nLab topological concrete category

Terminology warning. Here the term topological category is understood in a tradition motivated by examples of concrete categories in topology and does not refer to categories internal to or enriched in in Top (cf. topologically enriched category instead). The term topological groupoid (which is an internal groupoid in Top) is not taken by this tradition; indeed, the only groupoid that is a topological category over $Set$, in the sense used here, is trivial. On the other hand, there is use of the term ‘topological functor’, which we tend to avoid other than below.

Idea

A topological category is a concrete category with nice features matching the ability to form weak and strong topologies in Top.

Definition

The following definition relates to a functor $U\colon C \to D$ (such as the forgetful functor from $Top$ to Set), which one may think of as exhibiting $C$ as a bundle over $D$.

Sometimes, when $D =$ Set, the category $C$ satisfying the properties described below is called a topological construct Preuss 2002. Usually $C$ and $D$ will be large categories.

In this context one also says:

• spaces for the objects of $C$,

• algebras for the objects of $D$,

• maps for the morphism in $C$,

• homomorphisms for the morphisms in $D$.

This alludes to the typical example situation where $C$ is a category of spaces with some kind of topological structure while $D$ is, if not Set then some kind of algebraic category.

We now first state the default definition (Def. ) and then comment on some variants (Rem. and Rem. ).

Definition

Given a functor $U \colon C \longrightarrow D$, it exhibits its domain category $C$ as a topological category over $D$ if, given

1. any object $X \in D$ (an “algebra”)

2. any (possibly large) indexed set $\big\{S_i \in C\big\}_{i \in I}$ (of “spaces”)

3. homomorphisms$f_i \colon X \to U(S_i)$ (that is, a “$U$-structured source from $X$”),

then there exists an initial lift (think: “smallest topology rendering the $f_i$ continuous”), which is to say

1. a “space” $T \in C$ such that $U(T) = X$, and maps $m_i \colon T \to S_i$ such that $U(m_i) = f_i$,

2. given any space $T'$, homomorphism $g'\colon U(T') \to X$, and maps $m'_i\colon T' \to S_i$, if each composite $f_i \circ g$ equals $U(m'_i)$, then there exists a unique map $n\colon T' \to T$ such that $U(n) = g'$ and $m_i \circ n = m'_i$:

The conditions in Def. imply the following crucial further property (which is often included as an assumption in the definition, in which case the uniqueness of $n$ can be left out):

Proposition

If a functor $U\colon C \to D$ exhibits a topological category in the sense of Def. , then it is faithful and in this sense exhibits $C$ as a concrete category over $D$.

Proof

See Theorem 21.3 of Adámek, Herrlich & Strecker 1990.

Thus we may think of objects of a topological category $C$ as objects of $D$ equipped with extra structure. The idea is then that $T$ is $X$ equipped with the initial structure or weak structure determined by the requirement that the homomorphisms $f_i$ be structure-preserving maps.

The following says that the notion is “self-dual”:

Proposition

A functor $U\colon C \to D$ is topological, in the sense of Def. , if and only if the opposite functor $U^op\colon C^op \to D^op$ is so.

(This may be understood as a categorification of the theorem that any complete semilattice is a complete lattice.)

Thus, every topological category also has final (not usually called terminal) or strong structures, each determined by a family of homomorphisms $f_i\colon U(S_i) \to X$ (a $U$-structured sink to $X$).

Remark

Both of these results (faithfulness, Prop. , and self-duality, Prop. ) depend on the assumption in Def. that the family $\{S_i\}$ is allowed to be large.

Otherwise, there exists counterexamples: For instance, if $C$ is a large category with all (small) products, then the functor $C \to 1$ to the terminal category satisfies the lifting property in Def. for small families $\{S_i\}$. However, it need not satisfy the dual property (unless $C$ also has all small coproducts) nor need it be faithful.

It also follows that:

Proposition

If $U \colon C \longrightarrow D$ exhibits a topological category (Def. ), then $U$ is a Grothendieck fibration and an opfibration.

Remark

(amnestic version) Since initial lifts have a universal property, they are unique up to unique isomorphism. However, some authors (such as AHS90) ask that they be literally unique. This is tantamount to deciding that $U$ should be an amnestic functor. A drawback (from an nPOV) is that this condition violates the principle of equivalence, and arguably doesn’t add anything mathematically important.

Thus, although it occurs in the literature, here we will consider it purely optional. (It is possible that some results recorded here about topological categories will depend on this assumption, but only results not respecting the principle of equivalence could be affected.)

Remark

(weak version)
On the other hand, the default definition does already refer to equality of objects in the condition $U(T) = X$; thus as stated it already violates the principle of equivalence, just as the notion of Grothendieck fibration does. But (also as for Grothendieck fibrations) this other use of equality of objects is really more of a “typing judgment”, which can be made precise by working with displayed categories instead. (In the context of homotopy type theory, the amnestic condition is equivalent to “fiberwise univalence”.)

However, if we want to, we can also formulate a “fully isomorphism-invariant” version of the definition, corresponding to the weakened bicategorical notion of Street fibration. In this case, an initial lift consists of:

• a space $T$, an isomorphism $g\colon U(T) \to X$, and maps $m_i\colon T \to S_i$ such that $f_i \circ g = U(m_i)$ for all $i \in I$ and,

• given any space $T'$, homomorphism $g'\colon U(T') \to X$, and maps $m'_i\colon T' \to S_i$, if $f_i \circ g' = U(m'_i)$ for all $i \in I$, then there exists a unique map $n\colon T' \to T$ such that $g \circ U(n) = g'$ and $m_i \circ n = m'_i$:

Examples

• The name ‘topological category’ comes from these examples from point-set topology; these are all topological over Set:

  • the category Top of topological spaces,

  • the category of convergence spaces (or of pseudotopological or of pretopological spaces),

  • the category of uniform spaces or of Cauchy spaces,

  • lots more in this vein.

• In contrast, the category of locales is not topological over $Set$; not even the category of spatial locales (equivalent to the category of sober spaces) is topological, essentially because soberification of a topological space may not preserve the underlying set.

• Also, the category Diff of smooth manifolds is not topological but most categories of generalized smooth spaces are.

• Outside of topology, the category of measurable spaces is topological over $Set$.

• The category PreOrd of preordered sets (equivalent to the category of Alexandroff spaces) is topological over $Set$. Given a family of preordered sets $\{(S_i, \prec_i)\}$ and a family of $\{f_i \colon X \to S_i\}$, the induced preorder is $(\leq) \subseteq X^2 = \{(x,x') \mid \forall i \cdot f_i(x) \prec_i f_i(x') \}$.

• The category of topological groups is topological over Grp, the category of topological vector spaces is topological over $\mathbb{R}$-Vect, etc.

Further properties

• If $C$ is topological over $D$, then so is any full retract of $C$, as long as the functors involved live in $Cat/D$.

• In particular, a reflective or coreflective subcategory of $C$ is topological, as long as the reflectors or coreflectors become identity morphisms in $D$.

• The forgetful functor $U\colon C \to D$ is not only faithful but also (because every algebra must have an initial/indiscrete topology determined by the empty source) essentially surjective (in fact surjective on the nose for the non-weak definitions). Thus it is never full (except in the trivial case where $U$ is an equivalence, of course).

• If $D$ is complete or cocomplete, then so is $C$.

• If $D$ is total or cototal, then so is $C$; see solid functor.

• If $D$ is mono-complete or epi-cocomplete, then so is $C$.

• If $D$ is well-powered or co-well-powered, then so is $C$.

• If $D$ has a factorization structure for sinks $(E,M)$, then $C$ has one $(E',M')$, where $M'$ is the collection of morphisms in $C$ lying over $M$-morphisms in $D$, and $E'$ the collection of final sinks in $C$ lying over $E$-sinks in $D$. This generalizes the lifting of orthogonal factorization systems along Grothendieck fibrations.

• If $D$ is concrete, then so is $C$. More generally, if $D$ has a generator, then $C$ is concrete over $D$.

• In particular, if $D$ is Set, then $C$ is a concrete category that is complete, cocomplete, well powered, and well copowered and has a factorization structure for sinks.

Functors

• A functor $F\colon C\to C'$ between topological concrete categories $C/D$, $C'/D$ with the same base category $D$ preserves initial lifts iff it is right adjoint. It preserves final lifts iff it is left adjoint.

• More generally: If a functor $F\colon C\to C'$ between topological concrete categories $C/D$, $C'/D'$ with different base categories lying over a functor $F_0: D\to D'$. If $F$ is right (left) adjoint, then $F_0$ is right (left) adjoint and $F$ preserves initial (final) lifts. A partial converse holds: If $F_0$ is right (left) adjoint to $G_0$ and $F$ preserves initial (final) lifts, then there is functor $G$ lying over $G_0$ so that $F$ is right (left) adjoint to $G_0$.

Special cases

• If $X$ is any algebra, then there is a discrete space over $X$ induced by the empty family of maps. Similarly, we have an indiscrete space with the final structure induced by no maps. This defines functors $disc, indisc\colon D \to C$ that are respectively left and right adjoints of $U$.

• Suppose that $D$ has an initial object $0_D$. Then the discrete space $0_C$ over $0_D$ is initial in $C$. Similarly, the indiscrete space over a terminal object in $D$ is terminal in $C$.

• More generally, suppose that $D$ has products or coproducts (indexed by whichever cardinalities you may wish to consider). Then $C$ also has (co)products, lying over the (co)products in $D$, with structures induced by the product projections or coproduct inclusions.

• More general limits and colimits are constructed in a similar way. However, it is not typically the case that $U$ creates (co)limits in $C$ because creation of a limit requires that every preimage of the limiting cone is limiting. This fails for $U: \mathrm{Top} \to \mathrm{Set}$ since we can coarsen the topology on the limit vertex to obtain a counterexample.

• If a single algebra $X$ has been given the structure of several spaces, then there are a supremum structure and an infimum structure on $X$ induced (as the initial and final structures) by the various incarnations of its identity homomorphism. Exploiting this shows how to construct final structures out of initial ones and conversely.

• If $X$ is a regular subalgebra of some $U(S)$, then the inclusion homomorphism makes $X$ into a subspace of $S$, which is also a subobject in $C$. Every regular subobject of $S$ is of this form; note however that there may be nonregular subobjects in $C$ even if all subobjects in $D$ are regular.

Familiarly fibrations

The theory of topological functors can be developed along the lines of Grothendieck’s theory of fibrations, where cartesian morphisms are replaced by cartesian families. In this way just as by definition “A functor is a fibration if it creates cartesian morphisms and cartesian morphism compose”, there is the definition “A functor is topological if it creates cartesian families and cartesian families compose”.

References

• J. Martin Harvey: Topological functors from factorization, in: Categorical Topology, Lecture Notes in Mathematics 719, Springer (1979) [doi:10.1007/BFb0065263]

• Jiří Adámek, Horst Herrlich, George Strecker: Abstract and Concrete Categories – The Joy of Cats, Wiley (1990), reprinted as: Reprints in Theory and Applications of Categories 17 (2006) 1-507 [tac:tr17, book webpage, pdf]

• Gerhard Preuß: Foundations of Topology: An Approach to Convenient Topology; Kluwer (2002) [ISBN 1-4020-0891-0, doi:10.1007/978-94-010-0489-3]

• Richard Garner, Topological functors as total categories, Theory and Applications of Categories 29 15 (2014) 406-421 [tac:29-15]

• Eduardo J. Dubuc, Luis Español, Topological functors as familiarly fibrations [arXiv:0611701]