nLab
topological concrete category

Topological categories

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 (a topologically enriched category)! (Fortunately the term topological groupoid is not taken by this tradition; indeed, the only groupoid that is a topological category over SetSet 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

Default version

Most generally, the definition relates to a functor U:CDU\colon C \to D (such as the forgetful functor from TopTop to Set), but one can think of this as giving CC as a bundle over DD. Sometimes, when DD is in fact Set, the category CC satisfying the properties described belows is called a topological construct (Preuss). Usually CC and DD will be large categories.

By a space we will mean an object of CC, and by an algebra we will mean an object of DD. By a map we will mean a morphism in CC, and by a homomorphism we will mean a morphism in DD. (The reason is that, typically, CC will be a category of spaces with some kind of topological structure while DD will be, if not SetSet, then some kind of algebraic category.)

Then CC is a topological category over DD if, given any algebra XX and any (possibly large) family of spaces S iS_i and homomorphisms f i:XU(S i)f_i\colon X \to U(S_i) (that is, a “UU-structured” source from XX), there exists an initial lift (think: “smallest topology rendering the f if_i continuous”), which is to say

  • a space TT such that U(T)=XU(T)=X, and maps m i:TS im_i\colon T \to S_i such that U(m i)=f iU(m_i) = f_i, and

  • given any space TT', homomorphism g:U(T)Xg'\colon U(T') \to X, and maps m i:TS im'_i\colon T' \to S_i, if each composite g;f ig' ; f_i equals U(m i)U(m'_i), then there exists a unique map n:TTn\colon T' \to T such that U(n)=gU(n) = g' and n;m i=m in ; m_i = m'_i.

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

T nn m i T m i S iUU(T) U(n)U(n) g U(m i) U(T) = X f i U(S i) U(m i) \array { T' \\ n \downarrow \downarrow n' & \searrow^{m'_i} \\ T & \underset{m_i}\rightarrow & S_i } \;\;\; \stackrel{U}\mapsto \;\;\; \array { U(T') \\ U(n) \downarrow \downarrow U(n') & \searrow^{g'} & & \searrow^{U(m'_i)} \\ U(T) & = & X & \underset{f_i}\rightarrow & U(S_i) \\ & & \underset{U(m_i)}\longrightarrow }

It follows by a clever argument that U:CDU\colon 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 nn can be left out. Thus we may think of objects of CC as objects of DD equipped with extra structure. The idea is then that TT is XX equipped with the initial structure or weak structure determined by the requirement that the homomorphisms f if_i be structure-preserving maps.

The dual concept could be called a cotopological category. However, this is not actually anything new; U:CDU\colon C \to D is topological if and only if U op:C opD opU^op\colon C^op \to D^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)Xf_i\colon U(S_i) \to X (a UU-structured sink to XX).

Both of these results (faithfulness and self-duality) depend on the fact that we have allowed the family {S i}\{S_i\} to be potentially large. Counterexamples are easy to find. For instance, if CC is a large category with all (small) products, then the functor C1C \to 1 to the terminal category satisfies the above lifting property for small families {S i}\{S_i\}. However, it need not satisfy the dual property (unless CC also has all small coproducts) nor need it be faithful.

It also follows that UU is a Grothendieck fibration and opfibration.

Amnestic version

Since initial lifts have a universal property, they are unique up to unique isomorphism. However, it is traditional in some literature to ask that they be literally unique (this is done for instance in ACC). This is tantamount to deciding that UU 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 ‘evil’ results could be affected.)

Weak version

On the other hand, the default definition above does already refer to equality of objects in the condition U(T)=XU(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 TT, an isomorphism g:U(T)Xg\colon U(T) \to X, and maps m i:TS im_i\colon T \to S_i such that each composite g;f ig ; f_i equals U(m i)U(m_i) and,

  • given any space TT', homomorphism g:U(T)Xg'\colon U(T') \to X, and maps m i:TS im'_i\colon T' \to S_i, if each composite g;f ig' ; f_i equals U(m i)U(m'_i), then there exists a unique map n:TTn\colon T' \to T such that U(n);g=gU(n) ; g = g' and n;m i=m in ; m_i = m'_i.

T nn m i T m i S iUU(T) U(n)U(n) g U(m i) U(T) g X f i U(S i) U(m i) \array { T' \\ n \downarrow \downarrow n' & \searrow^{m'_i} \\ T & \underset{m_i}\rightarrow & S_i } \;\;\; \stackrel{U}\mapsto \;\;\; \array { U(T') \\ U(n) \downarrow \downarrow U(n') & \searrow^{g'} & & \searrow^{U(m'_i)} \\ U(T) & \overset{\sim}\underset{g}\rightarrow & X & \underset{f_i}\rightarrow & U(S_i) \\ & & \underset{U(m_i)}\longrightarrow }

Examples

Further properties

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

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

  • The forgetful functor U:CDU\colon 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 UU is an equivalence, of course).

  • If DD is complete or cocomplete, then so is CC.

  • If DD is total or cototal, then so is CC; see solid functor.

  • If DD is mono-complete or epi-cocomplete, then so is CC.

  • If DD is well-powered or co-well-powered, then so is CC.

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

  • If DD is concrete, then so is CC. More generally, if DD has a generator, then CC is concrete over DD.

  • In particular, if DD is Set, then CC is a concrete category that is complete, cocomplete, well powered, and well copowered.

Functors

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

  • More generally: If a functor F:CCF\colon C\to C' between topological concrete categories C/DC/D, C/DC'/D' with different base categories lying over a functor F 0:DDF_0: D\to D'. If FF is right (left) adjoint, then F 0F_0 is right (left) adjoint and FF preserves initial (final) lifts. A partial converse holds: If F 0F_0 is right (left) adjoint to G 0G_0 and FF preserves initial (final) lifts, then there is functor GG lying over G 0G_0 so that FF is right (left) adjoint to G 0G_0.

Special cases

  • If XX is any algebra, then there is a discrete space over XX 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:DCdisc, indisc\colon D \to C that are respectively left and right adjoints of UU.

  • Suppose that DD has an initial object 0 D0_D. Then the discrete space 0 C0_C over 0 D0_D is initial in CC. Similarly, the indiscrete space over a terminal object in DD is terminal in CC.

  • More generally, suppose that DD has products or coproducts (indexed by whichever cardinalities you may wish to consider). Then CC also has (co)products, lying over the (co)products in DD, 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 UU creates (co)limits in CC because creation of a limit requires that every preimage of the limiting cone is limiting. This fails for U:TopSetU: \mathrm{Top} \to \mathrm{Set} since we can coarsen the topology on the limit vertex to obtain a counterexample.

  • If a single algebra XX has been given the structure of several spaces, then there are a supremum structure and an infimum structure on XX 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 XX is a regular subalgebra of some U(S)U(S), then the inclusion homomorphism makes XX into a subspace of SS, which is also a subobject in CC. Every regular subobject of SS is of this form; note however that there may be nonregular subobjects in CC even if all subobjects in DD 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

  • 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).
  • Gerhard Preuss; 2002; Foundations of Topology: An Approach to Convenient Topology; Kluwer ISBN 1-4020-0891-0.

Revised on June 30, 2017 05:00:35 by Mike Shulman (76.167.222.204)