Reindexing, units and terminal probes

Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original text: CC0. Self-checked; no independent review.

A change of indices gives a specified map between colimits. A transformation between two index functors gives a triangle of these maps. Under a coherence condition, a transformation from the identity supplies the inverse comparison. An index adjunction supplies a useful three-map sequence, but the general conclusion concerns a composite of two of those maps. Terminal objects and formal colimits of representables provide two ways to recognize when reindexing is cofinal.

Compose from right to left. Category data and choices live in ambient set theory. Fix a working Grothendieck universe \(\mathcal U\); small means \(\mathcal U\)-small, with the full closure and relabelling proofs in Universes and small categories, Sections 1–3. Functor categories may require a larger ambient universe. General reindexing statements assume the indicated colimits exist; only the formal presheaf test in Section 5 requires its entire base and index to be small.

Retain the entire cocone representations, diagram-map laws and structural comparisons in Compatible families, Sections 2–3, all inverse cofinal cocone factors in Change an index, Sections 1–3, and the specified unit, counit and triangles in Passing maps across an adjunction, Sections 1–2. These are complete proof interfaces, including their arbitrary ambient and empty-index conventions.

1. A transformation gives a commuting triangle

Let \(D:I\to C\), let \(\phi_1,\phi_2:J\to I\), and let \(\theta:\phi_1\to\phi_2\) be natural. Suppose all three colimits below exist. Give them their chosen legs

The full retained structural comparison gives \(q_a:K_a\to L\) with \(q_a d_j^a=c_{\phi_a j}\). The full diagram-transformation construction gives \(r:K_1\to K_2\) with \(r d_j^1=d_j^2 D(\theta_j)\).

Proposition 1.1. These maps satisfy \(q_2r=q_1\), naturally in \(D\).

Proof. For every \(j\),

The middle equality is precisely the \(I\)-cocone equation at \(\theta_j:\phi_1j\to\phi_2j\). Colimit uniqueness gives the result. For a transformation \(v:D\to E\), each comparison square commutes after every structural leg, by naturality of \(v\) at \(\theta_j\) and the full retained diagram-map equations. Uniqueness proves the squares on the colimits. No cofinality hypothesis is needed. For empty \(J\), uniqueness is the empty-colimit factor property. \(\square\)

2. When reindexing has a specified inverse

Let \(T:I\to I\) and \(\theta:1_I\to T\) be natural. Suppose

For \(D:I\to C\), choose colimits \(L=\operatorname{colim}_I D\) and \(M=\operatorname{colim}_I DT\), with legs \(c_i\) and \(d_i\). Their two specified maps are

The first is the index comparison, and the second is induced by \(D\theta\). The complete retained cocone proof makes the second family compatible: for \(s:i\to j\), naturality of \(\theta\) gives \(d_jD(\theta_j)D(s)=d_jD(Ts)D(\theta_i)=d_iD(\theta_i)\).

Theorem 2.1. The maps \(e\) and \(x\) are inverse isomorphisms.

Proof of both composites. At each \(c_i\), the original cocone equation gives \(ex c_i=c_{T i}D(\theta_i)=c_i\). Hence \(ex=1_L\). At each \(d_i\), use (2.1) and the cocone of \(DT\):

Thus \(xe=1_M\). Each equality uses its specified leg, so the inverse is exactly the map induced by \(D\theta\). For a transformation of \(D\), the retained leg laws make both \(e\) and \(x\) natural, including their inverse squares. The proof requires neither \(T^2=T\) nor invertibility of \(\theta_i\). Empty indices use the same empty universal uniqueness. \(\square\)

Naturality of \(\theta\) alone gives \(T(\theta_i)\theta_i=\theta_{T i}\theta_i\). That equation cannot be cancelled without a suitable hypothesis on \(\theta_i\). Exercise 4 supplies a complete example in which dropping (2.1) destroys the second inverse identity.

3. Three maps from an index adjunction

Let \(\lambda:I\to J\) be left adjoint to \(\mu:J\to I\), with specified unit \(\theta:1_I\to\mu\lambda\) and counit \(\varepsilon:\lambda\mu\to1_J\). Write \(T=\mu\lambda\). Retain the full unit/counit Hom factors and triangles from Passing maps across an adjunction, Section 2.

Proposition 3.1. The right adjoint \(\mu\) is cofinal.

Full comma specialization. For \(i\in I\), the incoming comma \((i\downarrow\mu)\) has initial object \((\lambda i,\theta_i)\). Indeed, to an object \((j,a:i\to\mu j)\), its unique outgoing arrow is \[ \begin{gathered} b=\varepsilon_j\lambda(a):\lambda i\to j,\\ \mu(b)\theta_i=a. \end{gathered} \tag{3.1} \] These are exactly the inverse adjunction factor and its equation. The full initial-comma proof in Fibres, slices and index adjunctions, Section 5, with the adjoints renamed, checks existence and uniqueness for every object. A category with an initial object is nonempty and connected: every object is joined to that initial object by its unique arrow. Thus every incoming comma satisfies the complete cofinal definition. \(\square\)

Suppose \(C\) admits \(I\)-colimits and let \(D:I\to C\). The two \(I\)-colimits \(L=\operatorname{colim}D\) and \(M=\operatorname{colim}DT\) exist. Cofinality and the entire inverse-cocone theorem in Change an index, Section 2, give existence of \(N=\operatorname{colim}_J D\mu\), even though all \(J\)-colimits in \(C\) were not assumed. Write their legs as \(c_i,d_i,n_j\).

The relevant maps are completely specified by

The first family is compatible by naturality of the unit, as in Section 2. The second is the comparison for restriction of \(D\mu\) along \(\lambda\), and the third is the comparison for restriction of \(D\) along \(\mu\). The retained proofs verify every cocone equation and all these endpoints. Cofinality makes this particular \(w\) invertible, with the inverse obtained by extending the \(J\)-cocone.

Theorem 3.2. Always \(wvu=1_L\), and \(vu=w^{-1}\). If every \(\lambda(\theta_i)\) is invertible in \(J\), then \(u,v,w\) are all invertible.

Proof of the unconditional assertion. On each \(c_i\), \[ wvu c_i=c_{T i}D(\theta_i)=c_i. \tag{3.3} \] The last equality is the original cocone equation. Uniqueness gives \(wvu=1_L\), and composing with \(w^{-1}\) gives \(vu=w^{-1}\). Thus the composite \(vu\) and \(w\) are inverses. This calculation alone does not make the two individual factors \(u,v\) invertible.

The extra unit condition and its exact coherence consequence. Define \[ m_i=\mu(\varepsilon_{\lambda i}):T^2i\to Ti. \tag{3.4} \] The two adjunction triangles give \[ m_i T(\theta_i)=1_{T i}, \qquad m_i\theta_{T i}=1_{T i}. \tag{3.5} \] For the first equation, apply \(\mu\) to \(\varepsilon_{\lambda i}\lambda(\theta_i)=1_{\lambda i}\). For the second, use \(\mu(\varepsilon_{\lambda i})\theta_{\mu\lambda i}=1_{\mu\lambda i}\). Under the extra condition, \(T(\theta_i)=\mu(\lambda(\theta_i))\) is invertible, because the full functor inverse law sends inverses to inverses. The first equation forces \(m_i=T(\theta_i)^{-1}\). The second then forces \[ \theta_{T i}=T(\theta_i). \tag{3.6} \] Thus the hypotheses of Theorem 2.1 hold, with \(x=u\) and \(e=wv\): indeed \(wv d_i=c_{T i}\). That theorem gives \(u=(wv)^{-1}\). Since \(w\) and \(wv\) are invertible, \(v=w^{-1}(wv)\) is invertible too. This proves all three assertions with their canonical maps. \(\square\)

The unit condition has supplied (3.6), rather than an unsupported equality \(T^2=T\). All maps in (3.2) are natural under transformations of \(D\), by checking their stated structural legs and using the complete retained comparison laws. If either \(I\) or \(J\) is empty, existence of both adjoint functors forces both empty, and every comparison is between the chosen empty colimits. The \(I\)-colimit assumption then supplies the needed initial object of \(C\).

4. A terminal object tests cofinality

Let \(F:A\to B\), and suppose \(A\) has a terminal object \(t\). No smallness of the whole categories is required beyond the stated ambient set convention.

Theorem 4.1. The functor \(F\) is cofinal if and only if \(F(t)\) is terminal in \(B\).

Proof. Suppose \(F(t)\) is terminal. For every \(Y\in B\), the incoming comma \((Y\downarrow F)\) has object \((t,\ell_Y)\), where \(\ell_Y:Y\to F(t)\) is the unique arrow. From \((a,q:Y\to F(a))\), the unique arrow \(p_a:a\to t\) satisfies \(F(p_a)q=\ell_Y\), because both sides go from \(Y\) to the terminal object. Uniqueness of \(p_a\) makes \((t,\ell_Y)\) terminal in this comma. It is nonempty and connected, so \(F\) is cofinal.

Conversely, suppose all incoming commas are nonempty and connected. For a fixed \(Y\), attach to \((a,q)\) the arrow

For a comma arrow \(s:(a,q)\to(a',q')\), its equation is \(F(s)q=q'\). Terminality in \(A\) gives \(p_{a'}s=p_a\), so \(b_{a',q'}=F(p_{a'}s)q=b_{a,q}\). Read this equality in either orientation and propagate it along every zigzag. Connectedness makes all \(b_{a,q}\) equal; nonemptiness supplies one such arrow. Any arrow \(r:Y\to F(t)\) gives the comma object \((t,r)\), for which \(b_{t,r}=r\), since \(p_t=1_t\). Every arrow \(r\) therefore equals the one already supplied. There is exactly one arrow from each \(Y\), proving terminality of \(F(t)\). \(\square\)

Since \(t\) exists, \(A\) is nonempty, and existence of \(F\) then forces \(B\) nonempty. No empty-target exception is being hidden in the criterion.

For an object \(Z\in C\), let \(\delta_Z:\mathsf{Pt}\to C\) select it, where \(\mathsf{Pt}\) has one object and its identity arrow.

Proposition 4.2. The category \((X\downarrow\delta_Z)\) is isomorphic to the discrete category on \(C(X,Z)\). Consequently \(\delta_Z\) is cofinal if and only if \(Z\) is terminal.

Proof on every object and arrow. Its objects are the arrows \(q:X\to Z\). An arrow \(q\to q'\) must come from the unique identity of \(\mathsf{Pt}\), and its comma equation is \(q=q'\). Thus there is an identity at each \(q\) and no other arrow. Sending the comma object to \(q\), and its identity to the corresponding discrete identity, is a functor; sending them back is its literal inverse. A discrete category is nonempty and connected exactly when its object set has one element: its generated zigzag equivalence is equality. Hence cofinality is exactly the condition that \(C(X,Z)\) has one element for every \(X\). This is terminality. Its Hom values are the original ambient sets; local smallness supplies small encodings when needed. \(\square\)

The same conclusion follows from Theorem 4.1, since the unique object of \(\mathsf{Pt}\) is terminal. The explicit isomorphism above also identifies the entire comma category used in the original test.

5. The formal Yoneda colimit detects cofinality

Now let \(I,J\) be small and let \(\alpha:J\to I\). Use the complete Hom and Yoneda construction in Points and representations, Section 2. The representable \(h_i\) has value \(h_i(k)=I(k,i)\), with precomposition in \(k\). Postcomposition along \(\alpha(s)\) makes \(j\mapsto h_{\alpha j}\) a diagram of presheaves.

Let \[ Q=\operatorname{colim}_{j\in J}h_{\alpha j} \quad\text{in }\widehat I. \tag{5.1} \] This is the formal colimit of the original object diagram, rather than a claim that a representative in \(I\) exists. The entire arbitrary-base pointwise proof in Colimits as connected components, Section 4 supplies existence, all natural universal factors and the value formula \[ Q(k)=\operatorname{colim}_{j\in J}I(k,\alpha j). \tag{5.2} \] Smallness of \(I,J\), the full Hom encoding and the small tagged-union/quotient bounds make each value small. The transformation set may be placed in a larger ambient universe without changing these values.

The covariant element category of \(j\mapsto I(k,\alpha j)\) is exactly \((k\downarrow\alpha)\): an object is \((j,q:k\to\alpha j)\), and an arrow \(s:j\to j'\) satisfies \(\alpha(s)q=q'\). The entire component/class proof in that lesson, Sections 1–2, therefore gives the specified bijection

Its inverse sends that component back to \([j,q]\); equality of the generated relations proves independence and both inverse equations. The pointwise restriction along \(a:k'\to k\) sends \([j,q]\) to \([j,qa]\). On comma categories the same formula is the functor taking \((j,q)\) to \((j,qa)\) and retaining \(s\). Its comma equation, identities and compositions are preserved. Thus (5.3) is a natural comparison on the whole base, rather than only an objectwise cardinal identity.

Theorem 5.1. The functor \(\alpha\) is cofinal if and only if \(Q\) is a terminal presheaf.

Proof with the actual singleton comparison. The singleton presheaf \(1\) is terminal by the complete pointwise proof: every component function to it is unique and all naturality squares commute. If \(\alpha\) is cofinal, every incoming comma is nonempty and connected, so (5.3) makes every value \(Q(k)\) a singleton. The unique transformation \(q:Q\to1\) is then invertible. Its inverse at \(k\) selects the sole class; restriction takes that class to the sole class at \(k'\), proving naturality of the inverse. Both composites fix the sole elements. Hence \(Q\) is terminal.

Conversely, if \(Q\) is terminal, the unique transformations between \(Q\) and \(1\) compose to their identities, by terminal uniqueness. Thus \(Q(k)\) is a singleton at every \(k\). Equation (5.3) makes each incoming comma nonempty and connected, so \(\alpha\) is cofinal. \(\square\)

If \(J=\varnothing\) and \(I\) has an object, then \(Q(k)=\varnothing\) at that object and is not terminal; the incoming comma is empty. If \(I=\varnothing\), existence of \(\alpha\) forces \(J=\varnothing\), and \(\widehat I\) has its unique presheaf and transformation. The cofinality condition is vacuous and this presheaf is terminal, as the theorem asserts.

6. Four graded exercises with full solutions

Exercise 1 — the triangle retains each leg

Introductory. Let \(I=\{0<1\}\), \(J=\mathsf{Pt}\), with \(\phi_1(*)=0,\phi_2(*)=1\) and the unique transformation between them. Let \(D(0)=\{a,b\}\), \(D(1)=\{c,d,e\}\), and send both \(a,b\) to \(c\). Compute every map in Proposition 1.1 and its equality at all elements.

Solution. The colimit of \(D\) is \(D(1)\): a cocone is determined by its map from \(D(1)\), and its other map is the composite with \(D(0\to1)\). Its full unique factor is that \(D(1)\)-component. The two point-index colimits are \(K_1=\{a,b\}\) and \(K_2=\{c,d,e\}\), since a one-object identity-only diagram has its value as colimit.

Here \(q_1(a)=q_1(b)=c\); \(q_2\) is the identity of \(\{c,d,e\}\); and \(r(a)=r(b)=c\). Thus \(q_2r=q_1\) at both \(a,b\). For any target set, assigning a cocone on the arrow diagram is exactly assigning one map from \(\{c,d,e\}\); precomposing it with the computed triangle gives identical values on both original points. This checks the specified factor comparison, including the two points \(d,e\) which have no preimage from stage \(0\).

Exercise 2 — the three maps for an endpoint adjunction

Intermediate. Let \(I=\{0<1\}\), \(J=\{0<1<2\}\), with \(\lambda(0)=0,\lambda(1)=2\) and \(\mu(0)=\mu(1)=0,\mu(2)=1\). Let \(D:I\to\mathsf{Set}\) be \(\{a,b\}\to\{c,d\}\), with both \(a,b\mapsto c\). Verify the index adjunction, then calculate \(L,M,N,u,v,w\), each unit condition and every incoming comma initial object.

Solution. For \(i=0\), both \(\lambda i\le j\) and \(i\le\mu j\) always hold. For \(i=1\), both hold exactly at \(j=2\). These six pairs verify the natural order-category Hom bijections, with their unique arrows. The unit is the identity since \(\mu\lambda=1_I\); the counit is the identity at \(0,2\) and the arrow \(0\to1\) at the middle object of \(J\). Both triangles hold.

The incoming comma at \(i=0\) consists of all three objects of \(J\) and their order arrows, and has initial object \(0=\lambda(0)\). At \(i=1\) it contains only \(2=\lambda(1)\). These are the exact \((\lambda i,\theta_i)\) objects of Proposition 3.1.

The two \(I\)-diagrams \(D\) and \(DT\) agree, so \(L=M=\{c,d\}\). The \(J\)-diagram \(D\mu\) is \(\{a,b\}\xrightarrow{1}\{a,b\}\to\{c,d\}\). Its colimit is \(N=\{c,d\}\), by its terminal index \(2\) and the full endpoint factor proof. Each of \(u,v,w\) is the identity on \(c,d\), by (3.2) on the terminal legs. At stage \(0\) every relevant leg sends \(a,b\) to \(c\), and at stage \(1\) of \(I\) it is the identity on \(c,d\), verifying all other equations. Each \(\lambda(\theta_i)\) is the identity, so the extra condition holds and \(wvu=1\) with all three actual inverses.

Exercise 3 — one object need not be terminal

Advanced. Let \(I\) have one object \(t\), with automorphism group \(\{1,g\}\), \(g^2=1\). Compare the functor \(\delta_t:\mathsf{Pt}\to I\) with \(1_I:I\to I\). Compute each formal Yoneda colimit, its restriction action, and the incoming comma component sets. Decide cofinality and terminality.

Solution. For \(\delta_t\), the incoming comma at \(t\) is discrete on \(I(t,t)=\{1,g\}\), by Proposition 4.2. It has two components and is not connected, so \(\delta_t\) is not cofinal. Its formal Yoneda colimit is just \(h_t\), with two elements at the only test object. Restriction along \(g\) interchanges \(1,g\) by right multiplication, so it is not the singleton terminal presheaf. There is also no terminal object in \(I\): its only candidate has two endomorphisms.

For \(1_I\), its incoming comma at \(t\) has objects \(1,g:t\to t\). There is a unique comma arrow from \(q\) to \(q'\), namely \(q'q^{-1}\), so it is nonempty and connected. Its formal colimit at \(t\) is the quotient of \(\{1,g\}\) by the diagram's postcomposition action; \(g\) sends \(1\) to \(g\), identifying the two. There is one class, and the remaining restriction by right multiplication fixes that class. The resulting presheaf is exactly the terminal singleton presheaf.

The two calculations give their explicit class maps and inverses in (5.3). They show why a single object with extra arrows differs from the category \(\mathsf{Pt}\).

Exercise 4 — naturality does not give the coherence equation

Challenge. Let \(I\) have one object and endomorphism monoid \(\operatorname{End}(\mathbb N)\), all functions of nonnegative integers under composition. Let \(T:I\to I\) act on an arrow \(f\) by

Let \(D:I\to\mathsf{Set}\) be the natural action on \(\mathbb N\). Prove \(T\) is a functor and \(\theta:1_I\to T\) is natural. Compute both colimits in (2.2), their actual comparisons, and the failure of (2.1). Use full quotient factors for arbitrary target sets.

Solution. The identity function is fixed by \(T\), because its two defining cases give \(n\) at every \(n\). For functions \(f,h\), both \(T(hf)\) and \(Th\,Tf\) fix \(0\); at \(n+1\), both have value \(h(f(n))+1\). Thus \(T\) preserves composition. Naturality is \((Tf)\theta(n)=f(n)+1=\theta f(n)\) for every \(f,n\).

The original action diagram has colimit a singleton. A cocone to a set \(W\) is a function \(a:\mathbb N\to W\) invariant under every endomorphism. The constant endomorphism with value \(0\) forces \(a(n)=a(0)\) for all \(n\); conversely every constant function is invariant. Such a cocone therefore has its unique factor from a singleton, taking its point to \(a(0)\). This proves the entire colimit property even when \(W\) is empty, in which case neither map exists.

Under the action \(Tf\), the point \(0\) is fixed and positive integers map to positive integers. The arrows coming from constant functions connect every positive integer to \(1\), so there are exactly two quotient classes: \(\{0\}\) and \(\{1,2,\ldots\}\). A cocone function is precisely one with a constant value on the positive class and an independent value at \(0\). Thus it factors uniquely from the two-point class set, with both inverse factor assignments verified at every integer and naturally under postcomposition in \(W\). This proves the full colimit \(M\), rather than just its orbit count.

The canonical \(e:M\to L\) sends both classes to the singleton. The induced \(x:L\to M\) sends that point to the positive class, since every \(\theta(n)\) is positive. Hence \(ex=1_L\), whereas \(xe\) sends the zero class to the positive class and is not \(1_M\).

At the one object of \(I\), \(\theta_T=\theta\) sends \(0\) to \(1\), while \(T\theta\) fixes \(0\). They are unequal, so (2.1) fails exactly as needed. The monoid \(\operatorname{End}(\mathbb N)\) is a small function set by the complete universe closure proof, and its one-object category is small. Therefore the set colimits used here exist under the same working-universe conventions. This is an infinite example with a complete written proof; a finite sample of its integers cannot establish its full quotient or functor claims.

7. References and retained proof interfaces

• Schapira. Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Sections 2.3–2.5, for limits, colimits and their properties.
• [Stacks, Categories]. The pinned Categories structured source gives the full cofinal definition and the canonical colimit comparison declaration, with that comparison proof omitted. The entire owned inverse cocone proof in Change an index supplies the argument used here. The linked source retains GFDL 1.2 or later under its licence; no source expression is incorporated.
• Owned proofs. The complete compatible-cocone and reindexing laws, cofinal inverse factors, adjunction triangles and comma endpoint proofs are linked above. The full Hom/Yoneda and pointwise quotient/component proofs supply every value, naturality map, size bound and inverse in the formal presheaf criterion. Their existing versions retain their respective attribution and licences.