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Finite diagrams and comma objects

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

A map between two formal filtered objects can be represented after moving to later stages. A diagram asks for more: several maps must have representatives at the same stages, and the prescribed equations must hold there. Finiteness lets us arrange finitely many witnesses together. It also lets us compute natural transformations before or after a filtered completion.

Read Ind-objects through their elements for formal filtered colimits and functor extension, and Formal colimits and compact presentations for the full-faithfulness criterion using finitely presented pieces. The complete finite-set interchange proof is Filtered stages and finite limits, Section 4. Section 2 below proves finite acyclic strictification with all vertex-index arrows, stage relations and cofinal comparisons.

Fix an indexing universe \(\mathcal U\) and a larger ambient universe. Categories are locally \(\mathcal U\)-small, and all filtered diagrams are \(\mathcal U\)-small. A finite category has finitely many objects and finitely many morphisms. Having only finitely many objects would not suffice for the finite naturality argument below.

1. Naturality is a finite set of equations

For any \(\mathcal U\)-small category \(K\), the constant embedding \(\iota:\mathsf C\to\operatorname{Ind}(\mathsf C)\) gives a fully faithful functor

The target admits filtered colimits computed at each object of \(K\). An arrow of \(K\) acts on the colimit through the compatible maps at the stages; identities and composition follow by checking the maps from each stage. The universal filtered extension therefore gives a comparison

It sends a filtered family of diagrams to its pointwise formal colimit. This construction works for every small \(K\). Finiteness enters the next theorem's claim about maps between formal diagrams.

Theorem 1.1. For every finite category \(K\), the comparison \(\Phi\) is fully faithful. No restriction on the endomorphisms of \(K\) is required.

Proof. Take \(U:K\to\mathsf C\), viewed as a diagram of constant ind-objects, and a filtered diagram \((V_i)\) in the target. A natural transformation \(U\to V_i\) is a family \((\eta_a)_{a\in\operatorname{Ob}K}\) satisfying

Consequently its set is the equalizer of the two maps

The two maps give the two sides of (1.2). Both products are finite. Each constant object \(\iota U(a)\) is finitely presented in the ind-category, by the element lesson's filtered-colimit theorem. Thus its Hom functor commutes with a filtered colimit. Commuting the filtered colimit through the finite products and equalizer in (1.3) gives

The bijection is the canonical map: component maps are sent to their colimit maps, with the same naturality equations. It says that every \(\mathsf C\)-valued diagram is finitely presented in the target category. The original embedding of diagram categories is fully faithful. Apply the compact-presentation full-faithfulness theorem from the preceding lesson to this embedding and its realization extension. That extension is \(\Phi\), so \(\Phi\) is fully faithful. \(\square\)

Equation (1.4) expresses both existence and equality. A transformation into the colimit has representatives for finitely many components, and its finitely many naturality equations hold at one later stage. Equality of two transformations likewise requires only finitely many component equalities. These are precisely the two parts of finite presentation.

2. When every finite diagram comes from stages

For the construction, let \(J\) denote the finite diagram shape and reserve \(K\) for the common stage category. Write \(\Phi_J\) for the comparison (1.1) with shape \(J\). The original presentations are arbitrary small filtered categories.

2.1. Finite witnesses and compact constants

Fix an indexing universe \(\mathcal U\) and an ambient universe containing the coefficient categories. Let \(\mathsf C\) be locally \(\mathcal U\)-small. It need not have a \(\mathcal U\)-small skeleton, limits, or colimits. Every presenting category used below is \(\mathcal U\)-small and filtered. Write

Use Ind-objects through their elements, Sections 1–3 for the presheaf model: ind-objects are small filtered colimits of representables, their inclusion into presheaves is full, small filtered colimits exist there and are computed pointwise, and every \(\iota X\) is finitely presented. In particular, for a specified presentation

Yoneda gives the canonical identification

We need two finite-witness facts in their arbitrary-category form.

Finite graph cocones. A finite graph of specified objects and arrows in a filtered category has a commuting cocone. First bring its finitely many objects to one target, using the common-target condition successively. For each specified edge \(a:x\to y\), the proposed arrows \(r_x\) and \(r_y a\) are parallel. Postcompose the whole family by an arrow equalizing that pair. There are finitely many edges, and each already imposed equation survives postcomposition. The empty graph uses any object, supplied by nonemptiness. Distinct graph vertices may label the same index object; they still have separately specified cocone components. No closure of the graph under composition is needed.

Eventual equality by one arrow. If two elements of one stage \(S(i)\) in a filtered set diagram have the same colimit class, there is a single arrow \(s:i\to j\) making their images equal. Indeed the common-stage equality criterion initially gives \(a,b:i\to k\) with \(S(a)x=S(b)y\). Equalize the parallel index arrows \(a,b\) by \(t:k\to j\); then \(s=ta=tb\) works. A finite list of such equalities, even for finitely many different set diagrams on the same index, can be made true by one arrow. Choose one witness arrow for each equation, and take a cocone on the finite star of these arrows. Its common composite from the original stage witnesses every equation.

These are exactly the constructions proved in Filtered stages and finite limits, Sections 1–2. Its complete Theorem 4.1 also supplies the finite-limit interchange used in Theorem 1.1 above.

2.2. A category containing every stage map

A finite category means one with finitely many objects and morphisms. Suppose \(J\) has a degree function

Let \(D:J\to\operatorname{Ind}_{\mathcal U}(\mathsf C)\) be a diagram. Choose, and henceforth keep, presentations at its vertices:

The categories \(I_v\) may have parallel arrows, endomorphisms, and arbitrary small object sets. No replacement by posets is made.

Define \(K=K(D;(X_v,I_v))\) as follows. An object \(k\) consists of an index \(i_v(k)\in I_v\) at every vertex and an actual coefficient-category map

Require all three kinds of equations:

Thus every object is an actual \(\mathsf C\)-valued \(J\)-diagram, with a specified natural transformation from its constant image into \(D\).

A morphism \(u:k\to\ell\) is a tuple of index-category arrows

satisfying

Two such morphisms are equal precisely when their index-arrow tuples are equal, even if the functors \(X_v\) identify some arrows. Identities are the index identities. If \(u:k\to\ell\) and \(t:\ell\to m\), then

so coordinatewise composition is a morphism. The category laws follow from those of the \(I_v\). Projection on a vertex is the functor

There is also a functor

Its map on \(u\) has components \(X_v(u_v)\); equation (2a.2.4) is its naturality. All identities and compositions in both variables are included in (2a.2.3)–(2a.2.4).

The category \(K\) is \(\mathcal U\)-small. The vertex tuples form a finite product of small object sets. Over each tuple, arrow data belong to a finite product of locally small Hom sets; the equations select a subset. Between two levels, morphisms form a subset of a finite product of small Hom sets in the \(I_v\). Small tagged unions over the small family of levels then bound the entire arrow set. If labels for actual coefficient objects do not themselves lie in \(\mathcal U\), encode the data by the small vertex indices and the associated Hom-set elements, as in Universes and small categories. This does not require a small skeleton of \(\mathsf C\).

2.3. Refinement with prescribed index arrows

The next lemma supplies more than a common target of stages. It keeps the actual composites of index arrows, which is essential for filteredness and cofinality.

Refinement lemma. Take finitely many levels \(k_s\in K\). For every vertex \(v\), take any finite graph \(H_v\) of objects and arrows in \(I_v\), containing a marked vertex labeled \(i_v(k_s)\) for each \(s\). Extra vertices, extra arrows, parallel arrows, and repeated labels are allowed. Then there is a level \(\ell\in K\), morphisms \(q_s:k_s\to\ell\), and a cocone from \(H_v\) to \(i_v(\ell)\) whose marked components are exactly \((q_s)_v\).

Proof. Process the finitely many vertices of \(J\) in increasing degree; break ties in any order. All nonidentity sources of arrows into the vertex \(w\) being processed have already been processed. Their indices \(i_v(\ell)\), marked arrows \((q_s)_v\), and the maps between those earlier vertices are fixed.

First choose a cocone on \(H_w\), with target \(b\in I_w\). Denote its marked arrows by \(a_s:i_w(k_s)\to b\). If there are no marked vertices or other vertices, filtered nonemptiness still supplies such a \(b\).

For every nonidentity arrow \(\gamma:v\to w\), the formal map

has a representative \(r_\gamma:X_v(i_v(\ell))\to X_w(b_\gamma)\), by (2a.1.1). There are finitely many such arrows. Bring \(b\) and all \(b_\gamma\) to a common index \(c\in I_w\). Postcompose the cocone components and all representatives to \(c\). Call the resulting marked arrows \(a_s^0:i_w(k_s)\to c\) and incoming maps \(g_\gamma^0:X_v(i_v(\ell))\to X_w(c)\).

At this stage impose the following finite list of coefficient-map equations.

First, for every pair of nonidentity arrows \(\alpha:u\to v\), \(\beta:v\to w\), impose

Both maps have source \(X_u(i_u(\ell))\) and target \(X_w(c)\). They have the same image in \(D_w\), because the earlier arrow \(f^\ell_\alpha\) represents \(D(\alpha)\) and \(D(\beta)D(\alpha)=D(\beta\alpha)\).

Second, for every \(s\) and every nonidentity \(\gamma:v\to w\), impose

Here both maps have source \(X_v(i_v(k_s))\) and target \(X_w(c)\). Their images in \(D_w\) both equal \(D(\gamma)\lambda_{v,i_v(k_s)}\): the left uses validity of \(k_s\), and the right uses validity of \(g^0_\gamma\) and the presentation cocone equation for \((q_s)_v\).

For each equation, (2a.1.1) and the eventual-equality fact give an arrow out of \(c\) making it true. Synchronize these finitely many arrows by a finite graph cocone. This gives one arrow \(t:c\to i_w(\ell)\) making every equation true. Replace every cocone component by its composite with \(t\), put

and retain the already fixed data at earlier vertices.

Postcomposition preserves every equation of \(H_w\). Equations (2a.3.1) install all nontrivial composition equations ending at \(w\). Equations (2a.3.2) install every transition square into \(w\) for every \(q_s\). The identity transition square holds automatically. The incoming maps still represent their specified formal maps, since postcomposition by a presenting transition leaves the formal class unchanged. No earlier stage or earlier map is altered.

After all vertices have been processed, every composition equation between two nonidentity arrows has been installed at its target. Compositions involving identities follow from the assigned identities. Every \(q_s\) satisfies all of (2a.2.4), and every \(H_v\) has its claimed cocone. This proves the lemma. \(\square\)

This proof is valid without injective transitions, faithful presentation functors, or poset indices. In particular, it does not infer equality of index arrows from equality of their coefficient images. Index-arrow equations are installed in the cocones on \(H_v\) before the coefficient-map equations are installed.

2.4. Filteredness and the cofinal comma condition

Filteredness and cofinality. The category \(K\) is filtered, and every \(p_v:K\to I_v\) is cofinal. More precisely, each comma category \((i\downarrow p_v)\) is nonempty and filtered.

Proof of filteredness. With no old levels and empty graphs \(H_v\), the refinement lemma produces a level. This proves nonemptiness. With two old levels and a discrete graph of their marked indices at each vertex, it gives a common target and two morphisms in \(K\).

Let \(u,u':k\rightrightarrows\ell\) be parallel morphisms in \(K\). For each \(v\), filteredness of \(I_v\) supplies

Apply the refinement lemma to the one old level \(\ell\), with \(H_v\) containing the marked \(i_v(\ell)\), the extra \(b_v\), and the arrow \(e_v\). Its resulting morphism \(q:\ell\to m\) has component \(q_v=a_ve_v\), where \(a_v:b_v\to i_v(m)\) is the extra cocone component. Therefore \(q_vu_v=q_vu'_v\) as arrows in \(I_v\), at every vertex. Since equality of \(K\)-morphisms is coordinatewise equality of these arrows, \(qu=qu'\). This proves the parallel-arrow axiom.

Proof of cofinality. Fix \(v\) and \(i\in I_v\). Use no old levels, but put the prescribed stage \(i\) into \(H_v\). The refinement lemma gives \(k\) and an arrow \(a:i\to p_v(k)\), hence an object \((k,a)\) of \((i\downarrow p_v)\).

For two comma objects \((k,a)\), \((\ell,b)\), put both levels into the refinement lemma. At vertex \(v\), use the finite graph with the two marked stages, an extra vertex labeled \(i\), and the given arrows

At the other vertices use discrete graphs of the marked stages. The result is \(m\), arrows \(q:k\to m\), \(r:\ell\to m\), and an arrow \(c:i\to p_v(m)\) satisfying

Thus \((m,c)\) is an actual common target in the comma category, with both comma equations. This proves connectedness as well as the common-target axiom. Mere filteredness of \(K\), without these equations on the given \(a,b\), would not prove cofinality.

For parallel comma arrows, use an equalizing arrow in \(K\), whose existence was just proved. Postcomposing the target's arrow from \(i\) gives the new comma object; the original comma equations survive. Thus the comma category is filtered. In particular it is nonempty and connected, which is the cofinality criterion. \(\square\)

2.5. The specified pointwise comparison

The maps \(\lambda_{v,p_v(k)}\) and equations (2a.2.3) define a cocone

Let its pointwise colimit comparison be

We verify this actual map, including its elements and arrows, rather than merely assert that the objects have some isomorphic presentation. At a vertex \(v\) and a test object \(T\), it is

For surjectivity, represent an element of \(D_v(T)\) by \(a:T\to X_v(i)\). A comma object \((k,s:i\to p_v(k))\) gives the representative \(X_v(s)a\) on the left with that image.

For injectivity, suppose \(a:T\to X_v(p_v(k))\) and \(b:T\to X_v(p_v(\ell))\) have the same image. The filtered equality criterion in \(I_v\) gives an index \(i\) and arrows

Use the refinement lemma on \(k,\ell\), with \(H_v\) containing these two arrows and their common target \(i\). It supplies a level \(m\), morphisms \(q:k\to m\), \(r:\ell\to m\), and a cocone component \(h:i\to p_v(m)\) with

Consequently \(X_v(p_v(q))a=X_v(p_v(r))b\). These are equal representatives at the one \(K\)-stage \(m\), so the left colimit classes coincide. This proves bijectivity directly and also verifies the cofinal comparison's effect on classes.

The maps (2a.5.2) commute with precomposition by every test-object arrow. For \(\alpha:v\to w\), the map of left-hand colimits takes \([k,a]\) to \([k,f^k_\alpha a]\); equation (2a.2.3) says exactly that (2a.5.2) intertwines this with \(D(\alpha)\). Thus \(\rho\) is a natural isomorphism of \(J\)-diagrams. Its inverse is natural because it is the inverse of these specified bijections.

Finite acyclic strictification theorem. The chosen diagram \(D\), with its specified arbitrary small filtered vertex presentations, is the pointwise formal colimit of the actual diagrams \(Z_k:J\to\mathsf C\) on the one small filtered category \(K\). Its vertex projections are cofinal, and all identity, composition, and transition equations hold as actual coefficient maps.

A finite directed acyclic graph has only finitely many paths: a path cannot repeat a vertex, so its length is bounded by the number of vertices minus one. Its free category is therefore finite and has an increasing degree. If finitely many equations between parallel paths are prescribed and are true of the formal diagram, quotient this finite category by those relations. The degree remains increasing, the formal diagram descends, and the theorem applied to this quotient makes every prescribed equation true at every stage. This covers a commuting square, a composable pair with its chosen composite, simultaneous parallel maps, and parallel maps required to be equal. For the last case the two arrow labels map to one arrow in the quotient. The theorem imposes no inverse equations arising from a cycle; its hypothesis excludes such cycles.

2.6. Equivalence for identity endomorphisms

Theorem 2.1. If \(K\) is finite and

then \(\Phi\) is an equivalence. Distinct objects of \(K\) may be isomorphic.

Proof. Full faithfulness is Theorem 1.1. For essential surjectivity, first pass to a skeleton of \(K\). Restriction to a skeleton is an equivalence of diagram categories, for either \(\mathsf C\) or its ind-category. Indization preserves equivalences, since the induced functors of a pair of quasi-inverses still have those composite identifications. The comparison is compatible with these restriction functors. It is therefore enough to prove essential surjectivity for skeletal \(K\).

In that case, put \(a\le b\) when there is an arrow \(a\to b\). If arrows exist in both directions, their two composites are endomorphisms and hence identities by (2.1). They are inverse isomorphisms, so the skeletal objects coincide. Thus this is a partial order. The length of a longest chain ending at an object gives a degree that strictly increases on every nonidentity arrow. Finiteness guarantees these lengths are finite.

Now take a diagram \(D:K\to\operatorname{Ind}(\mathsf C)\). Choose small filtered presentations at its finitely many vertices. The construction in Sections 2.1–2.5 produces a common small filtered category \(R\), cofinal functors to those vertex indices, and a functor

Equivalently, this is a filtered diagram \(R\to\operatorname{Fun}(K,\mathsf C)\). Its formal colimit maps under \(\Phi\) to \(D\) through the canonical pointwise comparison proved in Section 2.5. This proves essential surjectivity. If \(K\) is empty, both sides of (1.1) are terminal categories, so the assertion holds there as well. \(\square\)

Condition (2.1) is used to obtain a degree function after passing to a skeleton. It is absent from the full-faithfulness theorem. These two assertions should be kept distinct. The condition is sufficient for essential surjectivity in an arbitrary coefficient category; Exercise 2 gives a setting where essential surjectivity also holds for a finite group category.

In particular, for the category with one arrow, write \(\mathsf{Arr}(\mathsf C)=\operatorname{Fun}(\{0\to1\},\mathsf C)\). We obtain

Objects, commuting squares, and their composition are all covered by this equivalence.

2.7. Transformations and composition

No functorial choice of individual representatives is required to define \(\Phi_J\) or its equivalence. Given selected strictifications of \(D\) and \(E\), full faithfulness gives a unique formal morphism between the selected objects mapping to a specified transformation \(D\to E\). These unique lifts respect identities and composition because \(\Phi_J\) is a functor and faithful. Likewise two selected strictifications of the same \(D\), with their specified comparison isomorphisms, have a unique isomorphism respecting those comparisons.

For an explicit simultaneous stage representation of a transformation, suppose \(J\) has a degree and apply the construction to \(J\times[1]\), where \([1]=\{0\to1\}\). The diagram whose two slices are \(D,E\) and whose vertical arrows are the transformation is a functor exactly by its naturality equations. The degree \(2d(v)+\epsilon\), \(\epsilon\in\{0,1\}\), increases on every nonidentity arrow. Strictification gives natural stage maps between the two stage diagrams, with their squares commuting. For two composable transformations use \(J\times[2]\), including the arrow \(0\to2\) assigned their composite, and degree \(3d(v)+\epsilon\). Its stage composition equation identifies the composite of the two stage maps with the stage map for that composite. For a shape satisfying (2.1), first perform this on its skeleton and then transport across restriction equivalences. This supplies actual finite-stage representatives as well as the intrinsic functorial map statement.

Applying any functor \(H:\mathsf C\to\mathsf E\) termwise also preserves the construction's equations. Every level and refinement on \(K\) maps to the level and refinement with arrow maps \(H(f^k_\alpha)\) and the same original index arrows. The result presents \(\operatorname{Ind}(H)D\), because that extension preserves filtered colimits. The comparison (2a.5.1) commutes with \(\operatorname{Ind}(H)\) under the specified constant and colimit identifications. No faithfulness or finite-limit preservation by \(H\) is required.

2.8. Reverse the construction for pro-objects

Define \(\operatorname{Pro}_{\mathcal U}(\mathsf C)=\operatorname{Ind}_{\mathcal U}(\mathsf C^{\mathrm{op}})^{\mathrm{op}}\). A pro-presentation will here be a functor \(X_v:I_v\to\mathsf C\) on a small cofiltered category, with projections

Suppose \(J\) is finite with an increasing degree and \(D:J\to\operatorname{Pro}(\mathsf C)\). Pass to the opposite diagram

with filtered vertex indices \(I_v^{\mathrm{op}}\). Put \(M=\max(\{0\}\cup d(\operatorname{Ob}J))\). Then \(d_{\mathrm{op}}(v)=M-d(v)\) increases on every nonidentity arrow of \(J^{\mathrm{op}}\); for empty \(J\) it is the empty degree function. Apply Sections 2.2–2.5, obtaining a filtered category \(K^+\); put \(K=(K^+)^{\mathrm{op}}\). It is small and cofiltered. The opposite projection functors

are coinitial: each comma category \((q_v\downarrow i)\) is nonempty and cofiltered, being the opposite of the corresponding cofinality comma category in Section 2.4.

After reversing the coefficient arrows, every \(k\in K\) gives actual maps

satisfying the same identity and composition equations and the pro representation equation

For \(a:k\to\ell\) in the cofiltered \(K\), the coefficient transitions \(X_v(q_v(a))\) satisfy

Thus they give a cofiltered diagram \(K\to\operatorname{Fun}(J,\mathsf C)\). The pointwise comparison

is an isomorphism by the exact dual of (2a.5.1). In the covariant-functor model, a vertex of the right side is represented by

Its comparison with the original covariant functor is precisely the filtered pointwise isomorphism of Section 2.5 for \(\mathsf C^{\mathrm{op}}\). The fully faithful target for pro-objects is the opposite of the covariant-functor category; (2a.9.3) is not a pointwise limit of the sets in (2a.9.4).

For every finite \(J\), the canonical comparison

is fully faithful. Indeed \((\operatorname{Fun}(J,\mathsf C))^{\mathrm{op}}\) identifies with \(\operatorname{Fun}(J^{\mathrm{op}},\mathsf C^{\mathrm{op}})\). Apply Theorem 1.1 to that finite shape and coefficient category and take opposites; this identifies the comparison, its maps, identities, and composition. Under the identity-endomorphism hypothesis of Theorem 2.1, then (2a.9.5) is an equivalence by the same operation on Theorem 2.1 and Section 2.7. The comma-category result in Section 3 likewise dualizes by reversing the comma arrow and exchanging its two sides. All size conditions are unchanged under opposites.

3. A map between two different functor images

Let \(F:\mathsf C_1\to\mathsf C_0\) and \(G:\mathsf C_2\to\mathsf C_0\) be arbitrary functors. Their comma category \((F\downarrow G)\) has objects

A morphism to \((X',Y',u')\) is a pair \((a:X\to X',b:Y\to Y')\) satisfying \(G(b)u=u'F(a)\). The comma category records a chosen map, not merely the existence of one.

Applying the constant embeddings gives a fully faithful functor from this category to

It is fully faithful because the component Hom sets and the equation between them are identified by the three constant embeddings.

Theorem 3.1. The canonical comparison is an equivalence:

No full-faithfulness or colimit assumption on \(F\) and \(G\) is needed.

Proof. Call the target \(\mathsf M\). It has filtered colimits computed on the two components. Indeed, \(\operatorname{Ind}(F)\) and \(\operatorname{Ind}(G)\) preserve filtered colimits, so the maps \(u_i\) induce a map between the corresponding colimit components. A pair of maps out of those components is a comma morphism exactly when its equation holds after precomposition with each stage. This proves the colimit universal property in \(\mathsf M\).

Fix an original comma object \(Z=(X,Y,u)\), regarded as a constant object of \(\mathsf M\). For a filtered diagram \(Z_i=(A_i,B_i,u_i)\), its Hom set into each stage is the fiber product

where

The first map to \(H_i\) sends \(a\) to \(u_i\operatorname{Ind}(F)(a)\); the second sends \(b\) to \(\operatorname{Ind}(G)(b)\iota u\). Their equality is the comma equation. Each of the three constant source objects is finitely presented in its ind-category. In the last Hom set we also use filtered-colimit preservation by \(\operatorname{Ind}(G)\). Taking a filtered colimit in (3.2) therefore gives the corresponding fiber product for \(\operatorname{colim}_i Z_i\), because a filtered colimit commutes with a fiber product of sets. Thus \(Z\) is finitely presented in \(\mathsf M\). The compact-presentation theorem makes the comparison in (3.1) fully faithful.

For essential surjectivity, write the two components of an arbitrary \(Z=(A,B,u)\) as \(A=\langle X_i\rangle\) and \(B=\langle Y_j\rangle\). Then \(u\) is a formal map

The one-arrow construction in Sections 2.1–2.5, applied to these specified \(\mathsf C_0\)-valued presentations, provides a common small filtered category \(R\), cofinal functors \(p:R\to I\), \(q:R\to J\), and maps

that represent \(u\) and commute with transitions. Hence \(r\mapsto(X_{p(r)},Y_{q(r)},v_r)\) is a filtered diagram in \((F\downarrow G)\). Its image under the comparison is \(Z\), up to the canonical cofinal identifications. This proves essential surjectivity. \(\square\)

As a useful special case, take \(\mathsf C_0\) to be the terminal category and take its unique functors from two categories. Its ind-category is also terminal, by the category-of-elements criterion. The comma categories are then products. Thus

4. Which pieces are allowed in a presentation?

Let \(\mathsf J\) be a full subcategory of \(\mathsf C\). Its ind-category embeds fully faithfully in \(\operatorname{Ind}(\mathsf C)\). The following test detects the objects in its image without choosing a presentation first.

Theorem 4.1. An ind-object \(A\) belongs to that essential image if and only if every map \(\iota X\to A\), for every \(X\in\mathsf C\), factors as

No essential-smallness hypothesis on \(\mathsf J\) is required.

Proof. If \(A=\langle Y_i\rangle\) with every \(Y_i\in\mathsf J\), the ind-object Hom formula says that a map from a constant \(X\) is represented by \(X\to Y_i\) at some stage. This is (4.1).

Conversely, let \(\mathsf E_A\) be the category of elements of \(A\). It is filtered and has a small full cofinal subcategory \(\mathsf D\), by the recognition theorem. Let \(\mathsf E_{\mathsf J}\) be the full subcategory of \(\mathsf E_A\) on the objects whose first component belongs to \(\mathsf J\). A map \(\iota X\to A\) is an element \(x\in A(X)\). Its factorization (4.1) is exactly an arrow from \((X,x)\) to an object of \(\mathsf E_{\mathsf J}\). Thus every object of \(\mathsf E_A\) maps to this full subcategory. It is filtered: move any common-target or equalizer witness in \(\mathsf E_A\) further into it, as in the recognition theorem.

For each \(d\in\mathsf D\), choose one arrow \(d\to e_d\) with \(e_d\in\mathsf E_{\mathsf J}\). The full subcategory on the small family \((e_d)\) is small by local smallness. Every object of \(\mathsf E_A\) maps to some \(d\), then to \(e_d\). This family is therefore cofinal in \(\mathsf E_A\), and also in \(\mathsf E_{\mathsf J}\). The recognition theorem now presents \(A\) as the formal colimit of the representables corresponding to these \(\mathsf J\)-objects. The same small diagram gives an object of \(\operatorname{Ind}(\mathsf J)\), whose image is \(A\). Only small many choices were made. \(\square\)

All test objects in \(\mathsf C\) occur in the criterion. Restricting the tests to \(\mathsf J\) can miss the obstruction, as the final exercise shows.

5. Exercises with solutions

Exercise 1 (introductory: a commuting square). Over a field \(k\), let \(f:k\to k^2\) send \(x\) to \((x,0)\). Describe every endomorphism of \(f\) in the arrow category. Compute the rule for composing two such endomorphisms.

Solution. An endomorphism is a scalar map \(a:k\to k\) and a matrix \(B:k^2\to k^2\) with \(Bf=fa\). This forces the first column of \(B\) to be \((a,0)\), so

Conversely, every such matrix satisfies the equation. If the second endomorphism has parameters \((a',c',d')\), its composite after the first has scalar \(a'a\) and matrix \(B'B\), with parameters

Thus the endomorphism algebra is the algebra of upper triangular two-by-two matrices, with the scalar component already determined by the upper-left entry.

Exercise 2 (intermediate: a shape with automorphisms). Let \(G\) be the cyclic group of order three, viewed as a one-object category. Prove that the comparison from the ind-category of finite \(G\)-sets to all \(G\)-sets is an equivalence. Explain why this does not follow merely from Theorem 2.1.

Solution. The category \(G\) is finite, so Theorem 1.1 gives full faithfulness after identifying \(\operatorname{Ind}(\mathsf{FinSet})\) with sets, as in the preceding lesson. For a \(G\)-set \(S\), take its finite invariant subsets. They form a filtered set under inclusion: the union of two is another such subset. Every point is in its orbit, which has at most three elements. Thus their union is \(S\). The action maps preserve each stage, so this union is a filtered diagram of finite \(G\)-sets whose image is the given action. This proves essential surjectivity. Theorem 2.1 requires only identity endomorphisms, whereas \(G\) has three endomorphisms at its object. Essential surjectivity here follows from finite orbits, an additional property of this coefficient category and action.

Exercise 3 (advanced: two maps as one comma object). Let \(\mathsf V_f\) be the category of finite-dimensional vector spaces over \(k\). Use \(F(V)=V\oplus V\) and \(G(W)=W\) to identify \((F\downarrow G)\) with pairs of linear maps \(f,g\colon V\rightrightarrows W\). Show directly that any such pair on arbitrary vector spaces is a filtered colimit of finite-dimensional pairs.

Solution. A map \(V\oplus V\to W\) is determined by its two restrictions, giving \(f,g\). A comma morphism \((a,b)\) satisfies the two equations \(bf=f'a\), \(bg=g'a\), exactly the morphism equations for a pair of maps. For arbitrary \(V,W\), index by pairs of finite-dimensional subspaces \((V_0,W_0)\) with \(f(V_0)+g(V_0)\subset W_0\). They form a nonempty small poset, including \((0,0)\), ordered by inclusion. Sums of two pairs give an upper bound. Every vector in \(V\) belongs to a one-dimensional choice of \(V_0\), whose two images span a finite-dimensional space that can be included in \(W_0\). Every vector of \(W\) is included by enlarging \(W_0\). Thus the component colimits are \(V,W\), and their maps are the original \(f,g\). This is also the essential-surjectivity construction predicted by Theorem 3.1 and the equivalence \(\operatorname{Ind}(\mathsf V_f)\simeq\mathsf{Vect}(k)\).

Exercise 4 (advanced: test objects cannot be restricted arbitrarily). In sets, let \(\mathsf J\) be the full subcategory of sets of cardinality at most one, and let \(A=\iota\{0,1\}\). Show that every map \(\iota X\to A\) with \(X\in\mathsf J\) factors through a \(\mathsf J\)-object. Nevertheless, prove that \(A\) is not in the image of \(\operatorname{Ind}(\mathsf J)\).

Solution. For a test \(X\in\mathsf J\), the map already factors through \(\iota X\), using its identity, so the restricted test always passes. With the test \(X=\{0,1\}\), however, the identity of the two-element set cannot factor through a set with at most one element. Theorem 4.1 therefore excludes \(A\). One can also verify the exclusion by realization: in a filtered colimit of sets of cardinality at most one, any two elements have representatives in stages that map to a common stage. That stage has at most one element, so the two colimit elements coincide. The realized colimit has cardinality at most one, whereas \(A\) realizes to a two-element set. The unrestricted criterion detects exactly this missing test.

References

• The Stacks Project, Categories, Section 4.19, for comparison with the complete owned filtered-set proof in Filtered stages and finite limits. No source text is copied.
• The complete independently written strictification construction is in Section 2 of this lesson. It proves the ind and pro statements for arbitrary small filtered and cofiltered presentations, including all finite acyclic relations and canonical comparisons.
• Pierre Schapira, Homological Algebra, Section 2.7, for ind-object foundations.
• Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Section 2.7, for ind-objects.