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TENSOR TOPOLOGY
PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Abstract. A subunit in a monoidal category is a subobject of the monoidal
unit for which a canonical morphism is invertible. They correspond to open
subsets of a base topological space in categories such as those of sheaves or
Hilbert modules. We show that under mild conditions subunits endow any
monoidal category with topological intuition: there are well-behaved notions
of restriction, localisation, and support, even though the subunits in general
only form a semilattice. We develop universal constructions completing any
monoidal category to one whose subunits universally form a lattice, preframe,
or frame.
1. Introduction
Categorical approaches have been very successful in bringing topological ideas
into other areas of mathematics. A major example is the category of sheaves over
a topological space, from which the open sets of the space can be reconstructed as
subobjects of the terminal object. More generally, in any topos such subobjects
form a frame. Frames are lattices with properties capturing the behaviour of the
open sets of a space, and form the basis of the constructive theory of pointfree
topology [34].
The goal of this article is to study this inherent notion of space in categories more
general than those with cartesian products. Specifically, it argues that a semblance
of this topological intuition remains in categories with mere tensor products. Its
aim is to lay foundations for this (ambitiously titled) ‘tensor topology’.
Boyarchenko and Drinfeld [9, 10] have already shown how to equate the open
sets of a space with certain morphisms in its monoidal category of sheaves of vector
spaces. This forms the basis for our approach. We focus on certain subobjects of
the tensor unit in a (braided) monoidal category that we call subunits, fitting with
other treatments of tensor units [30, 39, 18].
For subunits to behave well one requires only that monomorphisms and tensor
products interact well; we call a category firm when it does so for subunits and stiff
when it does so globally, after [43]. In a firm category subunits always form a (meet)
semilattice. They may have further features, such as having joins that interact with
the category through universal properties, and in the strongest case form a frame.
We axiomatise such spatial1categories. Aside from toposes, major noncartesian
examples are categories of Hilbert modules, with subunits indeed given by open
Chris Heunen and Sean Tull are supported by EPSRC Research Fellowship EP/L002388/2.
We thank Richard Garner, Marino Gran, Joachim Kock, Tim van der Linden, Bob Par´e, John
Power, Pedro Resende, Manny Reyes, Phil Scott, and Isar Stubbe for helpful discussions.
1There is a terminology clash here: the subunits of a spatial category form a frame, and not a
topological space in the sense of a spatial locale. (Locale is another word for frame.) We prefer
this clash over the one with ‘localic category’, which is already in use for categories internal to
the category of locales, as well as for categories enriched in locales.
1
arXiv:1810.01383v1 [math.CT] 2 Oct 2018
2 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
subsets of the base space. More generally, we show how to spatially complete any
stiff category.
There are at least two further perspectives on this study. First, it generalises
tensor triangular geometry [3], a programme with many applications including alge-
braic geometry, stable homotopy theory, modular representation theory, and sym-
plectic geometry [2, 4, 5]. We show that many results do not need any triangulation,
their natural home being mere monoidal categories [31]. For example, we will not
require our categories to be cocomplete [12].
Second, just as Grothendieck toposes may be regarded as a categorification of
frames [49], our results may be regarded as categorifying the study of central idem-
potents in a ring. Our algebraic examples include categories of firm nondegenerate
modules over a firm nonunital commutative ring, or more generally, over a nonunital
bialgebra in a braided monoidal category.
Structure of article. We set out the basics of subunits in Section 2, showing that
they form a semilattice in any firm category. Section 3 introduces our main exam-
ples: sheaves, Hilbert modules, modules over a ring, and order-theoretic examples
including commutative quantales, generalising frames [44].
In Section 4 we introduce the notion of a morphism ‘restricting to’ a subunit, and
show how to turn any subunit into a unit of its restricted category. These restriction
functors together are seen to form a graded monad. We also show that subunits
correspond to certain ideal subcategories and to certain comonads. Section 5 then
proves that restriction forms a localisation of our category, and more broadly that
one may localise to a category with only trivial subunits.
Section 6 introduces the notion of support of a morphism, derived from the
collection of subunits to which it restricts. This notion seems unrelated to earlier
definitions requiring more structure [36, 40].
In Sections 7 and 8 we characterise categories, such as toposes and categories of
Hilbert modules, whose subunits come with suprema satisfying universal proper-
ties and so form a lattice, preframe, or frame; the latter being spatial categories.
Finally, Sections 9 and 10 show how to complete a given monoidal category to one
with each kind of universal joins, including a spatial category, in a universal way.
This involves passing to certain presheaves, that we will call broad, under Day con-
volution, as detailed in Appendix A; but this completion is not a sheafification for
any Grothendieck topology.
Further directions. This foundation opens various directions for further investi-
gation. Applications to linear logic and computer science, as proposed in [17],
remain to be explored, including amending the graphical calculus for monoidal
categories [47] with spatial information. It would be interesting to examine what
happens to subunits under constructions such as Kleisli categories, Chu spaces,
or the Int-construction [37]. One could ask how much of the theory carries over
to skew monoidal categories [50], and how these notions relate to restriction cat-
egories [23]. Finally, it would be desirable to find global conditions on a category
providing its subunits with further properties, such as being a compact frame or
Boolean algebra, or with further structure, such as being a metric space.
TENSOR TOPOLOGY 3
2. Subunits
We work with braided monoidal categories [42], and will sometimes suppress the
coherence isomorphisms λA:I⊗A→A,ρA:A⊗I→A,αA,B,C :A⊗(B⊗C)→
(A⊗B)⊗C, and σA,B :A⊗B→B⊗A, and often abbreviate identity morphisms
1A:A→Asimply by A.
Recall that a subobject of an object Ais an equivalence class of monomorphisms
s:SA, where sand s0are identified if they factor through each other. When-
ever we talk about a subobject, we will use a small letter sfor a representing
monomorphism, and the corresponding capital Sfor its domain.
Definition 2.1. Asubunit in a braided monoidal category Cis a subobject s:S
Iof the tensor unit such that s⊗S:S⊗S→I⊗Sis an isomorphism2. Write
ISub(C) for the collection of subunits in C.
Note that, because sis monic, if s⊗Sis invertible then so is S⊗s.
Remark 2.2. We could have generalised the previous definition to arbitrary monoidal
categories by additionally requiring subunits to be central in the sense that there
is a natural isomorphism (−)⊗S⇒S⊗(−). Most results below still hold, but the
bureaucracy is not worth the added generality here.
Many results also remain valid when we require s⊗Snot to be invertible but
merely split epic, but for simplicity we stick with invertibility.
We begin with some useful observations, mostly adapted from Boyarchenko and
Drinfeld [9].
Lemma 2.3. Let m:A→Band e:B→Asatisfy e◦m=A, and s:SIbe a
subunit. If s⊗Bis an isomorphism, then so is s⊗A.
Proof. The diagram below commutes by bifunctoriality of ⊗.
S⊗A S ⊗B S ⊗A
I⊗A I ⊗B I ⊗A
S⊗m
s⊗A
S⊗e
s⊗B
's⊗A
I⊗m I⊗e
Both rows compose to the identity, and the middle vertical arrow is an isomorphism.
Hence s⊗Ais an isomorphism with inverse (S⊗e)◦(s⊗B)−1◦(I⊗m).
Recall that subobjects of a fixed object always form a partially ordered set, where
s≤tif and only if sfactors through t. The following observations characterises
this order in another way for subunits.
Lemma 2.4. A subunit sfactors through another tif and only if S⊗tis invertible,
or equivalently, s⊗Tis invertible.
Proof. Suppose s=t◦f. Set g= (S⊗f)◦(S⊗s)−1◦ρ−1
S:S→S⊗T. Then
ρS◦(S⊗t)◦g=ρS◦(S⊗s)◦(S⊗s)−1◦ρS−1=S.
Idempotence of tmakes S⊗T⊗t:S⊗T⊗T→S⊗T⊗Ian isomorphism. Hence,
by the right-handed version of Lemma 2.3, so is S⊗t. A symmetric argument
makes s⊗Tinvertible.
2Boyarchenko and Drinfeld call morphisms s:S→Ifor which s⊗Sand S⊗sare isomorphisms
open idempotents [9], with (the dual of ) this notion going back at least to [38, Example 4.2]. In [17]
subunits were called idempotent subunits.
4 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Conversely, suppose S⊗tis an isomorphism. Because the diagram
S⊗T I ⊗T T
S⊗I I ⊗I I
S⊗t
s⊗T
I⊗t
ρT
t
s⊗IρI
commutes, the bottom row s◦ρSfactors through the right vertical arrow t, whence
so does s.
It follows from Lemma 2.4 that subunits are determined by their domain: if
s, s0:SIare subunits, then s0=s◦ffor a unique f, which is an isomorphism.
This justifies our convention to use the same letter for a subunits and its domain.
For the theory to work smoothly, we impose a condition on the category.
Definition 2.5. A category is called firm when it is braided monoidal and s⊗
T:S⊗T→I⊗Tis a monomorphism whenever sand tare subunits.
This condition is very mild: Example 9.2 below gives a category that is not firm,
but we know of no other ‘naturally occurring’ categories that are not firm.
Lemma 2.6. Any co-closed braided monoidal category is firm.
Proof. Each functor (−)⊗Tis a right adjoint and so preserves limits and hence
monomorphisms. Hence whenever sis monic so is s⊗T.
In particular, a ∗-autonomous category is firm, as is a compact category.
Remark 2.7. In the following, we will completely disregard size issues, and pretend
ISub(C) is a set, as in our main examples.
Proposition 2.8. The subunits in a firm category form a semilattice, with largest
element I, meets given by
s:SI∧t:TI=λI◦(s⊗t): S⊗TI,
and the usual order of subobjects.
Proof. First observe that s⊗t= (I⊗t)◦(s⊗T) is monic, because I⊗t=λ−1
I◦t◦λT
is monic, and s⊗Tis monic by firmness. It is easily seen to be idempotent using
the braiding, and hence it is a well-defined subunit.
Next, we show that ISub(C) is an idempotent commutative monoid under ∧
and I. The subunit Iis a unit as I⊗s=λI◦(I⊗s) = s◦λSrepresents the
same subobject as s, and similarly I⊗srepresents the same subobject as sbecause
ρI=λI. An analogous argument using coherence establishes associativity. For
commutativity, use the braiding to observe that s⊗tand t⊗srepresent the same
subobject. For idempotence note that s⊗sand srepresent the same subobject
because λI◦(s⊗s) = s◦ρS◦(S⊗s).
Hence ISub(C) is a semilattice where sis below tif and only if s=s∧t. Finally,
we show that this order is the same as the usual order of subobjects. On the one
hand, if sand s⊗trepresent the same subobject, then S'S⊗T, making S⊗t
an isomorphism and so s≤tby Lemma 2.4.
S
I
T
s
t
⇐⇒ S I
S⊗T I ⊗I
s
s⊗t
λI
''
TENSOR TOPOLOGY 5
On the other hand, if s≤tthen by the same lemma S⊗tis an isomorphism with
s=λI◦(s⊗t)◦(S⊗t)−1⊗ρ−1
S, and so both subobjects are equal.
3. Examples
This section determines the subunits of four families of examples: cartesian
categories, like sheaves over a topological space; commutative unital quantales; firm
modules over a nonunital ring; and Hilbert modules over a nonunital commutative
C*-algebra.
Cartesian categories. We start with examples in which the tensor product is not
very interesting.
Example 3.1. Any cartesian category Cis firm, and ISub(C) consists of the
subobjects of the terminal object.
In particular, if Xis a topological space, then subunits in its category of sheaves
Sh(X) correspond to open subsets of X[8, Corollary 2.2.16].
Proof. Let s:S1 be a subterminal object. Let ∆ = hS, Si:S→S×Sbe the
diagonal and write πi:A1×A2→Aifor the projections. Then (s×S)◦∆◦π2=
π−1
2◦S◦π2= 1 ×S. Now, the unique map sof type S→1 is monic precisely when
any two parallel morphisms into Sare equal. Hence πi◦∆◦π2◦(s×S) = πi, and
so ∆ ◦π2◦(s×S) = hπ1, π2i=S×S. Thus s×Sis automatically invertible.
Finally, suppose si:Si1 for i= 1,2 are monic, and that f , g :A→S1×S2
satisfy (s1×s2)◦f= (s1×s2)◦g. Postcomposing with πishows that si◦πi◦f=
si◦πi◦g, whence πi◦f=πi◦gand so f=g. This establishes firmness.
Semilattices. Next we consider examples that are degenerate in another sense:
firm categories in which there is at most one morphism between two given objects.
Example 3.2. Any semilattice (L, ∧,1) forms a strict symmetric monoidal cat-
egory: objects are x∈L, there is a unique morphism x→yif x≤y, tensor
product is given by meet, and tensor unit is I= 1. Every morphism is monic so
this monoidal category is firm, and its (idempotent) subunits are (L, ∧,1).
This gives the free firm category on a semilattice. More precisely, this construc-
tion is left adjoint to the functor from the category Firm of firm categories with
(strong) monoidal subunit-preserving functors to the category SLat of semilattices
and their homomorphisms, which takes subunits.
SLat Firm⊥
ISub
Quantales. We move on to more interesting examples, namely special kinds of
semilattices like frames and quantales.
Definition 3.3. Aframe is a complete lattice in which finite joins distribute over
suprema. A morphism of frames is a function that preserves W,∧, and 1. Frames
and their morphisms form a category Frame.
The prototypical example of a frame is the collection of open sets of a topological
space [34]. Frames may be generalised as follows [45].
6 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Definition 3.4. Aquantale is a monoid in the category of complete lattices. More
precisely, it is a partially ordered set Qthat has all suprema, that has a multipli-
cation Q×Q→Q, and that has an element e, such that:
a_bi=_abi,_aib=_aib, ae =a=ea.
A morphism of quantales is a function that preserves W,·, and e. A quantale is
commutative when ab =ba for all a, b ∈Q. Commutative quantales and their
morphisms for a category cQuant.
Equivalently, a frame is a commutative quantale in which the multiplication is
idempotent.
Any quantale may be regarded as a monoidal category, whose objects are ele-
ments of the quantale, where the (composition of) morphisms is induced by the par-
tial order, and the tensor product is induced by the multiplication. This monoidal
category is firm, but only braided if the quantale is commutative.
Example 3.5. Taking subunits is right adjoint to the inclusion:
Frame cQuant
⊥
ISub
{q∈Q|q2=q≤e}Q
Proof. We first prove that ISub(Q) is a well-defined frame. If qi∈ISub(Q),
(_qi)2=_
i,j
qiqj≤_
i,j
qie=_
i
qi=_
i
qiqi≤_
i,j
qiqj= (_qi)2
and Wqi≤Wie=e, so Wqi∈ISub(Q). Moreover, if p, q ∈ISub(Q), then pq is
again below eand is idempotent by commutativity of Q. Moreover pq =p∧qin
ISub(Q): if o∈ISub(Q) has o≤pq then o≤pq ≤pe =pand similarly o≤q;
and conversely if o≤pand o≤qthen o=oo ≤pq. Since quantale multiplication
distributes over suprema, then so do finite meets.
For the adjunction, observe that if Fis a frame and Qis a commutative quantale,
then F= ISub(F) and any morphism F→Qof quantales restricts to a unique
morphism of frames F→ISub(Q).
Example 3.6. If Mis a monoid, then its (right) ideals form a unital quantale
Qwith multiplication IJ ={xy |x∈I , y ∈J}and unit Mitself. When Mis
commutative, so is Q, and ISub(Q) consists of all ideals satisfying I=II.
Example 3.7. If Ris a commutative ring, then its additive subgroups form a
unital commutative quantale Qwith multiplication GH ={x1y1+·· · +xnyn|
xi∈G, yi∈H}, supremum WGi={Pj∈Jxj|xj∈Gjfor J⊆Ifinite}, and unit
Z1 = {0,1,−1,1+1,−1−1,1+1+1,−1−1−1, . . .}. Then G≤Hiff G⊆Hand
ISub(Q) consists of those subgroups Gsuch that G⊆G·Gand G⊆Z1. The latter
means that Gmust be of the form nZ1 for some n∈N. The former then means that
n1 = n2y1 for some y∈Z. Thus ISub(Q) = {nZ1|n∈N,∃y∈Z:n1 = n2y1}.
Modules. Perhaps the example of a monoidal category known to most people is
that of modules over a ring. We have to take some pains to treat nonunital rings.
TENSOR TOPOLOGY 7
Definition 3.8. A commutative ring Ris firm when its multiplication is a bijection
R⊗R→R, and nondegenerate when r∈Rvanishes as soon as rs = 0 for all
s∈R. Any unital ring is firm and nondegenerate, but examples also include infinite
direct sums Ln∈NRnof unital rings Rn. Firm rings Rare idempotent: they equal
R2={Pn
i=1 r0
ir00
i|r0
i, r00
i∈R}. Let Rbe a nondegenerate firm commutative ring.
An R-module Eis firm when the scalar multiplication is a bijection E⊗R→E[43],
and nondegenerate when x∈Evanishes as soon as xr = 0 for all r∈R. If Ris
unital, then every R-module is firm and nondegenerate. Nondegenerate firm R-
modules and linear maps form a monoidal category FModR.
Example 3.9. The subunits in FModRcorrespond to nondegenerate firm idempo-
tent ideals: ideals S⊆Rthat are idempotent as rings, and nondegenerate and firm
as R-modules. Any ideal that is unital as a ring is a nondegenerate firm idempotent
ideal. The category FModRis firm.
Proof. Monomorphisms are injective by nondegeneracy, so every subunit is a non-
degenerate firm R-submodule of R, that is, a nondegenerate firm ideal. Because
the inclusion S⊗S→R⊗Sis surjective and Sis firm, the map S⊗S→Sgiven
by s0⊗s00 7→ s0s00 is surjective. Thus Sis idempotent.
Conversely, let Sbe a nondegenerate firm idempotent ideal of R. The inclusion
S⊗S→R⊗Sis surjective, as r⊗s∈R⊗Scan be written as r⊗s0s00 =rs0⊗s00 ∈
S⊗S. Hence Sis a subunit.
Next suppose ideal Sis unital (with generally 1S6= 1Rif Ris unital). Then
S⊗R→Sgiven by s⊗r7→ sr is bijective: surjective as 1S⊗s7→ 1Ss=s;
and injective as s⊗r= 1S⊗sr = 1S⊗0 = 0 if sr = 0. Hence Sis firm and
nondegenerate. Any s∈Scan be written as s=s1S∈S2, so Sis idempotent.
Finally, to see that the category is firm, let S, T ⊆Rbe nondegenerate firm
idempotent ideals. We need to show that the map S⊗T7→ R⊗Tgiven by
s⊗t7→ s⊗tis injective. Because Tis firm, it suffices that multiplication S⊗T→S
given by s⊗t7→ st is injective, which holds because Sis firm.
The previous example generalises to commutative nonunital bialgebras in any
symmetric monoidal category.
Example 3.10. Let Cbe a symmetric monoidal category. A commutative nonuni-
tal bialgebra in Cis an object Mtogether with an associative multiplication
µ:M⊗M→Mand a comonoid δ:M→M⊗M,ε:M→I, for which
and δare commutative and satisfy both ε◦µ=ε⊗εand the bialgebra law:
(µ⊗µ)◦(M⊗σ⊗M)◦(δ⊗δ) = δ◦µ
We define a braided monoidal category ModMwhere objects are α:M⊗A→A
satisfying α◦(µ⊗A) = α◦(M⊗α), with morphisms and ⊗all defined as for modules
over a (unital) commutative bialgebra (see e.g. [25, 2.2,2.3]). The category ModM
is firm when Cis, and its subunits correspond to firm ideals: monomorphisms
s:SMsuch that
M⊗S M ⊗M
S M
M⊗s
µ
s
and ε⊗Sand s⊗Sare isomorphisms.
8 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
We next instantiate the previous example in two special cases: in the monoidal
categories of semilattices and of quantales.
Example 3.11. Any semilattice Mis a nondegenerate nonunital bialgebra in SLat.
In ModMobjects are semilattices Awith functions α:M×A→Awhich respect
∧in each argument and satisfy α(x∧y, a) = α(x, α(y, a)). Subob jects of the
tensor unit correspond to subsets S⊆Mwhich are ideals under ∧, or equivalently
downward-closed. Because x⊗y= (x∧x)⊗y=x⊗(x∧y)∈S⊗S, we have
S⊗S=S⊗M, and every subobject of the tensor unit is a subunit.
Example 3.12. Any commutative unital quantale Mis a nondegenerate nonunital
bialgebra in the category of complete lattices. ModMthen consists of complete
lattices Awith functions α:M×A→Apreserving arbitrary suprema in each ar-
gument and with α(x, α(y, a)) = α(xy, a). Subobjects of the tensor unit are subsets
S⊆Mclosed under both arbitrary suprema and multiplication with elements of
M. Subunits furthermore have that for every r∈Sand x∈Mthere exist si, ti∈S
with r⊗x=Wsi⊗ti. For example, if M= [0,∞] under addition with the opposite
ordering, subunits include ∅,{∞},{0,∞}, (0,∞], and [0,∞].
Hilbert modules. The above examples of module categories were all algebraic in
nature. Our next suite of examples is more analytic. For more information we refer
to [41].
Definition 3.13. Fix a locally compact Hausdorff space X. It induces a commu-
tative C*-algebra
C0(X) = {f:X→Ccontinuous | ∀ε > 0∃K⊆Xcompact: |f(X\K)|< ε}.
AHilbert module is a C0(X)-module Awith a map h− | −i:A×A→C0(X) that is
C0(X)-linear in the second variable, satisfies ha|bi=hb|ai∗, and ha|ai ≥ 0 with
equality only if a= 0, and makes Acomplete in the norm kak2
A= supx∈Xha|ai(x).
A function f:A→Bbetween Hilbert C0(X)-modules is bounded when kf(a)kF≤
kfkkakAfor some kfk ∈ R. Here we will focus on contractions, i.e. those bounded
functions with kfk ≤ 1.
The category HilbC0(X)of Hilbert C0(X)-modules and contractive C0(X)-linear
maps is not abelian, not complete, and not cocomplete [26]. Nevertheless, HilbC0(X)
is symmetric monoidal [28, Proposition 2.2]. Here A⊗Bis constructed as fol-
lows: consider the algebraic tensor product of C0(X)-modules, and complete it to
a Hilbert module with inner product ha⊗b|a0⊗b0igiven by ha|a0ihb|b0i. The
tensor unit is C0(X) itself, which forms a Hilbert C0(X)-module under the inner
product hf|gi(x) = f(x)∗g(x).
Example 3.14. HilbC0(X)is firm, and its subunits are
(1) {f∈C0(X)|f(X\U)=0} ' C0(U)
for open subsets U⊆X.
Proof. If Uis an open subset of X, we may indeed identify C0(U) with the closed
ideal of C0(X) in (1): if f∈C0(U), then its extension by zero on X\Uis in
C0(X), and conversely, if f∈C0(X) is zero outside U, then its restriction to U
is in C0(U). Moreover, note that the canonical map C0(X)⊗C0(X)→C0(X) is
always an isomorphism as C0(X) is the tensor unit, and hence the same holds for
C0(U). Thus C0(U) is a subunit in HilbC0(X).
TENSOR TOPOLOGY 9
For the converse, let s:SC0(X) be a subunit in HilbC0(X). We will show
that s(S) is a closed ideal in C0(X), and therefore of the form C0(U) for some open
subset U⊆X. It is an ideal because sis C0(X)-linear. To see that it is closed, let
g∈s(S). Then
kgk4
S=khg|gi2
SkC0(X)=khg|giShg|giSkC0(X)
=khg⊗g|g⊗giC0(X)kC0(X)=kg⊗gk2
S
≤ kρ−1
Sk2kg2kS=kρ−1
Sk2khg|giSg∗gkC0(X)
≤ kρ−1
Sk2kgk2
Skgk2
C0(X)
and therefore kgkS≤ kρ−1
Sk2kgk2
C0(X). Because sis bounded, it is thus an equiva-
lence of normed spaces between (S, k−kS) and (s(S),k−kC0(X)). Since the former
is complete, so is the latter. Firmness follows from Example 4.10 later.
The category HilbC0(X)can be adapted to form a dagger category by considering
(not necessarily contractive) bounded maps between Hilbert modules to that are
adjointable. In that case only clopen subsets of Xcorrespond to subunits [28,
Lemma 3.3].
Another way to view a Hilbert C0(X)-module is as a field of Hilbert spaces
over X. Intuitively, this assigns to each x∈Xa Hilbert space, that ‘varies
continuously’ with x. In particular, for each x∈Xthere is a monoidal functor
Locx:HilbC0(X)→HilbC. For details, see [28]. This perspective may be useful in
reading Section 4 later.
Not every subobject of the tensor unit in HilbC0(X)is induced by an open subset
U⊆X, and so the condition of Definition 2.1 is not redundant.
Example 3.15. Let X= [0,1]. If f∈C0(X), write ˆ
f∈C0(X) for the map
x7→ xf(x). Then S={ˆ
f|f∈A}is a subobject of A=C0(X) in HilbC0(X)
under hˆ
f|ˆgiS=hf|giA, that is not closed under k−kA.
Proof. Clearly Sis a C0(X)-module, and h− | −iSis sesquilinear. Moreover Sis
complete: ˆ
fnis a Cauchy sequence in Sif and only if fnis a Cauchy sequence in A,
in which case it converges in Ato some f, and so ˆ
fnconverges to ˆ
fin S. Thus Sis
a well-defined Hilbert module. The inclusion S →Ais bounded and injective, and
hence a well-defined monomorphism. In fact, Ais a C*-algebra, and Sis an ideal.
The closure of Sin Ais the closed ideal {f∈C0(X)|f(0) = 0}, corresponding to
the closed subset {0} ⊆ X. It contains the function x7→ √xwhile Sdoes not, and
so Sis not closed.
4. Restriction
Regarding subunits as open subsets of an (imagined) base space, the idea of
restriction to such an open subset makes sense. For example, if Uis an open subset
of a locally compact Hausdorff space X, then any C0(X)-module induces a C0(U)-
module. This section shows that this restriction behaves well in any monoidal
category.
10 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Definition 4.1. A morphism f:A→Brestricts to a subunit s:S→Iwhen it
factors through λB◦(s⊗B).
A B
S⊗B I ⊗B
f
s⊗B
λB
As a special case, we can consider to which subunits identity morphisms re-
strict [11, Lemma 1.3].
Proposition 4.2. The following are equivalent for an object Aand subunit s:
(a) s⊗A:S⊗A→I⊗Ais an isomorphism;
(b) there is an isomorphism S⊗A'A;
(c) there is an isomorphism S⊗B'Afor some object B;
(d) the identity A→Arestricts to s.
Proof. Trivially (a) =⇒(b) =⇒(c). For (c) =⇒(d): because sis a subunit,
s⊗S⊗Ais an isomorphism, so if S⊗B'Athen also Ais an isomorphism by
Lemma 2.3. For (d) =⇒(a): if Afactors through s⊗A, then because sis a
subunit s⊗S⊗Ais an isomorphism, and hence so is s⊗Aby Lemma 2.3.
The following observation is simple, but effective in applications [17].
Lemma 4.3. Let s:S→Iand t:T→Ibe subunits in a firm category. If f
restricts to s, and grestricts to t, then f◦gand f⊗grestrict to s∧t.
Proof. Straightforward.
In particular, if Aor Brestrict to a subunit s, then so does any map A→B. It
also follows that restriction respects retractions: if e◦m= 1, then mrestricts to s
if and only if edoes.
Definition 4.4. Let sbe a subunit in a monoidal category C. Define the restriction
of Cto s, denoted by C|s, to be the full subcategory of Cof objects Afor which
s⊗Ais an isomorphism.
Proposition 4.5. If sis a subunit in a monoidal category C, then C|sis a core-
flective monoidal subcategory of C.
CC|s
>
The right adjoint C→C|s, given by A7→ S⊗Aand f7→ S⊗f, is also called
restriction to s.
Proof. First, if A∈C, note that S⊗Ais indeed in C|sbecause s⊗S⊗Ais
an isomorphism as sis a subunit. Similarly, C|sis a monoidal subcategory of C.
Finally, there is a natural bijection
C(A, B)'C|s(A, S ⊗B)
f7→ (s⊗f)◦(s⊗A)−1◦ρ−1
A
λB◦(s⊗B)◦g←[g
TENSOR TOPOLOGY 11
for A∈C|sand B∈C. So restriction is right adjoint to inclusion. For monoidality,
see [33, Theorem 5]; both functors are (strong) monoidal when C|shas tensor unit
Sand tensor product inherited from C.
Remark 4.6. The previous result motivates our terminology; a subunit sin Cis
precisely a subobject of Iwith the property that it may form the tensor unit of a
monoidal subcategory of C, namely C|s.
Example 4.7. Let Lbe a semilattice, regarded as a firm category as in Exam-
ple 3.2. For a subset U⊆Lwe define ↓U={x∈L|x≤ufor some u∈U}. Then
for s∈L, the restriction C|sis the subsemilattice ↓s=↓{s}.
Example 4.8. Let Lbe a frame. A subunit in Sh(L) is just an element s∈L, and
a morphism f:A⇒Brestricts to it precisely when A(x) = ∅for x6≤ s.
Example 4.9. Let Sbe a nondegenerate firm idempotent ideal of a nondegenerate
firm commutative ring R. Then FModR|Sis monoidally equivalent to FModS.
Proof. Send Ain FModR|Sto Awith S-module structure a·s:= as, and send an
R-linear map fto f. This defines a functor FModR|S→FModS. In the other
direction, a firm S-module B'B⊗SShas firm R-module structure (b⊗s)·r:=
b⊗(sr) because Sis idempotent, and if gis an S-linear map then g⊗SSis R-
linear. This defines a functor FModS→FModR|S. Composing both functors
sends a firm R-module Ato A⊗SS'A⊗RR'A, and a firm S-module Bto
B⊗SS'B.
Example 4.10. For any Hilbert C0(X)-module Aand subunit C0(U) induced by
an open subset U⊆X, the module A⊗C0(U) is isomorphic to its submodule
A|U={a∈A| ha|ai ∈ C0(U)}
again viewing C0(U) as a closed ideal of C0(X) via (1). Hence in HilbC0(X)a
morphism f:A→Brestricts to this subunit when hf(a)|f(a)i ∈ C0(U) for all
a∈A.
Restricting HilbC0(X)to this subunit thus gives the full subcategory of modules
Awith A=A|U. This is nearly, but not quite, HilbC0(U): any such module also
forms a C0(U)-module, but conversely there is no obvious way to extend the action
of scalars on a general C0(U)-module to make it a C0(X)-module. There is a so-
called local adjunction between HilbC0(X)|C0(U)and HilbC0(U), which is only an
adjunction when Uis clopen [14, Proposition 4.3].
Proof. Write S=C0(U). We first prove that A∈HilbC0(X)|Sif and only if
|a| ∈ C0(U) for all a∈A, where |a|2=ha, ai. On the one hand, if a∈Aand f∈S
then |a⊗f|(X\U) = |a||f|(X\U) = 0. Therefore |a| ∈ C0(U) for all a∈A⊗S.
Because A⊗S'Ais invertible, |a| ∈ C0(U) for all a∈A.
On the other hand, suppose that |a| ∈ C0(U) = 0 for all a∈A. We are to show
that the morphism A⊗S→Agiven by a⊗f7→ af is bijective. To see injectivity,
let f∈Sand a∈A, and suppose that af = 0. Then |a|·|f|=|af |= 0, so for all
x∈Ueither |a|(x) = 0 or f(x) = 0. So |a⊗f|(U) = 0, and hence a⊗f= 0. To
see surjectivity, let a∈A. Then |a|(x) = 0 for all x∈X\U. So a= lim afnfor an
approximate unit fnof S. But that means ais the image of lim a⊗fn.
Above we restricted along one individual subunit s. Next we investigate the
structure of the family of these functors when svaries.
12 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Definition 4.11. [21] Let Cbe a category and (E,⊗,1) a monoidal category.
Denote by [C,C] the monoidal category of endofunctors of Cwith F⊗G=G◦
F. An E-graded monad on Cis a lax monoidal functor T:E→[C,C]. More
concretely, an E-graded monad consists of:
•a functor T:E→[C,C];
•a natural transformation η: 1C⇒T(1);
•a natural transformation µs,t :T(t)◦T(s)→T(s⊗t) for all s, t in E;
making the following diagrams commute for all r, s, t in E.
T(t)◦T(s)◦T(r)
T(t)◦T(r⊗s)
T((r⊗s)⊗t)T(r⊗(s⊗t))
T(t⊗s)◦T(r)
µr,s ⊗1T(t)
µr⊗s,t
T(αr,s,t)
µr,s⊗t
1T(r)⊗µs,t
T(s)◦1C
T(s)T(1 ⊗s)
T(s)◦T(1)
η⊗1T(s)µ1,s
T(λs)
1C◦T(s)
T(s)T(s⊗1)
T(1) ◦T(s)
1T(s)⊗ηµs,1
T(ρs)
Theorem 4.12. Let Cbe a monoidal category. Restriction is a monad graded
over the subunits, when we do not identify monomorphisms representing the same
subunit. More precisely, it is an E-graded monad, where Ehas as objects monomor-
phisms s:SIin Cwith s⊗San isomorphism, and as morphisms f:s→t
those fin Cwith s=t◦f.
Proof. The functor E→[C,C] sends s:SIto (−)⊗S, and fto the natural
transformation 1(−)⊗f. The natural transformation ηE:E→E⊗Iis given by
ρ−1
E. The family of natural transformations µs,t : ((−)⊗S)⊗T→(−)⊗(S⊗T) is
given by α(−),S,T . Associativity and unitality diagrams follow.
We end this section by giving two characterisations of subunits in terms that
are perhaps more well-known. The first characterisation is in terms of idempotent
comonads.
Definition 4.13. Arestriction comonad on a monoidal category Cis a monoidal
comonad F:C→C:
•whose comultiplication δ:F⇒F2is invertible;
•whose counit ε:F→1Chas a monic unit component εI:F(I)I.
Proposition 4.14. Let Cbe a braided monoidal category. There is a bijection
between subunits in Cand restriction comonads on C.
TENSOR TOPOLOGY 13
Proof. If s:SIis a subunit, then F(A) = S⊗Adefines a comonad by Proposi-
tion 4.5. Its comultiplication is given by δA= (λS⊗A◦(s⊗S⊗A))−1, by definition
being an isomorphism. Its counit is given by εA=λA◦(s⊗A). Because ρI=λI,
its component εI=λI◦(s⊗I) = ρI◦(s⊗I) = s◦ρSis monic.
Conversely, if Fis a restriction monad, then εI:F(I)Iis a subobject of the
tensor unit. Writing ϕA,B :A⊗F(B)→F(A⊗B) for the coherence maps, and
ψA,B =F(σ)◦ϕB,A ◦σ:F(A)⊗B→F(A⊗B) for its induced symmetric version,
the insides of the following diagram commute:
F2(I⊗I)F(I⊗I)F(I⊗I)
F2(I⊗I)
F(F(I)⊗I)F(F(I)⊗I)
F(I)⊗F(I)F(I)⊗I
F(I)⊗εI
ϕF(I),I
F(ψI,I )
δ−1
I⊗I
δI⊗I
F(ψ−1
I,I )
εF(I⊗I)
εF(I)⊗I
ψ−1
I,I
But the long outside path is composed entirely of isomorphisms. Hence F(I)⊗εI
is invertible, and εIis a subunit.
These two constructions are clearly inverse to each other.
Remark 4.15. Monoidal comonads on Cform a category with morphisms of
monoidal comonads [48]. This category is monoidal as a subcategory of [C,C].
The monoidal unit is the identity comonad A7→ A. A subunit is a comonad
Fwith a comonad morphism λ:F⇒1Cwhose comultiplication is idempotent,
and such that λA:F(A)→Ais monic. But by coherence, the latter means that
εI=λI:F(I)Iis monic. It follows that subunits in Calso correspond bijec-
tively to subunits in [C,C] in the same sense as Definition 2.1, though we have
not strictly defined these since the latter category is not braided. See also [9,
Remark 2.3].
It also follows that restrictions monads automatically satisfy the Frobenius law
δ−1F◦F δ =F δ−1◦δF [27], matching the viewpoint in [29].
The second characterisation of subunits swe will give is in terms of the subcat-
egory C|s.
Definition 4.16. Let Cbe a monoidal category. A monocoreflective tensor ideal
is a full replete subcategory Dsuch that:
•if A∈Cand B∈D, then A⊗B∈D;
•the inclusion F:D→Chas a right adjoint G:C→D;
•the component of the counit at the tensor unit εI:F(G(I)) →Iis monic;
•F(B)⊗εIis invertible for all B∈D.
Proposition 4.17. Let Cbe a firm category. There is a bijection between ISub(C)
and the set of monocoreflective tensor ideals of C.
Proof. A subunit scorresponds to C|s, and a monocoreflective tensor ideal D
corresponds to εI. First notice that C|sis indeed a monocoreflective tensor ideal
14 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
by Proposition 4.5. Starting with s∈ISub(C) ends up with s◦λ:I⊗SI, which
equals squa subobject. Starting with a monocoreflective tensor ideal Dends up
with {A∈C|A⊗εIis invertible}. We need to show that this equals D. One
inclusion is obvious. For the other, let A∈C. If A⊗εI:A⊗F G(I)→A⊗Iis
invertible, then A'A⊗F(G(I)), and so A∈Dbecause Dis a tensor ideal.
We leave open the question of what sort of factorization systems are induced by
monocoreflective tensor ideals [13, 15].
5. Simplicity
Localisation in algebra generally refers to a process that adds formal inverses to
an algebraic structure [38, Chapter 7]. This section discusses how to localise all
subunits in a monoidal category at once, by showing that restriction is an example
of localisation in this sense.
Definition 5.1. Let Cbe a category and Σ a collection of morphisms in C. A
localisation of Cat Σ is a category C[Σ−1] and a functor Q:C→C[Σ−1] such
that:
•Q(f) is an isomorphism for every f∈Σ;
•for any functor R:C→Dsuch that R(f) is an isomorphism for all f∈Σ,
there exists a functor R:C[Σ−1]→Dand a natural isomorphism R◦Q'
R;
CC[Σ−1]
D
Q
R'
•precomposition (−)◦Q:C[Σ−1],D→[C,D] is full and faithful for every
category D.
Proposition 5.2. Restriction C→C|sat a subunit sis a localisation of Cat
{s⊗A|A∈C}.
Proof. Observe that S⊗(−) sends elements of Σ to isomorphisms because sis
idempotent. Let R:C→Dbe any functor making R(s⊗A) an isomorphism for
all A∈C. Define R:C|s→Dby A7→ R(A) and f7→ R(f). Then
ηA=R(ρA)◦R(s⊗A): R(s⊗A)→R(A)
is a natural isomorphism. It is easy to check that precomposition with restriction
is full and faithful.
The above universal property concerns a single subunit. We now move to local-
ising all subunits simultaneously.
Definition 5.3. A monoidal category is simple when it has no subunits but I.
In the words of Proposition 4.17, a category is simple when it has no proper
monocoreflective tensor ideals. Let us now show how to make a category simple.
Proposition 5.4. If Cis a firm category, then there is a universal simple category
Loc(C)with a monoidal functor C→Loc(C): any a monoidal functor F:C→D
TENSOR TOPOLOGY 15
into a simple category Dfactors through it via a unique monoidal functor Loc(C)→
D.
CLoc(C)
D
F
Proof. We proceed by formally inverting the collection of morphisms
Σ = {λA◦(s⊗A)|A∈C, s ∈ISub(C)}∪{A|A∈C}
To show that the localisation C[Σ−1] of Σ exists we will show that Σ admits a
calculus of right fractions [22]. Firstly, Σ contains all identities and is closed under
composition, since the composition of λA◦(A⊗t) and λA⊗T◦(A⊗T⊗s) is simply
λA◦(A⊗(s∧t)). It remains to show that:
•for morphisms s:A→Cin Σ and f:B→Cin C, there exist morphisms
t:P→Bin Σ and g:P→Ain Csuch that g◦s=t◦f;
• •
• •
f
s∈ΣΣ 3t
g
•if a morphism t:C→Din Σ and f, g :B→Cin Csatisfy t◦f=t◦g,
then f◦s=g◦sfor some s:A→Bin Σ.
It suffices to merely consider {λA◦(s⊗A)|A∈C, s ∈ISub(C)}by [20, Re-
mark 3.1]. The first, also called the right Ore condition, is satisfied by bifunctoriality
of the tensor:
S⊗A S ⊗B
I⊗A I ⊗B
A B
f
I⊗f
S⊗f
s⊗B
ρB
s⊗A
ρA
For the second, suppose that (s⊗B)◦f= (s⊗B)◦g. Then applying S⊗(−) and
using that S⊗sis invertible, it follows that S⊗f=S⊗g. But then
f◦λA◦(s⊗A) = λSB ◦(s⊗S⊗B)◦(S⊗f)
=λSB ◦(s⊗S⊗B)◦(S⊗g) = g◦λA◦(s⊗A),
so the second requirement is satisfied. As a result, C[Σ−1] exists; an easy con-
stuction may be found in [20]. It satisfies the universal property of localisation
on the nose. Moreover, the functor C→Loc(C) is monoidal because the class
Σ is closed under tensoring with objects of Cby construction [15, Corollary 1.4].
Finally, notice that Loc(C) is simple by construction.
16 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
6. Support
When a morphism frestricts to a given subunit s, we might also say that f‘has
support in’ s. Indeed it is natural to assume that each morphism in our category
comes with a canonical least subunit to which it restricts, which we may call its
support. But in general this requires extra structure.
Write Cfor the braided monoidal category whose objects are morphisms f∈C,
with f⊗gdefined as in C, tensor unit I, and a unique morphism f→gwhenever
(grestricts to s) =⇒(frestricts to s).
Definition 6.1. Asupport datum on a firm category Cis a functor F:C→L
into a complete lattice Lsatisfying
(2) F(f) = ^F(s): s∈ISub(C)|frestricts to s
for all morphisms fof C. A morphism of support data F→F0is one of complete
lattices G:L→L0with G◦F=F0.
Lemma 6.2. If F:C→Lis a support datum, and f, g morphisms in C:
•F(f) = V{F(A)|A∈C, f factors through A};
•F(f⊗g)≤F(f)∧F(g)for all f , g; so Fis colax monoidal.
This notion of support via objects is similar to that of [2, 40, 36].
Proof. For the first statement, it suffices to show that frestricts to a subunit s
iff it factors through some object Awhich does. But if ffactors through Athen
f=g◦A◦hfor some g, h and so if Arestricts to sso does f. Conversely if
f:B→Crestricts to sit factors over S⊗C, which always restricts to s.
For the second statement, Note that F(I)≤1 always, so colax monoidality
reduces to the rule above. But if frestricts to sthen so does f⊗g. Hence
F(f⊗g)≤F(f), and F(f⊗g)≤F(g) similarly.
Most features of support data follow from the associated map ISub(C)→L.
Proposition 6.3. Let Cbe a firm category and La complete lattice. Specifying a
support datum F:C→Lis equivalent to specifying a monotone map ISub(C)→
L.
Proof. In Cthere is a morphism s→tbetween subunits sand tprecisely when
s≤t. Hence any support datum restricts to a monotone map ISub(C)→L.
Conversely, let Fbe such a map and extend it to arbitrary morphisms by (2).
Both definitions of Fagree on subunits ssince a subunit restricts to another one
tprecisely when s≤t, so that F(s) = V{F(t)|s≤t}. Finally, for functoriality
suppose there exists a morphism f→gin C. If this holds then whenever g
restricts to sthen so does f, so that F(f)≤F(g).
This observation provides examples of support data. Recall that the free com-
plete lattice on a semilattice Lis given by its collection D(L) of downsets U=
↓U⊆Lunder inclusion, via the embedding x7→ ↓x[34, II.1.2].
Proposition 6.4. Any firm category Chas a canonical support datum, valued in
D(ISub(C)), given by
(3) supp0(f) = {s∈ISub(C)|frestricts to t=⇒s≤t}.
TENSOR TOPOLOGY 17
Moreover, supp0is initial: any support datum factors through it uniquely.
CD(ISub(C))
L
{si}
WF(si)
supp0
F
This generalises [2, 4, 5] from triangulated categories to firm ones.
Proof. Extend the embedding L→D(L) to a support datum via Proposition 6.3.
Initiality is immediate by freeness of D(L), with (3) coming from the description
of meets in terms of joins in a complete lattice.
Rather than require extra data, it would be desirable to define support internally
to the category. If Chas the property that ISub(C) is already a complete lattice
(or frame), then it indeed comes with a support datum given by the identity on
ISub(C). We may then define the support of a morphism as
supp(f) = ^s∈ISub(C)|frestricts to s.
Note that supp(f) = Wsupp0(f). It therefore follows from Proposition 6.4 that
supp also has a universal property: if ISub(C) is already a complete lattice, any
support datum Ffactors through supp via a semilattice morphism.
Example 6.5. Let Lbe a frame and consider Sh(L). A morphism f:A⇒Bhas
supp0(f) = ↓{t|A(t)6=∅}, and supp(f) = V{s|A(s)6=∅}.
Example 6.6. In HilbC0(X)the collection of subunits forms a frame, and each
morphism f:A→Bhas supp(f) = C0(Uf), where
Uf={x∈X| hf(a)|f(a)i(x)6= 0 for some a∈A}.
Letting Lbe the totally ordered set of cardinals below |X|, we may define another
support datum by F(f) = |Uf| ∈ L.
In the remaining sections we turn to categories coming with such an intrinsic
spatial structure. First, the following example shows that, even in case ISub(C) is
a frame, our notion of support differs from that of [2, Definition 3.1(SD5)] and [40,
Definition 3.2.1(5)]: without further assumptions, a support datum is only colax
monoidal.
Example 6.7. There is a firm category Cfor which ISub(C) is a frame but
supp(f)⊗supp(g)6= supp(f⊗g).
Proof. Let Qbe the commutative unital quantale with elements 0 ≤ε≤1, with
unit 1 and satisfying 0 = 0 ·0=0·ε=ε·ε. Then the frame of subunits is
ISub(Q) = {0,1}, and εsatisfies supp(ε) = 1 whereas supp(ε·ε) = 0.
7. Spatiality
In our main examples, the subunits satisfy extra properties over being a mere
semilattice, and they interact universally with the rest of the category. First, they
often satisfy the following property.
18 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Definition 7.1. A category is stiff when it is braided monoidal and
(4)
S⊗T⊗X T ⊗X
S⊗XX
s⊗T⊗X
S⊗t⊗Xt⊗X
s⊗X
is a pullback of monomorphisms for all objects Xand subunits s, t.
Any stiff category is firm: take X=Iand recall that pullbacks of monomor-
phisms are monomorphisms. More strongly, subunits often come with joins satis-
fying the following.
Definition 7.2. Let Cbe a braided monoidal category. We say that Chas univer-
sal finite joins of subunits when it has an initial object 0 whose morphism 0 →I
is monic, with X⊗0'0 for all objects X, and ISub(C) has finite joins such that
each diagram
(5)
S⊗T⊗X T ⊗X
S⊗X(S∨T)⊗X
is both a pullback and pushout of monomorphisms, where each morphism is the
obvious inclusion tensored with Xas in (4).
Lemma 7.3. Let Cbe braided monoidal with universal finite joins of subunits.
Then Cis stiff and ISub(C)is a distributive lattice with least element 0.
Proof. For stiffness, take t= 1Ito see that each morphism s⊗Xis monic. Then
since (s∨t)⊗Xis monic it follows easily that each diagram (4) is a pullback. By
assumption 0 →Iis indeed a subunit. Finally it follows from (5) with X=Rthat
subunits R, S, T satisfy (S∨T)∧R= (S∧R)∨(T∧R).
Example 7.4. Any coherent category Cforms a cartesian monoidal category with
universal finite joins of subunits.
Proof. Each partial order Sub(A) is a distributive lattice, and for subobjects S, T
Aeach diagram (5) with ∧replacing ⊗and X= 1 is indeed both a pushout and
pullback [35, A1.4.2, A1.4.3]. Moreover in such a category each functor X×(−)
preserves these pullbacks, since limits commute with limits, and preserves finite
joins and hence these pushouts since each functor (π2)∗: Sub(A)→Sub(X×A)
does so by coherence of C.
To obtain arbitrary joins of subunits from finite ones, it will suffice to also have
the following. Recall that a subset Uof a partially ordered set is (upward) directed
when any a, b ∈Uallow c∈Uwith a≤c≥b. A preframe is a semilattice in which
every directed subset has a supremum, and finite meets distribute over directed
suprema.
By a directed colimit of subunits we mean a colimit of a diagram D:J→C, for
which Jis a directed poset, all of whose arrows are inclusions SiSjbetween
TENSOR TOPOLOGY 19
a collection of subunits si:Si→I. In particular Dhas a cocone given by these
subunits, inducing a morphism colim D→Iif a colimit exists.
Definition 7.5. A stiff category Chas universal directed joins of subunits when
it has directed colimits of subunits, each of whose induced arrow colim S→Iis
again a subunit, and these colimits are preserved by each functor X⊗(−).
Lemma 7.6. If a stiff category Chas universal directed joins of subunits, then
ISub(C)is a preframe.
Proof. Any directed subset U⊆ISub(C) induces a diagram U→C, and its colimit
is by assumption a subunit which is easily seen to form a supremum of U. Taking
Xto be a subunit shows that ∧distributes over directed suprema.
Example 7.7. Any preframe L, regarded as a monoidal category under (∧,1), has
universal directed joins.
The rest of this section shows that the subunits of a category have a spatial
nature when it has both types of universal joins above. We unify Definitions 7.2
and 7.5 as follows. Let Cbe a braided monoidal category and U⊆ISub(C) a
family of subunits. For any object X, write D(U, X ) for the diagram of objects
S⊗Xfor s∈Uand all morphisms f:S⊗X→T⊗Xsatisfying (t⊗X)◦f=s⊗X.
If Cis stiff, there is a unique such ffor sand t.
S⊗X T ⊗X
X
s⊗Xt⊗X
Call such a set Uof subunits idempotent when U=U⊗U:= {s∧t|s, t ∈U}.
Definition 7.8. A category Cis spatial when it is stiff, ISub(C) is a frame, and the
canonical maps S⊗X→(WU)⊗Xform a colimit of D(U, X) for each idempotent
U⊆ISub(C) and X∈C.
Let us now see how this combines our earlier notions. In any poset P, an ideal
is a downward closed, upward directed subset. Let us call a subset U⊆Pfinitely
bounded when it has a finite set of maximal elements. If Uis downward closed then
equivalently it is finitely generated: U=↓{x1, . . . , xn}.
Proposition 7.9. A category Chas universal finite (directed) joins if and only if
ISub(C)has finite (directed) joins, and D(U, X)has colimit S⊗X→(WU)⊗X
for each idempotent U⊆ISub(C)that is finitely bounded (directed).
Proof. First consider finite joins. A colimit of D(∅, X ) is precisely an initial object
and the conditions on 0 in both cases are equivalent to 0 →Ibeing a subunit
with 0 ⊗X'0 for all X. Moreover in any stiff category it is easy to see that
cocones over the top left corner of (5) correspond to those over D(↓{s, t}, X). (See
also Lemma 8.1 below.) Hence the properties above provide each diagram with a
colimit (S∨T)⊗X, and so Cwith universal finite joins.
Conversely, suppose that Chas universal finite joins. For any idempotent U
we claim that any cocone csover D(U, X) extends to one over D(V, X), where
V={s1∨ ··· ∨ sn|si∈U}. Indeed for any s, t ∈Uthe following diagram
20 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
commutes, giving cs∨tas the unique mediating morphism.
S⊗T⊗X T ⊗X
S⊗X(S∨T)⊗X
C
ct
cs
cs∨t
Similarly define morphisms cs1∨···∨snfor arbitrary elements of V; these form a
cocone. Hence colim D(U, X ) = colim D(V , X). But if Uis bounded by some
s1, . . . , snthen clearly colim D(V, X) = (s1∨ · · · ∨ sn)⊗Xand we are done.
Next, consider directed joins. Let Dbe a directed diagram of inclusions between
elements of U⊆ISub(C). Then Umust be directed and therefore V={s1∧···∧sn|
si∈U}is idempotent and directed. Moreover, for each object X, any cocone cs
over D⊗Xextends to one over D(V, X ): for any s∈V, let s≤t∈Uand
set cs=ct◦(x⊗1X) where x:S→Tis the inclusion. Since R=WVhas
R⊗X= colim D(V, X ) then R⊗X= colim(D⊗X) as required.
Conversely, suppose Chas universal directed joins. Then ISub(C) is a preframe
by Lemma 7.6. If U⊆ISub(C) is directed and idempotent then for each Xwe
have R⊗X= colim D(U, I )⊗X, where R=WU. But any cocone over D(U, X)
certainly also forms one over D(U, I)⊗X, and so R⊗X= colim D(U, X ) also.
Corollary 7.10. A category is spatial if and only if it has universal finite and
directed joins of subunits.
Proof. Proposition 7.9 proves one direction. In the other direction, suppose C
has universal finite and directed joins of subunits. Then ISub(C) is a frame by
Lemmas 7.3 and 7.6, since a poset is a frame precisely when it is a preframe and
a distributive lattice. Let U⊆ISub(C) be idempotent. Then V={s1∨ ·· · ∨ sn|
si∈U}is idempotent by distributivity, as well as directed, so that colimD(V , X) =
(WV)⊗Xexists for any X. But colim D(U, X ) = colim D(V , X) as in the proof of
Proposition 7.9.
The previous corollary justifies saying that a category simply has universal joins
of subunits when it is spatial. The rest of this section shows that our main examples
are spatial.
Example 7.11. Any commutative unital quantale Qis spatial when regarded as
a category as in Example 3.5; in particular so is any frame under tensor ∧. Indeed
that example showed that ISub(Q) is a frame, and for any U⊆ISub(Q) and x∈Q
we have colim D(U, x) = Ws∈Usx = (Ws∈Us)x.
Example 7.12. Any cocomplete Heyting category Cis spatial under cartesian
products. This includes all cocomplete toposes, such as Grothendieck toposes.
Proof. Since a Heyting category is coherent, it has universal finite joins by Exam-
ple 7.4, with each change of base functor having a right adjoint and so preserv-
ing arbitrary joins of subobjects. In any cocomplete regular category with this
property, for any directed diagram Dand any cocone Cover Dall of whose legs
are monic, the induced map colim D→Cis again monic [24, Corollary II.2.4].
TENSOR TOPOLOGY 21
Hence whenever Uis directed, so is each map colimD(U, X )→X, ensuring that
colim D(U, X) = Ws∈Us×Xis in Sub(X). Since each functor X×(−) now pre-
serves arbitrary joins of subobjects furthermore Ws∈Us×X= colim D(U, I )×X,
establishing universal directed joins.
Next we consider Hilbert modules. In general HilbC0(X)is finitely cocomplete
but not cocomplete, and so lacks directed colimits by [42, IX.1.1]; this follows
from [1, Example 2.3 (9)] by taking Xto be trivial and so reducing to the category
of Hilbert spaces and contractive linear maps. Nonetheless, we have the following.
Example 7.13. HilbC0(X)is spatial.
Proof. Throughout this proof we again identify C0(U) with the submodule (1) of
C0(X), and identify the module A⊗C0(U) with A|U, for open U⊆X.
First let us show that HilbC0(X)has universal finite joins of subunits. For open
subsets U, V ⊆X, and any Hilbert C0(X)-module A, consider the diagram of
inclusions between A|U∩V,A|U,A|Vand A|U∪V. It is easily seen to be a pullback,
since A|U∩V=A|U∩A|Vas subsets of A. We verify that it is also a pushout.
Since any morphism AU∪V→Brestricts to C0(U∪V), it suffices to assume that
X=U∪V. We claim that
C0(U) + C0(V) = {gU+gv∈C0(X)|gU∈C0(U), gV∈C0(V)}
is a dense submodule of C0(X). To see this, let g∈C0(X) and ε > 0, and K
be compact with |g(x)| ≥ =⇒x∈K. Urysohn’s lemma for locally compact
Hausdoff spaces [46, 2.12] produces h∈C0(U) such that |h(x)| ≤ |g(x)|for x∈U
and h(x) = g(x) for x∈K\V. Then |(g−h)(x)| ≥ 2ε=⇒x∈Lfor some
compact L⊆K∩V. Again there is k∈C0(V) with |k(x)|≤|g(x)|for all x∈V
and k(x) = (g−h)(x) for x∈L. By construction kg−h−kk ≤ 4ε, establishing
the claim. It follows also that
A|U+A|V={aU+aV|aU∈A|U, aV∈A|V}
is dense in A, since A·C0(X) = {a·g|g∈C0(X)}is so too [41, p5].
Now suppose fU:A|U→Band fV:A|V→Bagree on A|U∩V. Then for
a=aU+aVwith aU∈A|Uand aV∈A|V, the assignment
f(a) = fU(aU) + fV(aV)
is a well-defined A-linear map. Hence it extends to a unique map f:A→Bwhich
is by definition the unique factorisation of fUand fVthrough the diagram.
Now we must check that fis contractive when fUand fVare. Let x∈X, and
without loss of generality say x∈U. Urysohn’s lemma again produces g∈C0(U)
with g(x) = 1 = kgk. Now a·g∈A|Ufor any a∈A. So, writing |a|2(x) for
|ha|ai(x)|, we find
|f(a)|(x) = |f(a)·g|(x)≤ kf(a)·gk=kfU(a)·gk≤kakkgk≤kak
using kfUk ≤ 1. Since xwas arbitrary, also kfk ≤ 1.
Next, let us consider universal directed joins of subunits. For this, let Wbe a
directed family of open sets in X; again it suffices to assume X=SW. We claim
that [
U∈W
C0(U) = {g∈C0(X)|g∈C0(U) for some U∈W}
22 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
is a dense submodule of C0(X). Again let g∈C0(X) and ε > 0, and let Kbe
compact with |g(x)| ≥ =⇒x∈K. Since Kis compact and Wis directed,
K⊆Ufor some U∈W. Urysohn again provides h∈C0(U) with |h(x)|≤|g(x)|
for all x∈Uand h(x) = g(x) for x∈K. Then |g−h|(x)≤ |g(x)|+|h(x)| ≤ 2εfor
x∈X\Kand so, since gand hagree on K, we have kg−hk ≤ 2ε, establishing
the claim. Similarly, for any Hilbert module A, since A·C0(X) is dense in A, so is
SU∈WA|U.
Finally, let fU:A|U→Bbe a cocone over D(W, A). It suffices to show that
there is a unique f:A→Bwith f(a) = fU(a) for all a∈A|U. But any a∈Ahas
a= lim(an)∞
n=1 with each an∈A|Unfor some Un. By directedness we may assume
Un⊆Un+1 for all n. Then f:A→Bmust satisfy f(a) = limfUn(an), making f
unique. Additionally, this limit is always well-defined since anis a Cauchy sequence
and so for n≤m:
kfUn(an)−fUm(am)k=kfUm(an−am)k≤kan−amk
and fUn(an) is also a Cauchy sequence. Clearly fis A-linear and kfk ≤ 1.
8. Universal joins from colimits
This section characterises each of the notions of universal joins purely categori-
cally, without order-theoretic assumptions on ISub(C). Instead, they will be cast
solely in terms of the diagrams D(U, X). When we turn to completions in the next
sections, we can therefore use the diagrams D(U, X) themselves as formal joins to
add.
Lemma 8.1. Let Cbe a stiff category. If U⊆ISub(C)is idempotent, then any
cocone over D(U, X)extends uniquely to one over D(↓U, X ).
Therefore, Chas colimits of D(U, X) for all downward-closed U⊆ISub(C) if
and only if it has them for idempotent U.
Proof. Let Ube idempotent and consider a cocone cs:S⊗X→Xover D(U, X).
Let r∈ ↓U, say r=s◦ffor s∈Uand f:R→S. Define cr=cs◦(f⊗X) : R⊗X→
X. This is clearly the only possible extension of csto D(↓U, X). We will prove
that it is a well-defined cocone. Suppose r0∈ISub(C) satisfies r0≤s0for s0∈U,
and r⊗X= (r0⊗X)◦g. Then the marked morphism in the following diagram is
an isomorphism:
R⊗XR0⊗X
R⊗R0⊗X
S⊗XS⊗S0⊗X S0⊗X
X
g
r⊗R0⊗X
'
R⊗r0⊗X
S⊗s0⊗X s ⊗S0⊗X
cscs0
The upper triangle and central squares commute trivially. The lower quadrilateral
commutes and equals cs⊗s0because s⊗s0∈Uand cis a cocone. Hence the outer
diagram commutes, showing cr=cr0◦gas required. In particular, taking R0=R
shows that cris independent of the choice of s.
TENSOR TOPOLOGY 23
Lemma 8.2. Let Cand Dbe stiff categories, U⊆ISub(C)be idempotent, and
cs:S⊗X→Cbe a cocone over D(U, X). If a functor F:C→Dpreserves
monomorphisms of the form s⊗XX, for subunits s, and the pullbacks (4),
then F(cs)is a cocone over DF(U), F (X), where F(U) = {F(s)|s∈U}.
Proof. Clearly, if s⊗X≤t⊗Xthen F(s⊗X)≤F(t⊗X), and F(cs) respects
the inclusion. Conversely, suppose that F(s⊗X)≤F(t⊗X) via some morphism
f, and consider the following diagram.
F(S⊗T⊗X)F(T⊗X)
F(S⊗X)F(C)
F(X)
F(s⊗T⊗X)
F(S⊗t⊗X)F(ct)
F(cs)
F(t⊗X)
F(s⊗X)
f
The outer rectangle commutes by bifunctoriality, and F(t⊗X)◦f=F(s⊗X) by
assumption. Hence the upper left triangle commutes because F(t⊗X) is monic
by stiffness and the assumption on F. The inner square commutes and is equal
to F(cs⊗t) by definition of D(U, X). Since the outer rectangle is a pullback, the
leftmost vertical morphism is invertible and hence F(ct)◦f=F(cs).
Now suppose a diagram D(U, X) has a colimit cX
s:S⊗X→colim D(U, X)
for each idempotent U⊆ISub(C) and object X. Then there are two canonical
morphisms. First, a mediating map colim D(U, I)→Ito the cocone s:S→I.
(6)
colim D(U, I)
I
S
s
cI
s
Second, in a stiff category it follows from applying Lemma 8.2 to (−)⊗Xthat
there is a unique map making the following triangle commute for all s∈U:
(7)
S⊗X
(colim D(U, I)) ⊗X
colim D(U, X)
cX
s
cI
s⊗X
If Chas universal joins of Uthen WU= colim D(U, I ) and (6) is monic, and (7)
is invertible by definition. We now set out to prove the converse.
Lemma 8.3. Let Cbe a stiff category, and let U⊆ISub(C)be idempotent. Suppose
that D(U, X)has a colimit for each object Xand that each morphism (7) is an
isomorphism. If the morphism colim D(U, I)→Iof (6) is monic, then it is a
subunit.
24 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Proof. Write sUfor this morphism, which is monic by assumption. For each s∈U,
we claim S⊗sU:S⊗colim D(U, I)→Sis an isomorphism. It is monic because
sU◦cs◦(S⊗sU) = s⊗sU=sU◦s⊗colim D(U, I)
where sUand s⊗colim D(U, I) are monic by stiffness. But it is also split epic since
(S⊗sU)◦(S⊗cs) = S⊗sis an isomorphism.
Now since s◦(S⊗sU) = sU◦(s⊗colim D(U, I)), bifunctoriality of ⊗shows that
for all s, t ∈U:
s⊗colim D(U, I)≤t⊗colim D(U, I)⇐⇒ s≤t
This gives an isomorphism of diagrams S⊗sU:S⊗colim D(U, I)→Sfrom
DU, colim D(U, I)to D(U, I ). Writing cs:S→colim D(U, I) for the latter col-
imit, cs⊗colim D(U, I) is a colimit for the former by assumption. Hence the unique
map making the following square commute
S⊗colim D(U, I)S
colim D(U, I)⊗colim D(U, I) colim D(U, I)
S⊗sU
cs⊗colim D(U, I)cs
is invertible. But this map is just colim D(U, I )⊗sU, so sUis a subunit.
We can now characterise spatial categories purely categorically.
Theorem 8.4. A stiff category Cis has universal (finite, directed) joins if and
only if for each idempotent (and finitely bounded, directed) U⊆ISub(C):
•the diagram D(U, X)has a colimit;
•the canonical morphism (6) is monic;
•the canonical morphism (7) is invertible.
Proof. The conditions are clearly necessary, as already discussed. Conversely, sup-
pose that they hold and let U⊆ISub(C) be as above. Lemma 8.1 lets us assume
U=↓U. Then sU: colim D(U, I )→Iis a subunit by Lemma 8.3, and by defini-
tion s≤sUfor all s∈U. Now suppose that tis also an upper bound in ISub(C)
of all s∈U. Then the inclusions is,t :S→Tform a cocone over D(U, I). Hence
there is a unique mediating map f: colim D(U, I)→Twith is,t =f◦cI
sfor all
s∈U. But then
t◦f◦cI
s=t◦is,t =s=sU◦cI
s
for all s∈U. Because the cI
sare jointly epic, t◦f=sU, so that sU≤t. There-
fore indeed colim D(U, I) = WU. Thus universal finite or directed joins follow by
Proposition 7.9, and so arbitrary ones by Corollary 7.10.
9. Completions
Our goal for this section is to embed a stiff category Cinto one with any given
kind of universal joins of subunits, including a spatial category. One might think to
work with the free cocompletion of C, the category of presheaves b
C= [Cop,Set].
Here, b
Cis endowed with the Day convolution b
⊗as tensor; for details see Appen-
dix A. Although b
Chas a complete lattice of subunits, we will see that it has two
problems: it is in general not firm, and it has too many subunits to be the spatial
TENSOR TOPOLOGY 25
completion. We will remedy both problems by passing to a full subcategory of
so-called broad presheaves.
First, note that any subunit sin a firm category Cinduces a subunit s◦(−): C(−, S)→
C(−, I) in b
Csince the Yoneda embedding is monoidal, full, and faithful, and pre-
serves all limits and hence monomorphisms.
Proposition 9.1. If Cis a cocomplete regular category, and for all objects Athe
functors A⊗(−)preserve colimits, then ISub(C)is a complete lattice. Thus, if C
is any braided monoidal category, then ISub( b
C)is a complete lattice.
Proof. In cocomplete regular categories, the subob jects of a fixed object form a
complete lattice [7, Proposition 4.2.6]. Explicitly, let si:SiIbe a family of
subunits. Choose a coproduct ci:Si→C. The unique mediating map C→I
factors through a monomorphism Wsi:SI, which is the supremum.
SiSj
C
S
I
cicj
sisj
e
s
Next we show that Wsiis a subunit. Let c=s◦e:C→I. We claim that
C⊗CC
`iSi⊗C
C⊗c
'`i(Si⊗c)
is a regular epimorphism. Since colimits commute with colimits, it suffices to check
that each Si⊗cis a regular epimorphism. But this is so: if Si⊗c=m◦f
for some regular epimorphism fand monomorphism m, then m◦f◦(Si⊗ci) =
(Si⊗c)◦(Si⊗ci) = Si⊗siis an isomorphism by idempotence of si, so that mis
split epic as well as monic and hence an isomorphism.
Now the topmost two rectangles in the following diagram commute.
Si
C
S
II⊗I
S⊗S
C⊗C
Si⊗Si
Si⊗si
C⊗c
λS◦(S⊗s)
λI
s
e
s⊗s
e⊗e
cici⊗ci
si⊗si
si
The left and right triangles commute by construction, and the bottom rectangle
commutes by bifunctoriality of the tensor and naturality of λ. Because eis a
coequaliser, so are C⊗eand e⊗S, and hence so is e⊗e. Therefore both vertical
morphisms factor as regular epimorphisms followed by monomorphisms, and the
26 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
mediating morphism, which must be λS◦(S⊗s) by uniqueness, is an isomorphism.
Thus S⊗sis an isomorphism, as required.
The second statement now follows, because b
Cis regular and cocomplete, and
the functors Fb
⊗(−) are cocontinuous [32].
However, the subunits in b
Care in general not well behaved.
Example 9.2. Consider the commutative monoid M= [0,1) ×[0,∞) under
(a, b)+(c, d) = ((a+c, b +d) if a+c < 1
(a+c−1, b +d+ 1) if a+c≥1
with unit (0,0). Then Mis a firm one-object category, but c
Mis not firm.
Proof. The identity (0,0) represents the only subunit of the one-object category
M, which is therefore firm. Appendix Aproves that c
Mis not firm.
Moreover, b
Cmay have subunits that are not suprema of subunits of C.
Remark 9.3. In general ISub( b
C) is not the free frame on ISub(C).
Proof. Consider a commutative unital quantale Qas a firm category. By their
description in Appendix A, any subunit in b
Qis given by a suitable downward
closed subset S⊆ ↓e⊆Qsuch that ∀x∈S∃y, z ∈S:x≤yz, and to be a subunit
it suffices for Sto be directed.
In particular, take Q= [0,∞] under the opposite order and addition. Then
ISub(Q) = {0,∞}, whose free completion to a frame is its collection of downsets
∅,{∞},{0,∞}. However, by the above description of subunits in b
Qit is easy to
see that ISub( b
Q)⊇∅,{∞},[0,∞],(0,∞].
Instead, to complete ISub(C) to a distributive lattice, preframe, or frame, we
will consider certain full subcategories of b
C.
Definition 9.4. A presheaf on a braided monoidal category Cis (finitely, direct-
edly) broad when it is naturally isomorphic to one of the form
hU, Xi:A7→ {f:A→X|frestricts to some s∈U}
for a (finitely bounded, directed) family Uof subunits and an object X.
Write b
Cbrd (b
Cfin,b
Cdir) for the full subcategory of (finitely, directedly) broad
presheaves. We will also write b
Ufor hU, Ii, and b
Xfor h{1}, Xi.
We will see below that the broad presheaves are precisely the colimits of the
diagrams D({ˆs|s∈U},ˆ
X), and leave open the possibility of characterising when
a given presheaf is broad in terms not referring to Uor X.
The following lemma shows that broad presheaves are closed under (Day) tensor
products and so form a monoidal category.
Lemma 9.5. For any objects X,Yand families of subunits U,Vin a stiff category
C, there is a (unique) natural isomorphism making
(8)
hU, Xib
⊗hV, Y i
b
Xb
⊗b
Y
hU⊗V, X ⊗Yi
\
X⊗Y
ub
⊗v
'
'
TENSOR TOPOLOGY 27
commute, where U⊗V={s∧t|s∈U, t ∈V}, and u, v are the inclusions.
Proof. See Appendix A.
We now describe the subunits in each completion.
Proposition 9.6. If Cis stiff, the subunits in b
Cbrd (b
Cfin,b
Cdir) are the presheaves
of the form b
Ufor (finitely bounded, directed) U⊆ISub(C).
Proof. Clearly b
Uis a subunit. Conversely, if η:hU, Xi → b
Iis a subunit then
sX=ηS⊗X(s⊗X): S⊗X→I
will be proven to be a subunit in Cfor each s∈U.
Given this, let U0={sX|s∈U}, noting that c
U0again belongs to each respective
category, and consider the function hU, X i(A)→ hU0, Ii(A) given by ((s⊗X)◦f)7→
sX◦f. It is surjective by definition of U0, clearly natural, and is well-defined and
injective since
sX◦f=s0
X◦f0⇐⇒ η(s⊗X)◦f=η(s0⊗X)◦f0
⇐⇒ η((s⊗X)◦f)) = η((s0⊗X)◦f0)
⇐⇒ (s⊗X)◦f= (s0⊗X)◦f0
by naturality and injectivity of η.
Let us show that sXis indeed a subunit. By stiffness of Ceach morphism (s⊗X)
is monic, and so by the above argument sXis, too.
Next we show sX⊗S⊗Xis invertible. Notice that hU, X i=h↓U, X i, so we
may assume that Uis idempotent. The fact that ηis a subunit means precisely
that each map
hU, X ⊗Xi(A)→ hU, X i(A)(∗)
(s⊗(X⊗X)) ◦f7→ (sX⊗X)◦f(9)
is a well-defined bijection, where f:A→S⊗X⊗Xand s∈U.
Now note that S⊗sX⊗Xis monic, since by injectivity of (∗), sX⊗Xis
monic, and it is easy to see from stiffness that for any subunit sand monic mthat
S⊗mis again monic. Moreover it is split epic and hence an isomorphism, since by
surjectivity of (∗) there is some fwith (sX⊗X)◦f=s⊗X, and S⊗(s⊗X) is
always split epic by idempotence of s.
For any semilattice, as well as its downsets forming its free completion to a frame,
recall that its free completion to a preframe is given by its collection of directed
downsets [51, Theorem 9.1.5]; and that its free completion to a distributive lattice
is given by its finitely bounded downsets [34, I.4.8], with (directed, finite) joins
given by unions.
Corollary 9.7. The subunits in b
Cfin,b
Cdir, and b
Cbrd, are the free completion of
ISub(C)to a distributive lattice, preframe, and frame, respectively.
Proof. For any U, V ⊆ISub(C) it is easy to see that b
U≤b
V⇐⇒ U≤ ↓ V. In
particular b
U=c
↓Uas we have already noted. Hence by Proposition 9.6, subunits
in each category correspond to the respective kinds of downset U⊆ISub(C).
Next let us note that each of our constructions are again stiff.
28 PAU ENRIQUE MOLINER, CHRIS HEUNEN, AND SEAN TULL
Lemma 9.8. If a monoidal category Cis stiff, then so are b
Cdir,b
Cfin and b
Cbrd.
Proof. For any object hU, X iand subunit V:b
V→b
Iin b
Cbrd we need to show
that the morphism hU, Xi ⊗ Vis monic. This holds since the obvious morphism
hU, Xi ⊗ b
V→b
Xfactors over it, and is itself monic by equation (8) of Lemma 9.5.
By the same result, for the pullback property we must show each diagram
hU⊗V⊗W, Xi hU⊗W, X i
hV⊗W, Xi hW, X i
to be a pullback in b
Cbrd. For this it suffices to check that applying the diagram
to each object Ayields a pullback in Set, or equivalently that any morphism
f:A→Xfactoring over u⊗w⊗Xand v⊗w0⊗Xfor some u∈U, v ∈Vand
w, w0∈Wfactors over u0⊗v0⊗w00 ⊗Xfor some u0∈U, v0∈V, w00 ∈W. But this
follows easily from the pullbacks (4) taking u0=u,v0=vand w00 =w∧w0, again
for convenience assuming Wto be idempotent.
The next lemma shows that b
Cbrd formally adds to Cthe colimits of the diagrams
D(U, X) for all suitable U⊆ISub(C) and objects X.
Lemma 9.9. Let Cbe firm, and let U, V ⊆ISub(C)be idempotent. Morphisms
α:hU, Xi→hV , Y iof broad presheaves correspond to cocones cs:S⊗X→Yover
D(U, X)for which each csrestricts to some t∈V.
Proof. Given αand s∈U, by naturality we may define such a cocone by cs=
αS⊗X(s⊗X). Conversely, given a cocone as above define
αA(s⊗X)◦g=cs◦g
for each g:A→S⊗X. This is clearly natural and is well-defined; indeed if
(s⊗X)◦g= (t⊗X)◦hthen since (4) is a pullback this morphism factors as
(s⊗t⊗X)◦kfor some k, then with cs◦g=cs∧t◦k=ct◦hsince the (cs) form a
cocone. Clearly these two assignments are inverses.
Finally we can prove that our free constructions have the desired properties.
Theorem 9.10. If Cis a stiff category, then:
•b
Cfin has universal finite joins of subunits;
•b
Cdir has universal directed joins of subunits;
•b
Cbrd is spatial.
Proof. Consider the final statement first. Lemma 9.8 makes b
Cbrd stiff. Let Ube
an idempotent family of subunits in b
Cbrd. By Proposition 9.6, its elements are of
the form b
Ufor some U⊆ISub(C). Also, its supremum in ISub( b
Cbrd) is given by
hSU, Iiwhere we write SU=S{U|b
U∈ U}.
Let V⊆ISub(C), and let Ybe an object in C. We have to prove that the
inclusions b
Ub
⊗hV, Y i → SUb
⊗hV, Y iare a colimit of the diagram