nLab
topological concrete category

Warning

The term ‘topological category’ is traditional, and comes from the frequent examples in topology. It does not mean an internal category or enriched category in Top! (Fortunately the term topological groupoid? is not taken by this tradition; indeed, the only groupoid that is a topological category over $\mathrm{Set}$ is trivial. On the other hand, they do seem to use the term ‘topological functor’, which here we avoid.)

Idea

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

Definition

Most generally, the definition relates to a functor $U:C\to D$ (such as the forgetful functor from $\mathrm{Top}$ to Set), but one can think of this as giving $C$ as a bundle over $D$. Usually $C$ and $D$ will be large categories. Let a space be an object of $C$, an algebra be an object of $D$, a map be a morphism in $C$, and a homomorphism be a morphism in $D$. (The reason is that, typically, $C$ will be a category of spaces with some kind of topological structure while $D$ will be, if not $\mathrm{Set}$, then some kind of algebraic category.)

Then $C$ is a topological category over $D$ if, given any algebra $X$ and any (possibly large) family of spaces $S_i$ and homomorphisms $f_i:X\to U(S_i)$ (that is a source from $X$), there exist

• a space $T$, an isomorphism $g:U(T)\to X$, and maps $m_i:T\to S_i$ such that each composite $g;f_i$ equals $U(m_i)$ and,

• given any space $T\prime$, homomorphism $g\prime :U(T\prime )\to X$, and maps $m{\prime }_i:T\prime \to S_i$, if each composite $g\prime ;f_i$ equals $U(m{\prime }_i)$, then there exists

  • a map $n:T\prime \to T$ such that $U(n);g=g\prime$ and
  • given any map $n\prime :T\prime \to T$, if $U(n\prime );g=g\prime$, then $n=n\prime$.

Here are some illustrative commutative diagrams (if you can read them):

It follows by a clever argument that $U:C\to D$ must be faithful; see Theorem 21.3 of ACC. That is also often included in the definition, in which case the uniqueness of $n$ can be left out. Thus we may think of objects of $C$ as objects of $D$ equipped with extra structure. The idea is then that $T$ is $X$ equipped with the intitial structure or weak structure? determined by the requirement that the homomorphisms $f_i$ be structure-preserving maps.

The dual concept could be called a cotopological category. However, this is not actually anything new; $U:C\to D$ is topological if and only if $U^{\mathrm{op}}:C^{\mathrm{op}}\to D^{\mathrm{op}}$ is. This is 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 $f_i:U(S_i)\to X$ (a sink to $X$).

Both of these results (faithfulness and self-duality) depend on the fact that we have allowed the family $\{S_i\}$ to be potentially large. Counterexamples are easy to find. For instance, if $C$ is a large category with all (small) products, then the functor $C\to 1$ to the terminal category satisfies the above lifting property 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 $U$ is a fibration and opfibration, in the weakened bicategorical sense of Street. One also often assumes in the definition $U(T)=X$ and that $g$ is the identity morphism, which in particular makes $U$ into a fibration in the original sense of Grothendieck. This is a bit evil, but it is convenient and satisfied in almost all examples, and any example not satisfying it is equivalent to one which does (via fibrant replacement by an isofibration).

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 $\mathrm{Set}$, apparently not even the category of spatial locales (equivalent to the category of sober spaces), 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.

  Andrew Stacey: All of the categories listed on generalized smooth space are concrete and are topological over Set: all admit discrete and indiscrete structures (only constant maps and all maps respectively).

  There are non-set based generalisations but the current flavour of generalized smooth space is set-based.

  Toby: Sorry, I should have checked there. Of course, these about concrete sheaves; I knew that once!

• Outside of topology, the category of measurable spaces is topological over $\mathrm{Set}$.

• The category of topological groups is topological over Grp, the category of topological vector spaces is topological over $ℝ$-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 $\mathrm{Cat}$/$D$.
• 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 $U:C\to D$ is not only faithful but also (for different reasons) essentially surjective. 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 well-powered or co-well-powered, then so is $C$.
• 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.

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 $\mathrm{disc},\mathrm{indisc}:D\to C$ that are respectively left and right adjuncts 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. We say that $U$ creates (co)limits in $C$.
• 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.

References

I should find some … I used HAF (Section 9.15) and the English Wikipedia (on topological vector spaces) as reminders, but there not good for much more than that.

The Wikipedia entry on Category of topological spaces contains some references, including

• Jiří Adámek, Horst Herrlich, & George E. Strecker; 1990; Abstract and Concrete Categories; originally published John Wiley & Sons ISBN 0-471-60922-6; free on-line edition (4.2MB PDF).