Theory and Applications of Categories, Vol. 29, No. 26, 2014, pp. 775–780.
DIAGRAMMATIC CHARACTERISATION OF
ENRICHED ABSOLUTE COLIMITS
RICHARD GARNER
Abstract. We provide a diagrammatic criterion for the existence of an absolute colimit in the context of enriched category theory.
An absolute colimit is one preserved by any functor; the class of absolute colimits was
characterised for ordinary categories by Paré [4] and for enriched ones by Street [5]. For
categories enriched over a monoidal category V or bicategory W, the appropriate colimits
are the weighted colimits of [6], and Street’s characterisation is in fact one of the class
of absolute weights: those weights ϕ such that ϕ-weighted colimits are preserved by any
functor. This is different to Paré’s result, which gives a diagrammatic characterisation of
when a particular cocone is absolutely colimiting. In this note, we give a result in the
enriched context which is closer in spirit to Paré’s than to Street’s. This result is very
useful in practice, but seems not to be in the literature; we set it down for future use.
1. The result
1.1. Background. We work in the context of bicategory-enriched category theory;
see [6], for example. W will denote a bicategory whose homs are locally small, complete
and cocomplete categories, and which is biclosed, meaning that for each 1-cell A : x → y in
W, the composition functors A⊗(–) : W(z, x) → W(z, y) and (–)⊗A : W(y, z) → W(x, z)
have right adjoints [A, –] and hA, –i respectively.
A W-category A comprises a set ob A of objects; for each a ∈ ob A an object
a ∈ ob W, the extent of a; for each pair of objects a, b, a hom-object C(b, a) ∈ W(a, b);
and identity and composition 2-cells ι : Ia → C(a, a) and µ : C(c, b)⊗C(b, a) → C(c, a) satisfying the expected axioms. A W-profunctor M : A −7→ B is given by objects M (b, a) ∈
W(a, b) and action maps µ : B(b0 , b) ⊗ M (b, a) ⊗ A(a, a0 ) → M (b0 , a0 ) satisfying unitality and associativity axioms. A profunctor map M → M 0 : A −7→ B comprises maps
M (b, a) → M 0 (b, a) compatible with the actions by A and B. The identity profunctor
A : A −7→ A has components A(b, a) with action given by composition in A. For profunctors M : A −7→ B and N : B −7→ C with B small, the tensor product N ⊗B M : A −7→ C
The support of Australian Research Council Discovery Projects DP110102360 and DP130101969 is
gratefully acknowledged.
Received by the editors 2014-09-30 and, in revised form, 2014-10-29.
Transmitted by Ross Street. Published on 2014-10-31.
2010 Mathematics Subject Classification: 18A30, 18D20.
Key words and phrases: Absolute colimits, enriched categories.
c Richard Garner, 2014. Permission to copy for private use granted.
775
776
RICHARD GARNER
has components given by coequalisers
P
P
0
0
b,b0 N (c, b) ⊗ B(b, b ) ⊗ M (b , a) ⇒
b N (c, b) ⊗ M (b, a) (N ⊗B M )(c, a)
and actions by C and A inherited from N and M . Small W-categories, profunctors and
profunctor maps comprise a bicategory W-Mod. There is a full embedding W → W-Mod
sending X to the W-category X with one object ? with (?) = X and X(?, ?) = IX .
If A and B are W-categories, then a W-functor F : A → B comprises an extentpreserving assignation on objects, together with 2-cells C(b, a) → D(F b, F a) subject to
two functoriality axioms. If F : A → C and G : B → C are W-functors then there is an
induced profunctor C(G, F ) : A −7→ B with components C(G, F )(b, a) = C(Gb, F a) and
action derived from the action of F and G on homs and composition in C.
Given profunctors M : A −7→ B, N : B −7→ C and L : A −7→ C with B small, a profunctor
map u : N ⊗B M → L is said to exhibit M as [N, L] if every map f : N ⊗B K → L is of
the form u ◦ (N ⊗B f¯) for a unique f¯: K → M ; while it is said to exhibit N as hM, Li if
every f : K ⊗B M → L is of the form u ◦ (f¯ ⊗B M ) for a unique f¯: K → N .
Given ϕ : A −7→ B in W-Mod and a functor F : B → C, a ϕ-weighted colimit of F is
a functor Z : A → C and profunctor map a : ϕ → C(F, Z) such that for each C ∈ C, the
map
µ
a⊗A 1
ϕ ⊗A C(Z, C) −−−
→ C(F, Z) ⊗A C(Z, C) −−−→ C(F, C)
(1)
exhibits C(Z, C) as [ϕ, C(F, C)]. A functor G : C → D preserves this colimit just when the
composite ϕ → C(F, Z) → D(GF, GZ) exhibits GZ as a ϕ-weighted colimit of GF ; the
colimit is absolute when it is preserved by all functors out of C. [5] proves that ϕ-weighted
colimits are absolute if and only if ϕ admits a right adjoint in W-Mod.
Dually, given ψ : B −7→ A in W-Mod and a functor F : B → C, a ψ-weighted limit of
F is a functor Z : A → C and map b : ψ → C(Z, F ) such that for each C ∈ C, the map
1⊗ b
µ
A
C(C, Z) ⊗A ψ −−−
→ C(C, Z) ⊗A C(Z, F ) −−−→ C(C, F )
exhibits C(C, Z) as hψ, C(C, Z)i. Absoluteness of limits is defined as before; now every
limit weighted by ψ : B −7→ A is absolute if and only if ψ has a left adjoint in W-Mod.
1.2. Theorem. Let ϕ : A −7→ B admit the right adjoint ψ : B −7→ A in W-Mod, and let
F : B → C and Z : A → C be W-functors. There is a bijective correspondence between
data of the following forms:
(a) A map a : ϕ → C(F, Z) exhibiting Z as a ϕ-weighted colimit of F ;
(b) A map b : ψ → C(Z, F ) exhibiting Z as a ψ-weighted limit of F ;
(c) Maps a : ϕ → C(F, Z) and b : ψ → C(Z, F ) such that the following two squares
commute in W-Mod(A, A) and W-Mod(B, B):
η
A
Z
ψ ⊗B ϕ
C(Z, Z) o
/
µ
b⊗B a
C(Z, F ) ⊗B C(F, Z)
ϕ ⊗A ψ
a⊗A b
/
ε
C(F, Z) ⊗A C(Z, F )
µ
/
B
F
C(F, F ) .
(2)
DIAGRAMMATIC CHARACTERISATION OF ENRICHED ABSOLUTE COLIMITS
777
Proof . Suppose first given (a); consider the composite profunctor map
a⊗ 1
µ
A
ϕ ⊗A C(Z, F ) −−−
→ C(F, Z) ⊗A C(Z, F ) −−−→ C(F, F ) .
(3)
Evaluating in the second variable at any a ∈ A yields the map (1) exhibiting C(Z, F a)
as [ϕ, C(F, F a)]; it follows easily that (3) exhibits C(Z, F ) as [ϕ, C(F, F )]. Applying this
universality to the composite ε ◦ F : ϕ ⊗A ψ → B → C(F, F ) yields a unique map b : ψ →
C(Z, F ) making the right square of (2) commute; we must show that the left one does
too. Arguing as before shows that
a⊗ 1
µ
A
ϕ ⊗A C(Z, Z) −−−
→ C(F, Z) ⊗A C(Z, Z) −−−→ C(F, Z)
(4)
exhibits C(Z, Z) as [ϕ, C(F, Z)]. It thus suffices to show that the left square of (2) commutes after applying the functor ϕ ⊗A (–) and postcomposing with (4); which follows by
a short calculation using commutativity in the right square and the triangle identities.
So from the data in (a) we may obtain that in (c), and the assignation is injective,
since b is uniquely determined by universality of a and commutativity on the right of (2).
For surjectivity, suppose given a and b as in (c); we must show that a exhibits Z as a ϕweighted colimit of F , in other words, that for each C ∈ C, the map (1) exhibits C(Z, C) as
[ϕ, C(F, C)], or in other words, that for each map f : ϕ ⊗A K → C(F, C), there is a unique
map f¯: K → C(Z, C) such that f = µ◦(a⊗A f¯) : ϕ⊗A K → C(F, Z)⊗A C(Z, C) → C(F, C).
For existence, we let f¯ be the composite
η⊗A 1
b⊗B f
µ
K∼
− C(Z, C) ;
= A ⊗A K −−−→ ψ ⊗B ϕ ⊗A K −−−→ C(Z, F ) ⊗B C(F, C) →
(5)
now rewriting with the right-hand square of (2) and using the triangle identities and F ’s
preservation of units shows that f = µ ◦ (a ⊗A f¯). For uniqueness, let g : K → C(Z, C)
also satisfy f = µ ◦ (a ⊗A g). Substituting into (5) shows that f¯ is the composite
η⊗A 1
b⊗B a⊗A g
µ
K∼
− C(Z, C) ;
= A ⊗A K −−−→ ψ ⊗B ϕ ⊗A K −−−−−→ C(Z, F ) ⊗B C(F, Z) ⊗A C(Z, C) →
which by rewriting with the left square of (2) and using Z’s preservation of identities is
equal to g. This proves the equivalence (a) ⇔ (c); now (b) ⇔ (c) follows by duality.
1.3. Examples. We first consider examples wherein W is the one-object bicategory corresponding to a monoidal category V.
• Let V = Set, and let ϕ be the weight for splittings of idempotents. The result
recovers the bijection, for an idempotent e : A → A, between: maps p : A → B
coequalising e and 1A ; maps i : B → A equalising e and 1A ; and pairs (i, p) with
pi = 1A and ip = e.
• Let V = Set∗ , and let ϕ be the weight for an initial object. The result recovers
the bijection in a pointed category between: initial objects; terminal objects; and
objects X with 1X = 0X .
778
RICHARD GARNER
• Let V = Ab, and let ϕ be the weight for binary coproducts. The result recovers the
bijection, for objects A, B in a pre-additive category, between: coproduct diagrams
i1 : A → Z ← B : i2 ; product diagrams p1 : A ← Z → B : p2 ; and tuples (i1 , i2 , p1 , p2 )
such that pj ik = δik and i1 p1 + i2 p2 = 1Z .
W
• Let V =
-Lat, and let ϕ be the weight for J-fold coproducts (for J a small
set). The result recovers the bijection, for objects (Aj : j ∈ J) in a sup-lattice
enriched category, between: coproduct diagrams (ij : Aj → Z)j∈J ; product
W diagrams
(pj : Z → Aj )j∈J ; and families (ij )j∈J and (pj )j∈J with pj ik = δjk and j ij pj = 1Z .
• Let V = k-Vect for k a field of characteristic zero, let G be a finite group, and let
ϕ : k −7→ kG be the trivial right kG-module k. By Burnside’s Lemma, ϕ has a right
adjoint kG −7→ k given by the trivial left kG-module k. Now the result recovers the
bijection, for a G-representation A in a k-linear category, between: maps p : A → Z
exhibiting Z as an object of coinvariants of A; maps i : Z → A exhibiting Z as an
1
Σg∈G g.
object of invariants of A; and pairs of maps (i, p) with pi = 1Z and ip = |G|
We conclude with two examples where W is a genuine bicategory.
• Let (C, j) be a subcanonical site, and let W denote the full sub-bicategory of
Span(Sh(C))op on objects of the form C(–, X). To any prestack p : E → C over C, we
may (as in [1]) associate a W-category with objects those of E, extents (a) = p(a),
and hom-object from a to b given by the span C(–, pa) ← E(a, b) → C(–, pb) in
Sh(C); here E(a, b)(x) is the set of all triples (f, g, θ) with f : pa ← x → pb : g in C
and θ : f ∗ (a) → g ∗ (b) in Ex (note that E(a, b) is a sheaf by the prestack condition).
For any cover (fi : Ui → U )i∈I in C, we have a W-category R[f ] with object set I,
extents (i) = Ui and hom-objects R[f ](j, i) = C(–, Uj ) ← C(–, Uj ×U Ui ) → C(–, Ui ).
There is a profunctor ϕ : U −7→ R[f ] with components given by the spans ϕ(i, ?) =
C(–, Ui ) ← C(–, Ui ) → C(–, U ). Writing ψ : R[f ] −7→ U for the reverse profunctor, it
is not hard to see that ϕ a ψ (in fact they are adjoint pseudoinverse).
The result now says the following. Given a prestack p : E → C, a cover (fi : Ui → U )
in C, and a family of spans pij : ai ← aij → aj : qij in E whose legs are cartesian
over the projections Ui ← Ui ×U Uj → Uj , there is a bijection between: cocones
(hi : ai → a) in E over the fi ’s that are colimiting for the diagram comprised of the
pij ’s and qij ’s; universal objects a ∈ EU equipped with vertical maps fi∗ (a) → ai
fitting into double pullback squares
fi∗ (a) o
ai o
/
·
pij
aij
qij
fj∗ (a)
/ aj
;
and objects a ∈ EU equipped with a family of maps (hi : ai → a) cartesian over the
fi ’s. This generalises [6, Proposition 5.2(b)]1 .
1
The proposition numbering here is taken from the TAC reprint.
DIAGRAMMATIC CHARACTERISATION OF ENRICHED ABSOLUTE COLIMITS
779
• Let W denote the bicategory whose objects are sets, and whose hom-category
W(X, Y ) is the category of finitary functors Set/Y → Set/X; note that W(X, Y ) '
[Fam(Y ) × X, Set], where Fam(Y ) has as objects, finite lists of elements of Y , and
as maps (y0 , . . . , ym ) → (z0 , . . . , zn ), functions f : [m] → [n] such that yi = zf (i) . To
any cartesian multicategory M (i.e., a Gentzen multicategory in the sense of [3]) we
may associate a W-category M whose objects of extent X are X-indexed families
of objects of M , and whose hom-object between families (ax )x∈X and (by )y∈Y is the
presheaf
M((by ), (ax ))(y0 , . . . , ym ; x) = M (by0 , . . . , bym ; ax )
in [Fam(Y ) × X, Set]; reindexing along maps in Y makes use of the cartesianness of
the multicategory structure. Composition and units in M follow from those in M .
Given a finite set X = {x0 , . . . , xn }, let ϕ : 1 −7→ X be the W-profunctor whose
unique component is the representable y(x0 , . . . , xn ; ?) ∈ [Fam(X)×1, Set]. This has
a right adjoint ψ : X −7→ 1 whose unique component is the presheaf Σx∈X y(?; x) ∈
[Fam(1) × X, Set]. The result now establishes a bijection, for any finite family
(a0 , . . . , an ) of objects in a cartesian multicategory M , between data of the following three forms: first, an object a and a multimap i ∈ M (a0 , . . . , an ; a), composition with which induces bijections between M (b0 , . . . , bk , a, c0 , . . . , c` ; d) and
M (b0 , . . . , bk , a0 , . . . , an , c0 , . . . , c` ; d); second, an object a and unary maps pj ∈
M (a; aj ), composition with which establishes bijections between M (b0 , . . . , bk ; a)
and Πj M (b0 , . . . , bk ; aj ); third, an object a and maps i and pj as above such that
pj ◦ i = πj ∈ M (a0 , . . . , an ; aj ) and i ◦ (p0 , . . . , pn ) = 1a ∈ M (a; a). This generalises
[2, Proposition 3.5].
References
[1] Betti, R., Carboni, A., Street, R., and Walters, R. Variation through
enrichment. Journal of Pure and Applied Algebra 29, 2 (1983), 109–127.
[2] Garner, R. Lawvere theories, finitary monads and Cauchy-completion. Journal of
Pure and Applied Algebra 218, 11 (2014), 1973–1988.
[3] Lambek, J. Multicategories revisited. In Categories in computer science and logic
(Boulder, 1987), vol. 92 of Contemporary Mathematics. American Mathematical Society, 1989, pp. 217–239.
[4] Paré, R. On absolute colimits. Journal of Algebra 19 (1971), 80–95.
[5] Street, R. Absolute colimits in enriched categories. Cahiers de Topologie et
Geométrie Différentielle Catégoriques 24, 4 (1983), 377–379.
[6] Street, R. Enriched categories and cohomology. Quaestiones Mathematicae 6
(1983), 265–283. Republished as: Reprints in Theory and Applications of Categories
14 (2005).
780
RICHARD GARNER
Department of Mathematics, Macquarie University, NSW 2109, Australia
Email: richard.garner@mq.edu.au
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