim (F \times G)$ . Hence the map is injective, so bijective. $□$

Advanced results Master

Theorem (right adjoints preserve limits; the dual). If $U : C \to D$ has a left adjoint $L ⊣ U$ , then $U$ preserves all limits that exist in $C$ . Dually, a left adjoint preserves all colimits. The proof is a hom-set computation: for a diagram $F : J \to C$ with limit $L$ , $$ \mathcal{D}(D, U\lim F)\cong\mathcal{C}(L D,\lim F)\cong\lim_j \mathcal{C}(L D, F(j))\cong\lim_j \mathcal{D}(D, U F(j))\cong\mathcal{D}(D,\lim_j U F(j)), $$ natural in $D$ , so $U lim F$ represents the same functor as $lim U F$ and the two agree. The middle isomorphism is the basic fact that $C (C, -)$ sends limits in $C$ to limits in $Set$ . The detailed adjunction development is 41.03.01, and the preservation theorem with its consequences for free–forgetful pairs is 41.03.02; the present unit supplies the completeness construction those proofs invoke.

Theorem (finitely presentable objects and filtered colimits). In a category $C$ with filtered colimits, an object $X$ is finitely presentable if and only if $C (X, -)$ preserves filtered colimits. In $Set$ the finitely presentable objects are the finite sets; in $R - Mod$ they are the finitely presented modules; in $Grp$ they are the finitely presented groups (finitely many generators and relations); in a variety of algebras they are the finitely presented algebras. Every object of any of these categories is a filtered colimit of finitely presentable ones, which is the defining property of a locally finitely presentable category in the sense of Gabriel–Ulmer ^{[Gabriel-Ulmer 1971]}: cocomplete, with a set of finitely presentable generators, every object their filtered colimit.

Theorem (filtered colimits commute with finite limits in $Set$ ; sifted colimits with finite products). For a filtered category $J$ and a finite category $I$ , and any $H : I \times J \to Set$ , the canonical comparison $$ \varinjlim_{j}\lim_{i} H(i,j);\longrightarrow;\lim_{i}\varinjlim_{j} H(i,j) $$ is a bijection. The same holds in any locally finitely presentable category and, more generally, in any category where filtered colimits are computed as in $Set$ on a generating set. Sifted colimits are exactly those commuting with finite products in $Set$ ; filtered colimits and reflexive coequalizers are the basic sifted shapes, and a functor preserving finite products and sifted colimits is determined by its restriction to finitely presentable objects.

Theorem (interchange of limits and colimits; the canonical map). For a bifunctor $H : I \times J \to C$ with $C$ (co)complete there is always a canonical comparison $lim_{j} lim_{i} H \to lim_{i} lim_{j} H$ from "colimit of limits" to "limit of colimits," built from the universal properties. This map is rarely invertible: limits commute with limits and colimits with colimits unconditionally, but a colimit commutes with a limit only under special hypotheses — the filtered/finite case above, the existence of a left adjoint to the limit functor, or distributivity conditions as in an $\infty$ -pretopos. The failure of the comparison is what derived functors and spectral sequences measure.

Synthesis. Putting these together, the completeness construction is the hinge from which the rest of the chapter swings. The foundational reason products-and-equalizers suffice is that a cone is a map into a product cut down by an equalizer, and this is exactly the structure that makes preservation modular: a functor is continuous as soon as it preserves the two stock shapes, which is why right adjoints, shown to preserve both by the hom-set computation, preserve every limit at once. This generalises the duality already met for cones: coproducts-and-coequalizers give cocompleteness and left adjoints preserve colimits, the whole theory reflected across $C \mapsto C^{op}$ . The central insight is that filteredness is the precise condition under which a colimit stops fighting finite limits — any finite diagram of data already lives in one object of a filtered system — so finitely presentable objects, defined by hom-functors preserving filtered colimits, become the atoms from which locally presentable categories are assembled. The bridge to what follows is that this same hom-set reasoning is the content of the adjoint-functor theorems in 41.03.01 and the representability machinery of 41.04.01: a limit exists, is preserved, or is created exactly when the relevant functor of points is representable, and the ind/pro-object completions $Ind (C)$ and $Pro (C)$ are the universal ways of freely adjoining filtered colimits and cofiltered limits, which is exactly how the calculus built here extends to the accessible and presentable categories that organise modern algebra and geometry.

Full proof set Master

Proposition 1 (the cone–equalizer correspondence is natural). With $P = \prod_{j} F (j)$ , $Q = \prod_{u} F (k)$ , and $s, t : P ⇉ Q$ as in the completeness theorem, the assignment sending a map $h : C \to P$ with $s \circ h = t \circ h$ to the cone $(pr_{j} \circ h)_{j}$ is a natural isomorphism $Eq (s, t) (C) ≅ Cone (C, F)$ of functors $C^{op} \to Set$ .

Proof. The universal property of $P$ gives a natural bijection $C (C, P) ≅ \prod_{j} C (C, F (j))$ , sending $h$ to $(pr_{j} \circ h)$ . Under it, the equation $s \circ h = t \circ h$ becomes, componentwise over each $u : j \to k$ , the equation $F (u) \circ (pr_{j} \circ h) = pr_{k} \circ h$ , since the $u$ -component of $s$ is $F (u) \circ pr_{j}$ and of $t$ is $pr_{k}$ . That is exactly the cone condition on the family $(pr_{j} \circ h)$ . Naturality in $C$ is precomposition: for $g : C^{'} \to C$ both the bijection and the cone condition are stable under $h \mapsto h \circ g$ , since $pr_{j} \circ (h \circ g) = (pr_{j} \circ h) \circ g$ . So the restriction to equalizing maps is a natural isomorphism onto the cone functor. $□$

Proposition 2 (a functor preserving products and equalizers is continuous). If $U : C \to D$ preserves all small products and all equalizers, and $C$ is complete, then $U$ preserves all small limits.

Proof. Let $F : J \to C$ be small with limit $E = eq (s, t)$ , $e : E \to P$ , built as in the theorem. Apply $U$ . Because $U$ preserves the two products $P, Q$ , the images $U P, U Q$ are the corresponding products in $D$ for the diagrams $U F (j)$ and $U F (k)$ , and $U s, U t$ are the two structure maps for $U F$ (their components are $U (F (u)) \circ pr_{j}$ and $pr_{k}$ , since $U$ preserves projections). Because $U$ preserves the equalizer, $U e : U E \to U P$ is the equalizer of $U s, U t$ . By the completeness theorem applied in $D$ , that equalizer is $lim U F$ , with cone $pr_{j} \circ U e = U (pr_{j} \circ e) = U π_{j}$ . Hence $(U E, U π)$ is a limit cone over $U F$ , i.e. $U$ preserves $lim F$ . $□$

Proposition 3 (right adjoints preserve limits). If $L ⊣ U$ with $L : D \to C$ , $U : C \to D$ , then $U$ preserves every limit existing in $C$ .

Proof. Let $F : J \to C$ have limit $lim F$ with cone $π$ . For each object $D$ of $D$ the adjunction gives a bijection $D (D, U C) ≅ C (L D, C)$ natural in $C$ . Apply it with $C = lim F$ and use that $C (L D, -)$ preserves limits (a representable sends limits to limits in $Set$ ): $$ \mathcal{D}(D,U\lim F)\cong\mathcal{C}(L D,\lim F)\cong\lim_j\mathcal{C}(L D,F(j))\cong\lim_j\mathcal{D}(D,U F(j)). $$ The composite is natural in $D$ , and the last set is $D (D, lim_{j} U F (j))$ when $lim U F$ exists, or otherwise exhibits the functor $D \mapsto lim_{j} D (D, U F (j))$ as represented by $U lim F$ . Either way $U lim F$ together with the cone $U π$ satisfies the universal property of $lim U F$ . Hence $U$ preserves the limit. $□$

Proposition 4 (filtered colimits commute with finite limits in $Set$ ). Let $J$ be filtered, $I$ finite, and $H : I \times J \to Set$ . Then the canonical map $θ : lim_{j} lim_{i} H (i, j) \to lim_{i} lim_{j} H (i, j)$ is a bijection.

Proof. A finite limit in $Set$ is the equalizer of two maps between finite products, so it suffices to treat finite products and equalizers; finite products are handled by Exercise 7 (and its $n$ -fold iterate, valid since $I$ is finite). For an equalizer, fix the parallel pair $f_{j}, g_{j} : A (j) ⇉ B (j)$ natural in $j$ , with $A = lim A (j)$ , $B = lim B (j)$ and induced $f, g : A ⇉ B$ . Surjectivity of $θ$ . An element of $eq (f, g)$ is a class $[a]$ , $a \in A (j)$ , with $f ([a]) = g ([a])$ , so $[f_{j} (a)] = [g_{j} (a)]$ in $B$ ; some $u : j \to k$ witnesses $B (u) (f_{j} (a)) = B (u) (g_{j} (a))$ , i.e. $f_{k} (A (u) (a)) = g_{k} (A (u) (a))$ by naturality. Thus $A (u) (a) \in eq (f_{k}, g_{k})$ represents a class mapping to $[a]$ , so $θ$ is onto. Injectivity. If $a \in eq (f_{j}, g_{j})$ and $a^{'} \in eq (f_{j^{'}}, g_{j^{'}})$ have $[a] = [a^{'}]$ in $A$ , a single later map merges them inside $A$ , and the merged element still equalizes $f, g$ stagewise; so the two classes in $lim eq$ already coincide. Hence $θ$ is bijective. $□$

Proposition 5 (finite sets are finitely presentable; the converse). In $Set$ , an object $X$ is finitely presentable if and only if $X$ is finite.

Proof. If $X$ is finite, $Set (X, -)$ preserves filtered colimits by Exercise 5: a map out of a finite set factors through a single stage and two such maps merge at a later stage. Conversely, suppose $X$ is finitely presentable. Write $X = lim_{S \subseteq X finite} S$ as the filtered colimit of its finite subsets along inclusions (the poset of finite subsets is directed). Finite presentability makes $id_{X} : X \to lim S$ factor through some finite $S_{0}$ , giving $r : X \to S_{0}$ with the inclusion $S_{0} ↪ X$ composed with $r$ equal to $id_{X}$ after passing to a later stage; since the transition maps are injections this forces $X \subseteq S_{0}$ , so $X$ is finite. $□$

Connections Master

Limits and colimits as universal cones 41.02.01. This unit is the construction half of the definition–construction pair begun in 41.02.01: that unit defines the universal cone and states the products-plus-equalizers reduction as an anchoring theorem, while the completeness proof, the explicit two-map presentation, and the cone–equalizer correspondence of Proposition 1 are carried out here. The inverse-limit and direct-limit examples of 41.02.01 are the cofiltered and filtered special cases whose general theory this unit develops.
Limits as adjoints to the diagonal and the adjoint-functor theorems 41.03.01. The limit functor is right adjoint to the constant-diagram functor $Δ$ , and the equalizer-of-products formula proved here is the explicit value of that adjoint; conversely the adjoint-functor theorems of 41.03.01 use exactly the completeness construction of this unit, since a continuous functor between locally presentable categories has a left adjoint precisely when it preserves the limits built from products and equalizers.
Right adjoints preserve limits; free–forgetful pairs 41.03.02. Proposition 3 is the headline preservation theorem, and its co-produced companion 41.03.02 develops the consequences for free–forgetful adjunctions — that forgetful functors of algebraic categories create limits while their left adjoints preserve colimits — together with the worked computation of why $Grp$ , $Ring$ , and $R - Mod$ limits are computed on underlying sets. This unit supplies the completeness construction and the hom-set proof those results invoke.
Representability, ind-objects, and presentable categories 41.04.01. Finitely presentable objects (defined here by hom-functors preserving filtered colimits) are the generators of locally presentable categories, and the ind-completion $Ind (C)$ that freely adjoins filtered colimits is the representability construction developed in 41.04.01; Proposition 5 and the filtered-colimit commutation are the technical core of that accessibility theory, with the direct-limit material of the algebra corpus 01.02.09 supplying the running module-theoretic examples.

Historical & philosophical context Master

The reduction of arbitrary limits to products and equalizers appears in Mac Lane's Categories for the Working Mathematician (1971; 2nd ed. 1998) ^{[Mac Lane 1998]} as the standard route to the completeness theorem, codifying a technique already implicit in the construction of inverse limits as submodules of products in the homological algebra of the 1950s. The preservation of limits by right adjoints is one of the first consequences Kan drew from his 1958 definition of adjoint functors, and it is the structural fact behind the older observation that the underlying set of a product of groups is the product of the underlying sets.

Filtered colimits and the finiteness conditions they characterise were given their general form by Pierre Gabriel and Friedrich Ulmer in Lokal präsentierbare Kategorien (1971) ^{[Gabriel-Ulmer 1971]}, where finitely presentable objects, locally presentable categories, and the equivalence between such categories and certain limit theories were established; the theory was extended to accessible categories by Michael Makkai and Robert Paré in 1989 and given a textbook treatment by Jiří Adámek and Jiří Rosický in 1994. The commutation of filtered colimits with finite limits in $Set$ is the categorical form of the algebraists' fact that a directed union of structures inherits any finitary property checkable on finitely many elements, and it underlies the exactness of the direct-limit functor in module categories that Grothendieck's 1957 Tôhoku axiom AB5 isolates. The ind- and pro-object constructions $Ind (C)$ and $Pro (C)$ were introduced by Grothendieck and Verdier in SGA 4 to handle the formal completions needed in étale cohomology.

Bibliography Master

@book{MacLane1998construct,
  author    = {Mac Lane, Saunders},
  title     = {Categories for the Working Mathematician},
  edition   = {2},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics 5},
  year      = {1998}
}

@book{Riehl2016construct,
  author    = {Riehl, Emily},
  title     = {Category Theory in Context},
  publisher = {Dover},
  year      = {2016}
}

@book{AdamekRosicky1994,
  author    = {Ad\'amek, Ji\v{r}\'i and Rosick\'y, Ji\v{r}\'i},
  title     = {Locally Presentable and Accessible Categories},
  publisher = {Cambridge University Press},
  series    = {London Mathematical Society Lecture Note Series 189},
  year      = {1994}
}

@book{Borceux1994construct,
  author    = {Borceux, Francis},
  title     = {Handbook of Categorical Algebra 1: Basic Category Theory},
  publisher = {Cambridge University Press},
  series    = {Encyclopedia of Mathematics and its Applications 50},
  year      = {1994}
}

@incollection{GabrielUlmer1971,
  author    = {Gabriel, Peter and Ulmer, Friedrich},
  title     = {Lokal pr\"asentierbare Kategorien},
  publisher = {Springer},
  series    = {Lecture Notes in Mathematics 221},
  year      = {1971}
}

@article{Kan1958construct,
  author  = {Kan, Daniel M.},
  title   = {Adjoint functors},
  journal = {Transactions of the American Mathematical Society},
  volume  = {87},
  year    = {1958},
  pages   = {294--329}
}

@book{MakkaiPare1989,
  author    = {Makkai, Michael and Par\'e, Robert},
  title     = {Accessible Categories: The Foundations of Categorical Model Theory},
  publisher = {American Mathematical Society},
  series    = {Contemporary Mathematics 104},
  year      = {1989}
}

Prerequisites

41.02.01

Tier anchors

beginner: Riehl 2016 *Category Theory in Context* (Dover) §3.2-3.3 (building limits out of products and equalizers, filtered colimits as unions); Leinster 2014 *Basic Category Theory* (Cambridge) §5.1, §6.3 (the completeness theorem and the directed-colimit picture)
intermediate: Mac Lane 1998 *Categories for the Working Mathematician* 2e (Springer GTM 5) Ch. V §2 (limits from products and equalizers, completeness), Ch. V §3-4 (preservation and creation of limits), Ch. IX §1-2 (filtered categories, filtered colimits commute with finite limits in Set); Riehl 2016 *Category Theory in Context* (Dover) §3.2-3.3, §3.8; Awodey 2010 *Category Theory* 2e (Oxford) §5.6
master: Mac Lane 1998 *Categories for the Working Mathematician* 2e (Springer GTM 5) Ch. V, IX in full; Adámek-Rosický 1994 *Locally Presentable and Accessible Categories* (Cambridge LMS 189) Ch. 1 (finitely presentable objects, filtered colimits, accessible categories); Borceux 1994 *Handbook of Categorical Algebra 1* (Cambridge) §2.13, Ch. 6 (flat functors, filtered colimits, ind-objects); Gabriel-Ulmer 1971 *Lokal präsentierbare Kategorien* (Springer LNM 221)

References

Mac Lane — Categories for the Working Mathematician · 2nd ed., Springer GTM 5, Ch. V §2-4, Ch. IX §1-2
Riehl — Category Theory in Context · Dover 2016, §3.2-3.3, §3.8
Adámek-Rosický — Locally Presentable and Accessible Categories · Cambridge LMS 189, 1994, Ch. 1
Borceux — Handbook of Categorical Algebra 1 · Cambridge 1994, §2.13, Ch. 6
Gabriel-Ulmer — Lokal präsentierbare Kategorien · Springer LNM 221, 1971

Estimated time

beginner: 20m
intermediate: 48m
master: 88m