Fibres, slices and index adjunctions

Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original text: CC0.

A diagram with maps to a fixed base carries more information than its underlying diagram. Colimits keep that information automatically. Limits keep it when their components force one common map to the base; connectedness provides exactly this mechanism. In sets, functions can also be formed separately in each fibre. Their evaluation adjunction explains why taking a new base preserves every small colimit. An adjunction between index categories gives a second useful simplification: universal extensions can be computed by restriction along the other adjoint.

Composition is from right to left. Fix a working Grothendieck universe \(\mathcal U\), with ambient choice and sufficiently large ambient universes for the categories and their functor categories. “Small” means \(\mathcal U\)-small; \(\mathsf{Set}\) denotes the category of such sets. Universes and small categories, Sections 1–3 prove all set closure and relabelling bounds used here. Arbitrary-index assertions below are conditional on the indicated diagrams, categories and chosen universal objects existing in that ambient setting. A larger ambient Hom set does not enlarge a small-colimit hypothesis.

Keep the entire structural-map factor proofs in Compatible families, Sections 2–4, the full relative product proofs in Universal forks, Sections 4–5, and the specified Hom correspondence, unit, counit and triangles in Passing maps across an adjunction, Sections 1–2. Colimits as connected components, Section 1 supplies existence of all small set colimits, with every quotient factor and its universe bound. References to these interfaces retain their whole proofs and size conventions.

1. Diagrams and their constant targets

Let \(I\) be any index category and suppose that \(C\) admits \(I\)-colimits. For each \(D:I\to C\), choose a colimit \(K_D\) with legs \(c_i^D:D(i)\to K_D\). The complete diagram-map construction in Compatible families, Section 2, gives a functor

Its proof uses every cocone equation to construct this map. The identity and composite obey the same leg equations, so uniqueness proves both functor laws. Thus the construction applies to all natural transformations, with no filteredness assumption.

Write \(\Delta_I:C\to\operatorname{Fun}(I,C)\) for the constant diagram functor. It sends \(T\) to the diagram with value \(T\) and every arrow \(1_T\), and sends \(b:T\to T'\) to the transformation with every component \(b\). These operations satisfy the functor laws componentwise.

Proposition 1.1. There is an adjunction \(\operatorname{Colim}_I\dashv\Delta_I\).

Proof by the retained universal factors. A transformation \(D\to\Delta_I T\) is exactly a cocone \((u_i:D(i)\to T)\): its naturality equation is \(u_jD(s)=u_i\) for \(s:i\to j\). The complete cocone representation therefore gives

Its inverse is the unique cocone factor; both inverse equations are part of that full representation proof. For \(v:D'\to D\) and \(b:T\to T'\), the image of \(b h\operatorname{Colim}_I(v)\) has component \(b h c_i^D v_i\), which is the component of \((\Delta_I b)(hc_i^D)_i v\). This proves naturality in both variables, with the inverse family natural as well. No additional set-valued preservation hypothesis is used. \(\square\)

For a projective-system index, take \(\beta:I^{\mathrm{op}}\to C\). Apply Proposition 1.1 to \(\beta^{\mathrm{op}}:I\to C^{\mathrm{op}}\) and reverse all arrows. The complete opposite-category proof in Compatible families, Section 3, identifies its colimit with \((\lim_{I^{\mathrm{op}}}\beta)^{\mathrm{op}}\). We obtain

Thus \(\Delta_{I^{\mathrm{op}}}\dashv\operatorname{Lim}_{I^{\mathrm{op}}}\). A transformation \(v:\beta\to\beta'\) has induced map characterized by \(p_i'\operatorname{Lim}(v)=v_i p_i\); the full retained factor proof gives the functor laws. The two naturality operations in (1.3) are precomposition on \(T\) and postcomposition by \(v_i\) on every cone component. These are exactly the reversed squares of (1.2).

When \(I\) is empty, its diagram category has one object and one transformation. Its colimit, if available, is initial in \(C\), and its limit is terminal. The cocone and cone families in (1.2) and (1.3) are empty, and their unique factors give the same adjunctions. The existence assumptions still require the respective initial or terminal object.

2. Colimits with a fixed map to the base

Fix \(Z\in C\). An object of \(C/Z\) is \(a:A\to Z\); a morphism \(a\to b\), for \(b:B\to Z\), is \(v:A\to B\) with \(bv=a\). The slice category laws are the full retained slice construction. Write \(U_Z:C/Z\to C\) for the forgetful functor.

Proposition 2.1. Every colimit of an underlying slice diagram has a unique lift to a colimit in \(C/Z\). Consequently, if \(C\) admits \(I\)-colimits, so does \(C/Z\), and \(U_Z\) preserves them.

Complete retained proof and exact lift. Use all of Initial objects, Section 1. For a slice diagram \((a_i:A_i\to Z)\), let \(c_i:A_i\to K\) be the chosen underlying colimit. Its structure map is the unique \(a:K\to Z\) with

That proof constructs the unique factor of every slice cocone and checks that it is over \(Z\). Uniqueness in (2.1) also proves uniqueness of the lifted structure, rather than merely existence of some slice colimit. Both the cocone legs and their factors are the original underlying maps. For the empty diagram, \(K\) is initial and \(a:K\to Z\) is its unique arrow; this is the empty slice colimit. For transformations of diagrams, the induced underlying map is over \(Z\) because both composites to \(Z\) have the same composites with all \(c_i\). Hence the lift respects the functorial maps of Section 1. \(\square\)

This is the precise creation statement used here: each specified underlying colimit cocone has one slice structure making it a universal slice cocone.

Suppose \(C\) has pullbacks. For \(u:Y\to Z\), the functor \(u^*:C/Z\to C/Y\) takes \(a:A\to Z\) to \(A\times_ZY\to Y\). On arrows it uses the unique pullback map retaining the given \(A\)-map and the identity of \(Y\). Checking the two projections proves identities and composition. The exact base-change definition in the retained Section 1 says that \(I\)-colimits are stable under base change precisely when these functors preserve them. Its comparison is

The map is induced by the pullbacks of \(c_i\), with their actual two projections. An abstract isomorphism between the two objects would not identify this comparison.

3. Functions in each fibre and base change

Let \(X\to Z\) and \(Y\to Z\) be small sets over the small base \(Z\). Write \(X_z\) and \(Y_z\) for their fibres. The full set construction in Universal forks, Section 5, provides all small products in \(\mathsf{Set}/Z\). For a family \((A_i\to Z)\), it is the set of pairs \((z,(a_i)_i)\) satisfying \(a_i\in(A_i)_z\), with projection \(z\). Its complete inverse factor sends an over-\(Z\) family to exactly this tuple. All index sets and values are small by the universe closure proofs. For no indices the product is \(1_Z:Z\to Z\). In particular the binary product is the usual fibre product.

Define the fibrewise function object

For \(b:X\to X'\) over \(Z\), its action is \(H_Y(b)(z,\varphi)=(z,b_z\varphi)\). Identity and composition follow by evaluation at every \(y\in Y_z\), so this is a functor on the slice.

Theorem 3.1. The functor \(-\times_ZY:\mathsf{Set}/Z\to\mathsf{Set}/Z\) is left adjoint to \(H_Y\).

Proof with every inverse specified. Evaluation is the over-\(Z\) map \(\mathrm{ev}:H_Y(X)\times_ZY\to X\) given by \(((z,\varphi),y)\mapsto\varphi(y)\). For \(T\to Z\), currying and its inverse are

The first map is defined because \(v(t,y)\in X_{z_t}\) whenever \(y\in Y_{z_t}\); the second is defined because \(w\) is over \(Z\). Substitution recovers \(v(t,y)\) at every pair, and recovers \(\varphi_t\) at every argument, hence recovers \(w(t)\). These are both inverse equations.

For \(a:T'\to T\) and \(b:X\to X'\) over \(Z\), either naturality route evaluates at \(t',y\) to \(b(v(a(t'),y))\). This proves both-variable naturality, including the naturality of the inverse bijection. If \(d:Y'\to Y\) is over \(Z\), precomposition gives \((z,\varphi)\mapsto(z,\varphi d_z)\) from \(H_Y(X)\) to \(H_{Y'}(X)\); the corresponding square evaluates to \(b(v(a(t'),d(y')))\). Thus the correspondence is also contravariantly compatible with the fibre argument \(Y\).

The unit at \(T\) sends \(t\) to \((z_t,y\mapsto(t,y))\) in \(H_Y(T\times_ZY)\), and the counit at \(X\) is evaluation. The first triangle at \((t,y)\) returns \((t,y)\); the second at \((z,\varphi)\) returns the function \(y\mapsto\varphi(y)\). They are the specified triangles of this adjunction. \(\square\)

Each \(X_z\) and \(Y_z\) is a small subset of its underlying set. The function-set and small indexed tagged-union proofs in Universes and small categories, Sections 2–3, therefore show that (3.1) is small. They include arbitrary small encodings and the needed transported bijections. There is one empty function when \(Y_z=\varnothing\), even when \(X_z=\varnothing\); there is no function when \(Y_z\ne\varnothing\) and \(X_z=\varnothing\). If \(Z=\varnothing\), all sets over it are empty and its slice has just the empty object and its identity, up to the stated set convention.

Complete canonical preservation interface. The pinned Categories, adjoint-colimit lemma proves that a left adjoint sends every existing colimit cocone to a colimit cocone. Retain its entire left-adjoint Hom-factor proof: transposing a compatible family to the right adjoint gives a cocone on the original diagram, whose unique factor transposes back. Its Hom squares supply both inverse factors and preserve the actual structural maps. Apply it with \(-\times_ZY\dashv H_Y\), whose full correspondence and triangles were checked above. Thus \(-\times_ZY\) preserves all small colimits in \(\mathsf{Set}/Z\).

Corollary 3.2. Every small colimit of sets is stable under arbitrary small-set base change.

Proof of the required comparison. A small diagram \(A_i\to Z\) has its slice colimit \(K\to Z\) by Proposition 2.1 and existence of small set colimits. Apply the preceding preservation interface: the induced cocone \(A_i\times_ZY\to K\times_ZY\), regarded over \(Z\), is a colimit. Proposition 2.1 for \(Z\) says that its underlying set cocone is a colimit. The very same maps commute with the projections to \(Y\). Proposition 2.1 for \(Y\) now makes it a colimit in \(\mathsf{Set}/Y\). This proves that \(u^*\) preserves the slice colimit. Its underlying comparison is exactly (2.2), since its legs are the two-projection pullbacks of the original \(c_i\); universal uniqueness makes that specified comparison invertible. Empty indices and empty fibres are already included in each step. \(\square\)

This argument uses the fibre product endofunctor on \(\mathsf{Set}/Z\) and the two forgetful creation statements. It provides preservation for \(u^*:\mathsf{Set}/Z\to\mathsf{Set}/Y\) with the required target slice.

4. Limits in slices and what forgetting preserves

Theorem 4.1. Every slice \(C/Z\) has finite limits if and only if \(C\) has pullbacks.

Proof. Each slice has the terminal object \(1_Z\): its unique arrow from \(a:A\to Z\) is \(a\) itself. Suppose \(C\) has pullbacks. For a cospan in \(C/Z\), with underlying arrows \(A\to D\leftarrow B\), take its pullback \(P\) in \(C\), with projections \(p_A,p_B\). Its map to \(Z\) is the common composite \(a p_A=b p_B\), because the original arrows were over \(Z\). For a compatible pair of slice maps from \(t:T\to Z\), the complete base pullback property gives a unique \(h:T\to P\). Its composite to \(Z\) is \(a p_Ah=t\), so \(h\) is a slice map. This proves the full slice pullback property. Terminality and pullbacks now give all finite slice limits by the entire finite-limit criterion, including its product and equalizer constructions and empty diagram.

Conversely, a binary product of \(A\to Z\) and \(B\to Z\) in \(C/Z\) is their pullback in \(C\). A compatible base pair \(T\to A,T\to B\) equips \(T\) with their common map \(t:T\to Z\), and the full product factor in the slice is its unique base pullback factor. This is the exact retained equivalence in Universal forks, Section 5. Finite limits in every slice supply these binary products for each actual target \(Z\). Therefore every base cospan has a pullback. \(\square\)

The theorem does not supply a terminal object in \(C\). For example, take the category of nonnegative integers ordered by \(\le\). A cospan \(a\le z\ge b\) has pullback \(\min(a,b)\), since the common lower bounds are exactly the integers below this minimum. It has no terminal object, since no nonnegative integer is largest. Each slice over \(z\) is the finite chain \(0\le\cdots\le z\), and Theorem 4.1 gives its finite limits. This is a complete written infinite-order example.

Call an index category connected when it is nonempty and all its objects belong to one equivalence class generated by its arrows. In other words, any two objects are joined by a finite zigzag of arrows with either orientation.

Theorem 4.2. Let \(D:I\to C/Z\), where \(I\) is connected. An underlying base limit lifts uniquely to a slice limit. Conversely, every slice limit of this diagram is an underlying base limit. Thus \(U_Z\) preserves every connected limit that exists in the slice.

Forward construction. Write \(a_i:A_i\to Z\) for its structure maps and \(p_i:L\to A_i\) for the chosen base limit. If \(s:i\to j\), the cone equation and the slice equation give

Equality propagates in either direction along a finite zigzag. Hence these composites have one common value \(a:L\to Z\); nonemptiness ensures that this value is specified. The \(p_i\) are slice maps. A slice cone from \(t:T\to Z\) has its unique base factor \(h:T\to L\). At any index, \(a h=a_i p_i h=t\), so it is a slice factor. Base uniqueness gives slice uniqueness, and (4.1) forces the same structure map for any lift.

Complete canonical reverse interface. Retain all of the pinned connected slice-limit lemma, including its nonempty convention and full factor proof. In its notation take the base object \(X=Z\) and the diagram \(M=D\). Every base cone from an arbitrary \(T\) has one common map \(T\to Z\), by the same zigzag equations. This equips \(T\) with the slice structure to which the universal slice factor applies. That proof therefore gives the unique base factor for every base cone, including all possible structure maps on \(T\); no base limit is assumed in this direction. It proves exactly the preservation assertion. These factors are natural in \(T\) by composition. \(\square\)

Theorem 4.3. If \(C\) has all finite, respectively all small, limits, then each \(C/Z\) has all finite, respectively all small, limits. If \(U_Z\) preserves all limits of that class, then \(Z\) is terminal in \(C\).

Proof of existence by one extra vertex. For a finite or small \(I\), adjoin an object \(*\) and one arrow \(\sigma_i:i\to *\) for each \(i\), with \(\sigma_j s=\sigma_i\) for every \(s:i\to j\). There are no arrows from \(*\) to an old object, and its only endomorphism is its identity. These rules define a category \(I^\triangleright\): every new nonidentity composition ends at \(*\), and the prescribed relation makes its composite the unique relevant \(\sigma_i\), proving associativity and both identity laws. Adding one object and one arrow per old object preserves the finite or small bound.

The diagram \(D\) extends to a base diagram \(\widetilde D:I^\triangleright\to C\) with values \(A_i\) and \(Z\), sending \(\sigma_i\) to \(a_i\). Its new composition equations are precisely \(a_jD(s)=a_i\), so it is a functor. Take its base limit, with projections

A cone from \(T\) to \(\widetilde D\) consists of a map \(t:T\to Z\) and a compatible family \(q_i:T\to A_i\) satisfying \(a_iq_i=t\). For each fixed \(t\), this is exactly a slice cone. The unique base factor satisfies both \(p_i h=q_i\) and \(r h=t\), so it is the unique slice factor. Conversely every slice factor has these components. This gives the whole inverse cone-factor correspondence and proves that \(r:R\to Z\) is the slice limit.

For \(I=\varnothing\), the augmented category has just \(*\); its limit is \(Z\) with \(r=1_Z\). Thus the empty slice limit has underlying object \(Z\). If \(U_Z\) preserves all finite or all small limits, it preserves this one, forcing \(Z\) to be terminal in \(C\). Conversely, if \(Z\) is terminal, every object and arrow of \(C\) has a unique structure over \(Z\); forgetting and supplying that structure are inverse functors, with the same cone factors. They preserve every existing limit. \(\square\)

Connected preservation and unrestricted slice existence are therefore separate assertions. For a disconnected family, (4.2) imposes equality of the different maps to the base; its underlying object generally differs from the base limit of the unaugmented diagram.

5. An index adjunction computes Kan extensions

Let \(\phi:J\to I\) be left adjoint to \(\psi:I\to J\). Denote its specified unit and counit by

Retain the whole definition and pointwise proofs of left and right Kan extensions in Extending diagrams, Sections 1–4. In particular, its restriction operation acts on every transformation by precomposition.

Theorem 5.1. For any target category \(C\), the two Kan extension functors exist and have specified natural isomorphisms

No general (co)completeness of \(C\), or smallness of all the indices, is needed.

Left extension and full transformation factors. For \(\beta:J\to C\) and \(D:I\to C\), define

The forward family is natural in \(j\): use naturality of \(h\) at \(\phi(s)\) and of \(\eta\) at \(s:j\to j'\). The inverse family is natural in \(i\): for \(a:i\to i'\), naturality of \(\varepsilon\) and \(u\) gives \(D(a)D(\varepsilon_i)u_{\psi i} =D(\varepsilon_{i'})u_{\psi i'}\beta(\psi a)\).

Starting from \(h\), its reconstructed component is

The first equality is naturality of \(h\), and the second is the second triangle in (5.1). Starting from \(u\), reconstruction at \(j\) is \(D(\varepsilon_{\phi j})u_{\psi\phi j}\beta(\eta_j) =D(\varepsilon_{\phi j})D(\phi\eta_j)u_j=u_j\). Here use naturality of \(u\) at \(\eta_j\) and the first triangle.

For \(z:\beta'\to\beta\), \(v:D\to D'\), the forward correspondence satisfies \(\Psi(vh(z\psi))=(v\phi)\Psi(h)z\); evaluate its \(j\)-component and use naturality of \(z\) at \(\eta_j\). This proves naturality in both variables, and hence naturality of its inverse. Thus \(\psi^*\dashv\phi^*\), with left-extension unit \(\beta(\eta_j)\) and left-extension counit \(D(\varepsilon_i)\). The universal definition gives the first comparison in (5.2), compatible with these specified transformations.

The pointwise formula has the same comparison. In \((\phi\downarrow i)\), the object \((\psi i,\varepsilon_i)\) is terminal: the unique arrow from \((j,t:\phi j\to i)\) is

These are precisely the full adjunction factor and inverse equations. A diagram over a terminal index object has its colimit at that object: every cocone is uniquely determined by its component there, since the unique arrows into it force all other components; conversely those arrows give a cocone, with uniqueness following from the same terminal component. Hence the comma colimit is \(\beta(\psi i)\), with leg \(\beta(b_{j,t})\). If a different comma colimit was chosen, its canonical map to this one has these legs, and its inverse is the leg at \((\psi i,\varepsilon_i)\); checking all legs proves both inverse equations.

For \(a:i\to i'\), the pointwise Kan arrow corresponds to \(\beta(\psi a)\): its composite with the \((j,t)\)-leg is \(\beta(\psi a\,\psi t\,\eta_j)=\beta(b_{j,at})\), exactly the defining leg equation. For \(z:\beta\to\beta'\), the comparison sends the induced Kan component to \(z_{\psi i}\), by naturality at \(b_{j,t}\). Thus this is a natural isomorphism both in the output index and in the whole input diagram, retaining the actual Kan maps.

Right extension and full transformation factors. For \(\alpha:I\to C\) and \(B:J\to C\), define

Both families are natural by the naturality of the unit, counit and input transformation. Starting from \(v\), reconstruction at \(i\) is \(\alpha(\varepsilon_i)v_{\phi\psi i}B(\eta_{\psi i}) =v_iB(\psi\varepsilon_i)B(\eta_{\psi i})=v_i\). Starting from \(k\), reconstruction at \(j\) is \(\alpha(\varepsilon_{\phi j})k_{\psi\phi j}B(\eta_j) =\alpha(\varepsilon_{\phi j})\alpha(\phi\eta_j)k_j=k_j\). These use the two triangles, respectively. For \(z:\alpha\to\alpha'\), \(b:B'\to B\), the forward map obeys \(\Xi((z\phi)kb)=z\Xi(k)(b\psi)\), by naturality of \(z\) at \(\varepsilon_i\). This checks both-variable naturality and its inverse. Consequently \(\psi^*\dashv\phi^*\) now supplies the right Kan extension of \(\alpha\) along \(\psi\). Its right-extension counit is \(\alpha(\varepsilon_i)\), and its right-extension unit is \(B(\eta_j)\).

For the pointwise right extension at \(j\), the outgoing comma \((j\downarrow\psi)\) has initial object \((\phi j,\eta_j)\). The unique arrow from it to \((i,t:j\to\psi i)\) is

The adjunction again proves both the equation and uniqueness. A diagram over an initial index object has its limit at that object: a cone is determined by its component there, and the unique outgoing arrows give all remaining components; each factor is the initial component. Thus this comma limit is \(\alpha(\phi j)\), with projection \(\alpha(a_{i,t})\). The canonical comparison from it to any chosen comma limit and the projection back at \((\phi j,\eta_j)\) are inverse, by checking every projection.

For \(s:j\to j'\), the induced right Kan arrow is \(\alpha(\phi s)\), since each target projection composes to \(\alpha(\varepsilon_i\phi(t')\phi(s))=\alpha(a_{i,t's})\), the exact reindexed projection. A transformation \(z:\alpha\to\alpha'\) gives \(z_{\phi j}\), by naturality at every \(a_{i,t}\). This proves the whole natural comparison in the second part of (5.2). \(\square\)

Each relevant comma category has its stated endpoint, so its required pointwise (co)limit exists even if the index is not small and \(C\) has no empty (co)limit. If either \(I\) or \(J\) is empty, the existence of both \(\phi\) and \(\psi\) forces both to be empty. Their diagram categories then have one object and one transformation, and (5.3)–(5.7) are the same vacuous universal comparisons; no initial or terminal object of \(C\) is introduced.

6. Four graded exercises with full solutions

Exercise 1 — count functions in two fibres

Introductory. Let \(Z=\{r,b\}\). Let \(X_r=\{0,1\}\), \(Y_r=\{u,v\}\), \(X_b=Y_b=\varnothing\). Let \(T_r=\{t\}\) and \(T_b=\{s_0,s_1\}\). Calculate \(H_Y(X)\) and both Hom sets in (3.2). Give evaluation and both inverse factors explicitly.

Solution. The fibre over \(r\) consists of the four functions labelled \(00,01,10,11\), with digits recording their values at \(u,v\). The fibre over \(b\) has exactly the empty function, denoted \(e_b\). Thus \(H_Y(X)\) has five elements altogether. Its evaluation sends \(((r,ab),u)\) to \(a\) and \(((r,ab),v)\) to \(b\); there is no evaluation pair over \(b\).

The product \(T\times_ZY\) has just \((t,u),(t,v)\), both over \(r\). An over-\(Z\) map to \(X\) chooses their two values independently, so there are four maps, labelled \(ab\). Its curry sends \(t\) to \((r,ab)\) and both \(s_0,s_1\) to \((b,e_b)\). Conversely, every over-\(Z\) map \(T\to H_Y(X)\) is forced at the two \(b\)-points and chooses one of the four \(r\)-points at \(t\); evaluating gives back exactly the two chosen values. These four assignments prove both inverse equations concretely. Empty fibres contribute a forced empty function, rather than an absent point.

Exercise 2 — the two products have different factors

Intermediate. Let \(Z=\{0,1\}\), and let both \(A\to Z\) and \(B\to Z\) be copies of \(1_Z\), labelled \(a_0,a_1\) and \(b_0,b_1\). Compare their product in \(\mathsf{Set}/Z\) with the product of their underlying sets. Calculate both factor tests for a singleton \(T=\{t\}\), and explain what happens to the empty slice limit.

Solution. The slice product has the two pairs \((a_0,b_0),(a_1,b_1)\), with their common projection to \(Z\). The ordinary product has four pairs, adding \((a_0,b_1),(a_1,b_0)\). If \(t:T\to Z\) is fixed at \(0\), there is exactly one slice map to each of \(A,B\), and exactly one factor into the slice product. The same holds with fixed value \(1\). When the structure on \(T\) is allowed to vary, there are therefore two compatible slice-factor choices.

There are four unrestricted base pairs of maps from \(T\) to the underlying \(A,B\), and hence four maps into the base product. The two off-diagonal choices have no common structure \(T\to Z\), so they do not give slice cones. This is why forgetting the disconnected two-object slice limit fails to preserve it.

The empty slice limit is \(1_Z\), whose underlying set has two elements. It is not terminal in sets: there are two maps from \(T\) to it, whereas a terminal set allows one. This checks the necessity in Theorem 4.3 using the actual factor count.

Exercise 3 — compute both extensions without completeness

Advanced. Let \(J=\{0<1\}\), \(I=\{0<1<2\}\), with \(\phi(0)=0,\phi(1)=2\), and \(\psi(0)=\psi(1)=0,\psi(2)=1\). Verify \(\phi\dashv\psi\). Take the set diagrams

Compute \(\operatorname{Lan}_\phi\beta\), \(\operatorname{Ran}_\psi\alpha\), their specified unit/counit components, and both transformation sets in (5.3) and (5.6), taking \(D=\alpha\) and \(B=\beta\).

Solution. For \(j=0\), both \(\phi(j)\le i\) and \(j\le\psi(i)\) always hold. For \(j=1\), both hold exactly when \(i=2\). These are all six pairs, proving the natural order-category Hom bijections. The unit is the identity at both objects of \(J\). The counit is the identity at \(0,2\), and the arrow \(0\to1\) at \(i=1\). Their order-category composites give both triangles.

The left extension is \(\{x\}\xrightarrow{1}\{x\}\to\{r,s\}\), with the last map \(x\mapsto r\). Its left-extension unit is the identity at both \(j\). The right extension is \(\{a,b\}\to\{d,e\}\), with both elements sent to \(d\). Its right-extension counit at \(i=0,2\) is the identity; at \(i=1\) it is the map \(\{a,b\}\to\{c\}\) sending both elements to \(c\).

A transformation \(\beta\to\alpha\phi\) chooses the image of \(x\) freely in \(\{a,b\}\). Commutativity forces the image of \(r\) to be \(d\), while the image of \(s\) is free in \(\{d,e\}\). Thus it has four possibilities. A transformation \(\beta\psi\to\alpha\) has those same two choices at indices \(0,2\), and its index-\(1\) map to \(\{c\}\) is forced. Its naturality at \(0\to1\) and \(1\to2\) is then satisfied. There are again four possibilities, and (5.3) sends the free choices to themselves.

For (5.6), its left set is \(\operatorname{Nat}_J(\beta,\alpha\phi)\), already calculated, and its right set is \(\operatorname{Nat}_I(\beta\psi,\alpha)\), also already calculated. Its forward formula supplies the forced middle component by the counit map \(\alpha(0\to1)\); its inverse retains the endpoint choices because the unit is the identity. Both full correspondences therefore have four elements and the stated inverse maps. The general proof used only the index adjunction, rather than a completeness property of this target.

Exercise 4 — pull back a quotient with its legs

Challenge. Let \(Z=\{0,1\}\), \(A=\{a,b,c\}\) with \(a,b\) over \(0\) and \(c\) over \(1\). Let \(E=\{q\}\) over \(0\), with \(f(q)=a,g(q)=b\). Let \(Y=\{r,s,t\}\to Z\) send \(r,s\) to \(0\) and \(t\) to \(1\). Calculate the coequalizer and its pullback. Prove, for every target set \(W\), that the canonical map (2.2) for this coequalizer diagram is invertible.

Solution. The coequalizer \(K\) has the classes \(\{a,b\}\) and \(\{c\}\), over \(0,1\), respectively. Its structure is well defined because the generator pair \(a\sim b\) lies in one fibre. The set \(K\times_ZY\) has the three points \((\{a,b\},r),(\{a,b\},s),(\{c\},t)\).

The pulled-back \(A\) has five points \((a,r),(b,r),(a,s),(b,s),(c,t)\). The pulled-back \(E\) has \((q,r),(q,s)\). Its two maps impose precisely \((a,r)\sim(b,r)\) and \((a,s)\sim(b,s)\). Thus the quotient has the three classes

The canonical comparison sends \(R,S,T\) to the three points of \(K\times_ZY\) listed above; its inverse sends those points to their corresponding classes. Both composites are identities, and these maps retain the \(Y\)-projection and the original quotient legs.

For any \(W\), a map from the five-point set equalizing the two pulled-back maps is exactly a choice of one value for \(R\), one for \(S\), and one for \(T\). The unique factor on the three-point quotient uses those values. Composing with the displayed inverse gives the same unique factor through \(K\times_ZY\). These operations are inverse on every point and natural under postcomposition \(W\to W'\). They prove the full canonical coequalizer property, rather than only agreement of cardinalities. For a two-element target there are \(2^3=8\) such factors.

7. References and retained proof interfaces

• Riehl. Emily Riehl, Category Theory in Context, Sections 1.3 and 6.1, for comma and slice categories and the adjoints of restriction along a functor.
• [Stacks, Categories]. The pinned Categories structured source retains GFDL 1.2 or later under its licence. The complete connected slice-limit proof at lines 1668–1697 supplies the reverse direction of Theorem 4.2; the complete left-adjoint colimit proof at lines 3624–3660 supplies the preservation interface in Section 3. Their mathematical interfaces and full factors are retained without copying their expression.
• Owned prerequisites. The full cone/cocone and opposite proofs, set-size closure, ordinary currying, slice-colimit computation, relative products and finite-limit criterion are the linked lessons above. The entire pointwise Kan factor proofs supply the exact structural comparisons in Theorem 5.1. Existing versions retain their own attribution and licences.