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 topological and lax-topological conditions in this tradition quantify over large families. We therefore work in a set theory with classes (e.g. NBG). An index set $J$ may be a proper class, and “family $(A_j)_{j\in J}$” means class-indexed. Readers who prefer Grothendieck universes may uniformly replace “class” by “$\mathcal U$-small” for a fixed universe $\mathcal U$.

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 (Creation of (co)limits). Topologicity is a creation statement: whenever a limit (resp. colimit) exists in $\mathcal{X}$ and is presented by a cone (resp. cocone) built from base maps, the initial (resp. final) lifting produces the corresponding limit (resp. colimit) in $\mathcal{A}$ lying over it. Thus topological functors create all small limits and colimits that exist in the base (and, in the present “large family” setting, create the corresponding large ones as well).

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 (quotient) topologies.

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 (and must) take $q=1_Y$ and $P(B)=Y$, recovering an on-the-nose $P$-final lifting in the sense of Definition 2.3.

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 $\mathcal I$.

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 $P$.

A dual “lax duality” theorem (Bremen 1976) compares lax-final liftings of sinks with a rigid lax-initial notion for sources; its standard corollary is the following.

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. Theorem 5.8 is due to Tholen (Tho79); it explains why the lax comparison arrow $q$ in Proposition 5.2 becomes strict under the additional fibration hypothesis.

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 of morphisms in $\mathcal E$ exist and remain in $\mathcal E$.

Remark 5.11. Theorem 5.10 is proved in Tho79, BorTho90. A category $\mathcal{A}$ satisfying (ii)–(iii) for a given $\mathcal E$ is sometimes called $\mathcal E$-cocomplete.

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 canonical 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 lax-topological functor can be expressed as a composite of topalg functors, and there exist lax-topological functors that are not topalg.

Remark 6.6. See BorTho79, Wis79 for the original analysis of topalg functors and their relationship with semi-topological/solid functors.

Characterizations via completion

Completion constructions are a core organizing principle in this theory. From now on assume that $P$ is amnestic.

Very roughly, a completion of $(\mathcal{A},P)$ is a fully faithful functor $\mathcal{A}\hookrightarrow \widehat{\mathcal{A}}$ over $\mathcal{X}$ with a reflector $\widehat{\mathcal{A}}\to\mathcal{A}$ such that the completed structure functor $\widehat{P}\colon \widehat{\mathcal{A}}\to\mathcal{X}$ enjoys stronger lifting properties than $P$ itself.

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 completion.

Remark 7.2. Part (a) is treated by Porst and (in enriched form) by Garner; see Por78, Gar14. Part (b) is developed by Herrlich–Strecker in terms of semi-universal maps and universal initial completions; see HerStr79.

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 $y\colon \mathcal{A}\to \widehat{\mathcal{A}}$ admits a left adjoint.

Remark 8.2. Theorem 8.1 is proved in Gar14. 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.

Acknowledgments

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

References

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