Topological, Lax-Topological (Semi-Topological/Solid), and Topologically Algebraic Functors

Abstract. We record three closely related notions of “structure functor” $P\colon \mathcal{A}\to\mathcal{X}$ that play a central role in categorical topology: topological functors, lax-topological (a.k.a. semi-topological or solid) functors, and topologically algebraic functors. The common language is that of structured sources and sinks—(possibly large) families of maps in the base—and corresponding lifting properties formulated as universal properties of cones and cocones. We collect characteristic equivalences and stability properties (closure under composition and pullback, reflective restrictions, and source-factorization axioms), two completion characterizations (Dedekind–MacNeille and universal completion), and the quantaloid-enriched reformulation in which “topological = total” in the sense of Street–Walters.

Keywords. categorical topology, concrete category, topological functor, solid functor, semi-topological functor, topologically algebraic functor, total category, quantaloid-enriched category, MacNeille completion

It has been a year since the Brümmer 90th anniversary conference at UCT, and I’ve had a lot of time to digest Walter Tholen’s presentation. The following series of notes are based on a “wishlist” of sorts that Tholen gave and are composed of my putting some serious thought into the questions presented by Tholen. I have tried to represent the ideas as faithfully as possible to the presentation. All faults are my own and please let me know if you have any corrections or find any issues.

A recurring thread in categorical topology runs from Bourbaki’s initial-structure intuition (Bou57), through attention to large families of maps (in the tradition of Čech–Hušek), into Grothendieck’s (co)fibration viewpoint, and finally to the enriched characterization of topological functors as total categories. Two weakenings of topologicity appear naturally: lax-topological functors (also called semi-topological or solid) and topologically algebraic functors.

The aim of this note is to give statements of the core definitions and equivalences, with careful attention to what is quantified (notably, large families) and to what is universal at the level of sources and sinks rather than individual arrows.

Ambient conventions

Size

The lifting conditions below quantify over possibly large families, including the empty family. In the classical class-based formulation, the index class $J$ may be proper. Collections of such families or of presheaves on a large category can exceed the size of a class in that same framework; when forming these categories we therefore work at a larger ambient size. Equivalently, one may use universes, keeping hom-sets small while allowing object collections and indexing families to be large, and enlarging the ambient universe for presheaf constructions. Restricting every indexing family to be small in the hom-set universe would change the lifting conditions; see Gar14, §2.

Concrete categories and structure functors

Fix a locally small category $\mathcal{X}$ and a functor

In the classical categorical-topology literature $P$ is treated as a structure functor. A standing convention in this note is that $P$ is faithful. (Under this assumption, local smallness of $\mathcal{X}$ implies local smallness of $\mathcal{A}$.)

When strict uniqueness statements are desired one often assumes that $P$ is amnestic:

Definition 1.1 (Amnestic). A functor $P \colon \mathcal{A} \to \mathcal{X}$ is called amnestic if every isomorphism in $\mathcal{A}$ lying over an identity in $\mathcal{X}$ is itself an identity.

Equivalently, whenever $f \colon A \to B$ is an isomorphism in $\mathcal{A}$ with $P(f) = 1_{P(A)}$, then $f = 1_A$ (and hence $A = B$).

Remark 1.2 (On dropping faithfulness). Faithfulness is not essential for the existence of the theory, but it is the cleanest setting for the “cone/cocone lifting” formulations used below. Removing faithfulness typically requires additional hypotheses or separate reduction theorems; we do not pursue those variants here.

Structured sources and sinks; initial and final liftings

The basic objects of quantification are families of maps in the base. The correct mental model is a diagonal filler for an entire cone/cocone, rather than componentwise existence of individual lifts.

$P$-structured sources

A $P$-structured source consists of an object $X\in\mathcal{X}$, a family $(A_j)_{j\in J}$ of objects of $\mathcal{A}$, and a family of arrows

in $\mathcal{X}$.

A lifting of this structured source is an object $A\in \mathcal{A}$ with $P(A)=X$ together with arrows

in $\mathcal{A}$ such that $P(\bar f_j)=f_j$ for all $j$.

Definition 2.1 ($P$-initial lifting). A lifting $(A,(\bar f_j)_{j\in J})$ of the structured source $(X,(A_j)_{j\in J},(f_j)_{j\in J})$ is $P$-initial if it satisfies the following diagonal property.

For every object $B\in\mathcal{A}$, every arrow $h\colon P(B)\to X$ in $\mathcal{X}$, and every family $(g_j\colon B\to A_j)_{j\in J}$ in $\mathcal{A}$ with $P(g_j)=f_j\circ h$ for all $j$, there exists a unique arrow $g\colon B\to A$ in $\mathcal{A}$ such that

Remark 2.2 (Bijection form). Assume $P$ is faithful. A lifting $(A,(\bar f_j)_{j\in J})$ of a $P$-structured source $(X,(A_j)_{j\in J},(f_j)_{j\in J})$ is $P$-initial if and only if for every object $B\in\mathcal{A}$ and every arrow $h\colon P(B)\to X$ in $\mathcal{X}$ the assignment

is a bijection between arrows $g\colon B\to A$ with $P(g)=h$ and families $(g_j)_{j\in J}$ with $P(g_j)=f_j\circ h$.

$P$-structured sinks

A $P$-structured sink consists of a family $(A_j)_{j\in J}$ of objects of $\mathcal{A}$, an object $X\in\mathcal{X}$, and a family of arrows

in $\mathcal{X}$.

A lifting of this structured sink is an object $A\in\mathcal{A}$ with $P(A)=X$ together with arrows

in $\mathcal{A}$ such that $P(\bar u_j)=u_j$ for all $j$.

Definition 2.3 ($P$-final lifting). A lifting $(A,(\bar u_j)_{j\in J})$ of the structured sink $((A_j)_{j\in J},X,(u_j)_{j\in J})$ is $P$-final if it satisfies the following diagonal property.

For every object $B\in\mathcal{A}$, every arrow $h\colon X\to P(B)$ in $\mathcal{X}$, and every family $(g_j\colon A_j\to B)_{j\in J}$ in $\mathcal{A}$ with $P(g_j)=h\circ u_j$ for all $j$, there exists a unique arrow $g\colon A\to B$ in $\mathcal{A}$ such that

Remark 2.4 (Initial-vs-terminal language). With the standard convention that a morphism of sinks $(A_j\to A)\to (A_j\to A')$ is a map of codomains $A\to A'$ commuting with all legs, a universal cocone is initial in the corresponding sink category. Accordingly, a $P$-final lifting is initial among liftings of the given structured sink (while a $P$-initial lifting is terminal among liftings of the structured source).

Topological functors

Definition 3.1 (Topological functor). A faithful functor $P\colon \mathcal{A}\to\mathcal{X}$ is topological (or initially/finally complete) if every $P$-structured sink admits a $P$-final lifting for all (possibly class-indexed) families. Equivalently, every $P$-structured source admits a $P$-initial lifting.

Remark 3.2 (Self-duality). The equivalence of “final liftings of all sinks” and “initial liftings of all sources” is a nontrivial self-duality theorem in the classical theory; see Bru76, Her74, Gar14 for proofs and variants. In particular, topologicity is stable under passing to opposite categories.

Remark 3.3 (Lifting of (co)limits). Let $D\colon\mathcal I\to\mathcal A$ be a diagram. An initial lifting of a limit cone of $PD$ is a limit cone of $D$: faithfulness gives the cone equations, and the initial lifting property lifts the unique mediating base map. Dually, a final lifting of a colimit cocone of $PD$ is a colimit cocone of $D$. This applies to small diagrams and to the large diagrams admitted by our size conventions whenever the corresponding base (co)limit exists.

These universal liftings are unique up to a unique isomorphism over the identity, and literally unique if $P$ is amnestic. This should not be confused with the stronger convention for creation requiring every lift of a base limit cone to be limiting. Even $\mathbf{Top}\to\mathbf{Set}$ fails that condition: the identity function from a discrete two-point space to the indiscrete two-point space lifts the identity cone of the one-object base diagram, but is not a limit cone in $\mathbf{Top}$.

Example 3.4. The forgetful functor $\mathbf{Top}\to\mathbf{Set}$ is topological: $P$-initial liftings are initial topologies and $P$-final liftings are final topologies. A quotient topology is the special case of a single surjective map.

External and universal characterizations in $\mathbf{CAT}$

Topologicity admits robust “external” formulations that do not mention topological spaces. We record a representative diagonal characterization. Throughout this section we assume that $P$ is faithful and amnestic.

Theorem 4.1 (External/diagonal characterization). Let $P\colon \mathcal{A}\to\mathcal{X}$ be faithful and amnestic. Then $P$ is topological if and only if it is injective with respect to full functors over $\mathcal{X}$, in the following sense: for every commutative square in $\mathbf{CAT}$

with $Q$ full, there exists a (not necessarily unique) diagonal filler $H\colon \mathcal{C}\to\mathcal{A}$ such that $HQ=F$ and $PH=G$.

Remark 4.2. Even for familiar examples (e.g. $\mathbf{Top}\to\mathbf{Set}$) the diagonal filler in Theorem 4.1 is generally not unique. Nevertheless, one often has distinguished extremal fillers (“finest” and “coarsest”), mirroring the existence of finest/coarsest induced structures (e.g. discrete/indiscrete).

Theorem 4.1 goes back to work of Brümmer–Hoffmann and Wolff; see BruHof76, Wol77. Universal characterizations in $\mathbf{CAT}$ (e.g. in terms of canonical universal extensions) appear in work of Wischnewsky and Tholen; see ThoWis79.

Lax-topological functors (semi-topological/solid)

The lax-topological condition relaxes the strict requirement (in Definition 2.3) that a final lifting of a sink have a vertex $A$ lying over the prescribed base object $X$, i.e. with $P(A)=X$. Instead one allows a comparison morphism in the base. The clean formulation is via left adjoints on categories of sinks.

Sink categories and the adjoint-on-sinks criterion

Fix a family $(A_j)_{j\in J}$ of objects of $\mathcal{A}$. Write $(A_j)\!\downarrow\!\mathcal{A}$ for the category of sinks under $(A_j)$: objects are families $(g_j\colon A_j\to A)_{j\in J}$ with codomain $A\in\mathcal{A}$, and morphisms are maps of codomains commuting with all legs. Similarly, $(P A_j)\!\downarrow\!\mathcal{X}$ denotes the category of sinks under $(P A_j)$ in $\mathcal{X}$.

Applying $P$ componentwise induces a functor

Definition 5.1 (Lax-topological / solid). A functor $P\colon \mathcal{A}\to\mathcal{X}$ is lax-topological (or semi-topological, or solid) if for every (possibly class-indexed) family $(A_j)_{j\in J}$ in $\mathcal{A}$, the induced functor $P_J$ has a left adjoint.

Unwinding the adjunction gives the concrete “lax final lifting” data.

Proposition 5.2 (Lax $P$-final liftings). Assume $P$ is lax-topological. Let $(u_j\colon P(A_j)\to Y)_{j\in J}$ be a structured sink in $\mathcal{X}$ with fixed domain family $(A_j)_{j\in J}$. Writing $L_J\dashv P_J$ for the left adjoint, the unit at $u=(u_j)$ exhibits a morphism in $(P A_j)\!\downarrow\!\mathcal{X}$

hence an arrow $q\colon Y\to P(B)$ and morphisms $\bar u_j\colon A_j\to B$ in $\mathcal{A}$ such that $P(\bar u_j)=q\circ u_j$ for all $j$. Moreover, this data satisfies the expected universal property: given any $r\colon Y\to P(C)$ and any sink $(g_j\colon A_j\to C)$ with $P(g_j)=r\circ u_j$, there is a unique $t\colon B\to C$ in $\mathcal{A}$ such that $t\bar u_j=g_j$ for all $j$ and $P(t)\,q=r$.

Remark 5.3 (Topological = lax + “strictness”). In Proposition 5.2 the comparison arrow $q$ need not be an identity. In the topological case, one may choose $q=1_Y$ and $P(B)=Y$, recovering an on-the-nose $P$-final lifting in the sense of Definition 2.3. An arbitrary choice of the left adjoint need not make $q$ an identity; it makes $q$ an isomorphism, by uniqueness of universal arrows. Conversely, if every structured sink has a lax-final lifting with identity comparison, then $P$ is topological.

Basic consequences

Proposition 5.4 (Empty family gives a left adjoint). If $P$ is lax-topological, then $P$ has a left adjoint.

Proof. For the empty family $J=\varnothing$, the sink categories satisfy $(\varnothing)\!\downarrow\!\mathcal{A}\simeq \mathcal{A}$ and $(\varnothing)\!\downarrow\!\mathcal{X}\simeq \mathcal{X}$, and $P_{\varnothing}$ identifies with $P$. A left adjoint to $P_{\varnothing}$ is therefore a left adjoint to $P$.

Proposition 5.5 (Existence of colimits). If $P$ is lax-topological and $\mathcal{X}$ has colimits of a given (small) diagram shape $\mathcal I$, then $\mathcal{A}$ has colimits of shape $\mathcal I$.

Proof. For $D\colon\mathcal I\to\mathcal A$, take a colimit cocone $(u_i\colon PD_i\to Y)$ in $\mathcal X$ and a lax-final lifting $(q\colon Y\to PB,(\bar u_i))$. Faithfulness makes $(\bar u_i)$ a cocone. Any cocone $(g_i\colon D_i\to C)$ induces a unique $r\colon Y\to PC$ with $r u_i=P g_i$, and Proposition 5.2 gives its unique factorization through $B$. Uniqueness holds among all cocone factorizations because the $u_i$ are jointly epic.

Remark 5.6. Proposition 5.5 is classically due to Hoffmann and Tholen; see BorTho90, §1.1. The point is existence of colimits in $\mathcal{A}$, not their creation by $P$.

The corresponding limit-existence result follows from the source formulation of solidity; see BorTho90, §1.7. This is not an assertion that the opposite of every lax-topological functor is lax-topological.

Corollary 5.7 (Existence of limits). If $P$ is lax-topological and $\mathcal{X}$ has limits of a given (small) diagram shape $\mathcal I$, then $\mathcal{A}$ has limits of shape $\mathcal I$.

Relationship with fibrations and reflective restrictions

The Grothendieck (co)fibration viewpoint re-enters via a sharp characterization.

Theorem 5.8 (Topological iff lax-topological + fibration). A functor $P\colon \mathcal{A}\to\mathcal{X}$ is topological if and only if it is lax-topological and a Grothendieck fibration.

Remark 5.9. To see the nontrivial implication, take a cartesian lifting $k\colon A\to B$ of the comparison $q\colon Y\to PB$ in Proposition 5.2. Each $\bar u_j$ factors through $k$, giving a sink into $A$ over the original $u_j$. Lax-final universality gives $t\colon B\to A$ with $P(t)q=1_Y$ and $kt=1_B$; faithfulness then gives $tk=1_A$. Thus $k$ is invertible and the sink into $A$ is a final lifting over $Y$. Conversely, a topological functor is lax-topological, and its singleton initial liftings are cartesian. See also Tho79.

A second fundamental characterization explains the ubiquity of lax-topological functors.

Theorem 5.10 (Reflective restriction and $\mathcal E$-cocompleteness). For a functor $P\colon \mathcal{A}\to\mathcal{X}$ the following are equivalent:

• (a) $P$ is lax-topological.
• (b) $P$ is (equivalent to) the restriction of a topological functor to a full reflective subcategory.
• (c) $P$ has a left adjoint and there exists a class $\mathcal E$ of morphisms in $\mathcal{A}$ such that:
  • (i) all counits of the adjunction lie in $\mathcal E$;
  • (ii) pushouts of morphisms in $\mathcal E$ along arbitrary morphisms exist and remain in $\mathcal E$;
  • (iii) cointersections (wide pushouts) of arbitrary families $(e_i\colon A\to B_i)$ in $\mathcal E$ exist, and the canonical arrow $A\to C$ to the wide-pushout vertex belongs to $\mathcal E$ (for the empty family, this is $1_A$).

Remark 5.11. For (a)–(b), see HerStr79, Theorem 2.8; for (a)–(c), see BorTho90, Theorem 1.2. A category $\mathcal{A}$ satisfying (ii)–(iii) for a given $\mathcal E$ is called $\mathcal E$-cocomplete. The reflector in (b) is an ordinary left adjoint; it need not commute with the functors to $\mathcal X$.

Standard examples

Example 5.12. The forgetful functor $\mathbf{Haus}\to \mathbf{Set}$ is lax-topological (indeed a reflective restriction of $\mathbf{Top}\to\mathbf{Set}$) but not topological.

Example 5.13. The forgetful functor $\mathbf{Grp}\to\mathbf{Set}$ is lax-topological. More generally, monadic functors over $\mathbf{Set}$ are lax-topological; over an arbitrary base $\mathcal{X}$ this may fail.

Topologically algebraic functors (topalg)

Topologically algebraic functors are defined by a factorization property for structured sources. The factorization is of a family/cone, not a single morphism.

$P$-epimorphisms

Definition 6.1 ($P$-epimorphism). An arrow $q\colon X\to P(A)$ in $\mathcal{X}$ is $P$-epic (or a $P$-epimorphism) if for all arrows $s,t\colon A\to C$ in $\mathcal{A}$,

Definition of topalg

Definition 6.2 (Topologically algebraic). A functor $P\colon \mathcal{A}\to\mathcal{X}$ is topologically algebraic (topalg, also written algtop) if every $P$-structured source $(f_j\colon X\to P(B_j))_{j\in J}$ factors as

where $q$ is $P$-epic and $(g_j\colon A\to B_j)_{j\in J}$ is a $P$-initial structured source (Definition 2.1) with domain object $P(A)$.

Remark 6.3. If $P$ is topological then it is topalg: one may take $q=1_X$ and use the $P$-initial lifting of the given source. Since $P$ is faithful, $1_X$ is $P$-epic.

Stability behaviour and relation to lax-topological functors

The three classes behave differently with respect to composition and pullback in $\mathbf{CAT}$.

Proposition 6.4 (Qualitative stability). In general:

• (a) Topological functors are closed under composition and stable under pullback.
• (b) Lax-topological functors are closed under composition but need not be pullback-stable.
• (c) Topalg functors are, in general, neither closed under composition nor pullback-stable.

Corollary 6.5 (Compositional hull). Every topalg functor is lax-topological, and the lax-topological functors form the smallest class containing the topalg functors and closed under composition. There exist lax-topological functors that are not topalg.

Remark 6.6. The compositional-hull statement is HerStr79, Theorem 2.8. For an explicit solid functor that is not topalg, and hence the failure of closure under composition, see HerNakStrTit80, Theorem 2.1 and Corollary 2.2. The failure of pullback stability already has a small example: the reflective inclusion $\{1\}\hookrightarrow(0<1)$ is topalg, but its pullback along $\{0\}\hookrightarrow(0<1)$ is $\varnothing\to\mathbf 1$, which has no left adjoint and is not lax-topological. See also BorTho79, Wis79.

Characterizations via completion

Completion constructions are a core organizing principle in this theory. In this section all structure functors are assumed faithful and amnestic.

An initial completion is a full embedding $E\colon\mathcal A\hookrightarrow\widehat{\mathcal A}$ over $\mathcal X$, where $\widehat P\colon\widehat{\mathcal A}\to\mathcal X$ is topological and $\widehat P E=P$. It is reflective if $E$ has an ordinary left adjoint $R$. This is an additional property, and $R$ need not be a functor over $\mathcal X$.

The Dedekind–MacNeille completion is characterized by $E$ being both initially and finally dense: every completed object admits a $\widehat P$-initial source into objects of $E\mathcal A$ and a $\widehat P$-final sink from such objects. The universal initial completion instead preserves initial sources and is universal for initiality-preserving functors over $\mathcal X$ into topological categories. Such functors extend uniquely along $E$, again preserving initial sources. These are distinct completion properties; neither definition includes reflectivity.

Theorem 7.1 (Completion characterizations). Assume $P$ is amnestic.

• (a) $P$ is lax-topological if and only if it admits a reflective Dedekind–MacNeille completion.
• (b) $P$ is topalg if and only if it admits a reflective universal initial completion.

Remark 7.2. For (a), see Hoffmann Hof78 and HerStr79, Theorem 2.8; for (b), see HerStr79, Theorem 2.7. Porst Por78, §3 characterizes MacNeille completions, while Garner Gar14, §7 identifies them with enriched MacNeille completions. The reflectivity in Theorem 7.1 concerns the embedding of $\mathcal A$ into its completion, not the separate reflection used to construct a MacNeille completion inside an enriched presheaf category.

Quantaloid enrichment: “topological = total”

Garner’s enriched reformulation explains why lifting axioms behave like completeness and why completion constructions look like enriched completions.

The quantaloid $\mathcal Q_{\mathcal{X}}$

Assume $\mathcal{X}$ is locally small. Define a quantaloid $\mathcal Q_{\mathcal{X}}$ as follows:

• objects are the objects of $\mathcal{X}$;
• for $X,Y\in\mathcal{X}$, the hom-object is the powerset
  $$\mathcal Q_{\mathcal{X}}(X,Y)=\mathcal{P}\big(\mathcal{X}(X,Y)\big),$$
  ordered by inclusion;
• composition is induced by composition in $\mathcal{X}$: for $U\subseteq \mathcal{X}(X,Y)$ and $V\subseteq \mathcal{X}(Y,Z)$,
  $$V\circ U=\{\,v\circ u \mid u\in U,\ v\in V\,\}\subseteq \mathcal{X}(X,Z);$$
• identities are singletons $\{1_X\}$, and joins are unions.

Faithful functors as $\mathcal Q_{\mathcal{X}}$-enriched categories

A concrete category $(\mathcal{A},P)$ over $\mathcal{X}$ with $P$ faithful corresponds to a $\mathcal Q_{\mathcal{X}}$-enriched category whose objects are those of $\mathcal{A}$, whose extent of $A$ is $P(A)$, and whose hom from $A$ to $B$ is the subset

viewed as an element of $\mathcal Q_{\mathcal{X}}(PA,PB)$. The enriched axioms amount precisely to closure under identities and composition:

Garner’s theorem

Theorem 8.1 (Garner: topological = total). For a faithful functor $P\colon \mathcal{A}\to\mathcal{X}$ with $\mathcal{X}$ locally small, the following are equivalent:

• (a) $P$ is topological (Definition 3.1).
• (b) The associated $\mathcal Q_{\mathcal{X}}$-enriched category $\mathcal{A}$ is total (totally cocomplete), i.e. its enriched Yoneda embedding $y\colon \mathcal{A}\to \mathcal P_{\mathcal Q_{\mathcal X}}\mathcal A$ into its enriched presheaf category admits a left adjoint in $\mathcal Q_{\mathcal X}\text{-}\mathbf{CAT}$.

Remark 8.2. Theorem 8.1 is Gar14, Theorem 5.2. The adjoint here preserves extents; concretely, it is an adjoint over $\mathcal X$. This is stronger than merely having an ordinary adjoint, and the totality asserted here is enriched totality, not ordinary $\mathbf{Set}$-enriched totality of $\mathcal A$. The quantaloid-enriched viewpoint is further developed and compared with older fibration/cofibration decompositions in SheTho16, Stu05.

Further structural consequences

Lax-topological functors are strong enough to lift categorical completeness properties defined by existence of certain (possibly large) colimits. A prominent example is totality in the sense of Street–Walters (StrWal78): a (locally small) category $\mathcal{C}$ is total if the Yoneda embedding $\mathcal{C}\to[\mathcal{C}^{\mathrm{op}},\mathbf{Set}]$ admits a left adjoint.

Theorem 9.1 (Lifting of totality). If $P\colon \mathcal{A}\to\mathcal{X}$ is lax-topological and $\mathcal{X}$ is total, then $\mathcal{A}$ is total.

Remark 9.2. Theorem 9.1 and related lifting results (compactness, hypercompleteness, mono-completeness) appear in Tho80, BorTho90, Theorem 3.1.

Acknowledgments

These notes were initially assembled from W. Tholen’s “Brümmer ’90” lecture slides (Tho24) and the classical literature cited below.

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