LOCALISATION AND QUOTIENT CATEGORIES
ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
Contents
1. Localisation of Categories
1.1. Category of Fractions
1.2. Calculus of Fractions
2. Quotient Category
3. Relation between Localisation and Quotienting
4. Symmetric Monoidal Categories
4.1. Internalisation
4.2. Enriched Categories
References
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Different notions in mathematics are invariant with respect to some class of morphisms between the studied
objects, for instance most algebraic topological invariants are invariant under homotopy equivalences. In such
situation, the homotopy category with respect to that class of morphisms becomes the category of main interest,
as the natural framework to consider such notions. The homotopy category of a pair (C , S) of a category C
and a class S of its morphisms is, roughly, the ‘minimal’ category obtained from C in which morphisms of S
are inverted. The homotopy category of the pair (C , S) is referred to by the localisation of C with respect to
S.
On the other hand, when one is interested in studying mathematical morphisms up-to certain equivalent
relation between parallel morphisms, it is natural to identify the related morphisms. That gives rise to the
quotient category with respect to the relation in consideration, as a ‘minimal’ category that identify related
morphisms. For instance, identifying homotopic continuous maps between topological spaces gives rise to the
classical homotopy category of topological spaces, not to be confused with the homotopy category of topological
spaces. The former is the quotient category of topological spaces with respect to homotopy relation, whereas
the latter is the localisation of the category of topological spaces with respect to homotopy equivalences.
Although the two notions of localisation and quotienting are priorly different, it turns out that they are
related, and in ‘nice’ situation they give rise to equivalent categories. For instance the classical homotopy
category of CW−complexes is equivalent to the localisation of CW−complexes with respect to weak homotopy
equivalences.
In this file, we recall the notions of localisation, and quotienting, their relations, and related notions. This
treatment is mainly based on [3], [4], [6], [7], and [11] .
1. Localisation of Categories
Localisation of categories is motivated and was given rise to by the more traditional notions of localisations
in algebra and topology, [12], and it is useful to first recall these notions. However, eager readers can skip to
localisation of categories.
Recall that in different branches of mathematics, and particularly in algebraic geometry, some questions can
be considered locally at prime ideals (points of schemes). Then, one can ask the local-to-global question if the
local solutions can be glued together into a global solution. Problems that can be attacked this way are said to
Date: August 16, 2015.
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ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
satisfy the local-to-global principle, also known as Hasse principle. The part of the story that we are concerned
with here is the local consideration, known as localisation.
Localisation of Rings. Let R be a unitary ring, and let S be a multiplicative subset in R. Then, the localisation
of R away from S, if exists, is an initial universal morphism of unitary rings
ϕS ∶ R → R[S −1 ]
that sends elements of S to invertible elements. To be more precise, call an R−algebra R → A S−local if the
structure morphism send elements of S to invertible elements in A. Then, ϕS is a universal morphism from
idR ∶ R → R to the inclusion of the full subcategory of S−local R−algebras in R−Alg, if exists, see [7, §.III.1].
Notice that the definition of initial universal property translates precisely to the requirement of R[S −1 ] being
S−local and ϕS being S−equivalence, where a morphism of R− algebras f ∶ A → B is called an S−equivalence if
the pre-composition with f induces a bijection
f ∗ ∶ R−Alg(B, C) Ð→ R−Alg(A, C)
for every S−local R−algebra C.
Recall that the category of (small) unitary rings and their morphisms is equivalent to the category of (small)
Ab−monoids and their additive functors
Ring Ð→ Ab−Mon.
A unitary ring R is sent to the Ab−monoid R with one object ∗ and the ring of endomorphisms EndR (∗) = R,
i.e. every element x of R is an endomorphism x ∶ ∗ → ∗ in R. The composition of morphisms in R is given by
multiplication of the original ring, whereas the Ab−enriched structure of R is given by the underlying abelian
group structure of the ring R. Then, a morphism x ∶ ∗ → ∗ is an isomorphism in R if and only if the element x
is invertible in R.
The localisation morphism
ϕS ∶ R → R[S −1 ],
if exists, is sent by the above equivalent of categorises to an additive functor of Ab−monoids
ϕ S ∶ R → R[S −1 ],
where R[S −1 ] is the image of the ring R[S −1 ] along the above equivalence. In fact, ϕ S is also a universal
morphism, analogue to the one above. To see that, notice that ϕ S is an R ↓ Ab−monoid, i.e. object in in the
R ↓ Ab−Mon, let S denote the class of morphisms in R that corresponds to the elements in S, and call an
R ↓ Ab−monoid S−local if it sends morphism of S to isomorphisms. Since additive functors between Ab−monoid
are completely determined by the ring morphisms between the rings of endomorphisms of their unique object,
then ϕ S is a universal morphism from idR to the inclusion of the full subcategory of S−local R ↓ Ab−monoid in
the category R ↓ Ab−Mon.
Localisation of Modules. Similar, but more interesting, argument applies to localisation of left R−modules M
away from a multiplicative subset S ⊂ R, for a unitary ring R, if exists. This localisation is defined to be the
initial universal morphism
ϕS ∶ M → M [S −1 ]
from M to the inclusion of the full subcategory of S−local left R−modules in R−Mod, if exists, where a left
R−module M ′ is called S−local if the R−linear maps induced by the action of elements of R
x×−∶
M′
m
→
↦
M′
xm
are isomorphisms for every x ∈ S. In other words, it is an S−equivalence from M to an S−local left R−module
M [S −1 ], where an R−linear map f ∶ M → N is called S−equivalence if the pre-composition with f induces a
bijection
f ∗ ∶ R−Mod(N, L) Ð→ R−Mod(M, L)
for every S−local left R−module L.
Recall that when R is commutative then the localisation of rings R → R[S −1 ] exists, and it is isomorphic
(as R−modules) to the localisation of R away from S as a left R−module. In general, there is a isomorphism
of R−modules M [S −1 ] ≅ M ⊗R R[S −1 ] for every R−module M , and under the above assumption that R is
LOCALISATION AND QUOTIENT CATEGORIES
3
commutative, there exists a localisation of every left R−module away from M , given by the canonical R−linear
map M → M [S −1 ] ≅ M ⊗R R[S −1 ]. The tensor functor
− ⊗R R[S −1 ] ∶ R−Mod Ð→ R−Mod
sends R−linear maps x ⋅ − ∶ M → M to isomorphism for every x ∈ S and M ∈ R−Mod, and hence its image
is contained in the full subcategory of S−local R−modules. Since the full subcategory of S−local R−modules in
R −Mod is isomorphic to the category of R[S −1 ]−modules, then the above tensor functor factorises into the
pre-composition of the inclusion functor
US ∶ R[S −1 ]−Mod Ð→ R−Mod.
with a functor
ϕ S ∶ R−Mod Ð→ R[S −1 ]−Mod.
Theorem [7, §.IV.1.Th.2] implies that the functor ϕ S is a left adjoint of US , and hence it is a reflector of
R−Mod in R[S −1 ]−Mod. Notice that the components of the unit of this adjunction are isomorphic to ϕM
for every M ∈ R−Mod. Since R[S −1 ] ⊗R R[S −1 ] ≅ R[S −1 ] then the adjunction monad US ϕ S = − ⊗R R[S −1 ] is
2−idempotent. Also, since R[S −1 ] is a flat module over R, i.e. − ⊗R R[S −1 ] is exact, then ϕ S is also an exact
functor.
Call a category under R −Mod S−local if the R−linear maps (x ⋅ −)M ∶ M → M are sent to isomorphisms
by the structure functors for every M ∈ R−Mod and x ∈ S. Since US is the inclusion functor, and the tensor
functor − ⊗R R[S −1 ] sends (x ⋅ −)M to isomorphisms for every M ∈ R−Mod and x ∈ S, then so does the functor
ϕ S , and hence ϕ S ∶ R−Mod → R[S −1 ]−Mod is an S−local category under R−Mod. In fact, it is 2−universal
S−local category under R−Mod, and is called the localisation functor of R−modules away from S, and it sends
an R−module M to its localisation M [S −1 ] ≅ M ⊗R R[S −1 ] away from S.
Observe the difference between the two cases of localisation of rings and localisation of R−modules. On
contrary to the case of rigs, the multiplicative subset is fixed for the base ring and hence for all its modules,
then the above localisation functor localises all R−modules at ones, when it exists.
Also, notice that for any monoidal functor F ∶ R−Mod → C into a monoidal category C , F sends the R−linear
maps (x ⋅ −)M to isomorphism for every x ∈ S and M ∈ R−Mod if and only if it sends R−linear maps (x ⋅ −)R to
isomorphism for every and for every x ∈ S. To see that, recall that the monoid of endomorphisms of the unitary
object of a monoidal category is commutative and acts on the monoids of endomorphisms of other objects in
the category. R is the unit object of R−Mod, and for every f ∈ EndR−Mod (R) and M ∈ R−Mod, one has an
endomorphism
M ⊗f ∶
M
ρM
/ M ⊗R
idM ⊗f
/ M ⊗R
ρ−1
M
/ M.
In fact (x ⋅ −)M ∶ M → M coincides with M ⊗ (x ⋅ −)R . Since left unit morphisms ρM is an isomorphism for every
M ∈ R−Mod, then the monoidal functor F sends (x ⋅ −)M to isomorphisms for every x ∈ S and M ∈ R−Mod if
and only if it sends (x ⋅ −)R to isomorphisms for every x ∈ S.
Localisation of Topological Spaces. A similar theory of localisation for ‘nice’ (pointed topological homotopy)
types was developed by Sullivan in 1970, see [11]. All the homotopy groups, homology groups, generalised
cohomology groups and other algebraic invariants of some ‘nice’ types are naturally educed with abelian group
structures, i.e. Z−modules. Then, those invariants can be localised away from multiplicative subsets of Z.
Localising those algebraic invariants simplifies some problems in homotopy theory, like computing the homotopy
groups of n−spheres. The main idea of localising types is to seek a type that is ‘close enough’ for the given one
and whose algebraic invariants are the localisations of those of the original type; and then attack the problem
in consideration for the localised types. The words nice and close will be made precise in the below argument.
For more on the motivation for the localisation of topological types with respect to homological functors, see
for instance the thread of Mike Shulman’s question on mathoverflow.net.
The existence of a canonical, unique, abelian group structure on the set of path-connected components of
a type depends on the cardinality of the set. The canonical choice of that cardinality that guarantees the
existences of a unique such structure is one! Hence, one restricts attention to path-connected types whose
fundamental groups are abelian, in order to have all homotopy groups being Z−modules. We will restrict our
attention further when recalling Sullivan’s results. However, there exists other constructions of localisation of
more general types, like nilpotent spaces. We call types that satisfy the above assumptions ‘abelian’ spaces, not
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ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
to be confused with abelian spaces in [9, p.131]. ‘Abelian’ spaces are general enough so the above localisation
question makes sense. Denote the full subcategory of ‘abelian’ spaces in hTop by AS.
Let l be a set of primes, not necessary non-empty, denote the multiplicative subset of Z generated by prime
numbers that are not in l by S, and let Zl ∶= Z[S −1 ]. Then, an ‘abelian’ space is called l−local if all its homotopy
groups are Zl−modules, and a morphism f ∶ X → Y in AS is called l−equivalence if the pre-composition with f
induces a bijection
f ∗ ∶ AS(Y, Z) Ð→ AS(X, Z)
for every l−local ‘abelian’ space Z. Then, the l−localisation of X ∈ AS, or the localisation of X at l, is defined
to be an l−equivalence
ϕ l ∶ X → Xl
for l−local ‘abelian’ space Xl , if exists. That is evidently equivalent to ϕ l being a universal morphism from X
to the inclusion of the full subcategory of l−local ‘abelian’ spaces in AS.
When X is an ‘abelian’ space of a homotopy type of a CW−complex whose fundamental group has a trivial
canonical action on the homotopy and homology groups, then the l−localisation ϕ X ∶ X → Xl exists, and Xl has
the desired l−localised homotopy and reduced homology groups, i.e.
π∗ (Xl ) ≅ π∗ (X) ⊗ Zl
and
̃∗ (Xl ) ≅ H
̃∗ (X) ⊗ Zl
H
̃∗ (−) is the functor of reduced integral homology groups. In fact, under the above assumptions, having
where H
either of the above two isomorphisms is equivalent to the morphism X → Xl being an l−localisation of X, see
[11, Th.2.1]. ‘Abelian’ spaces that satisfy the above additional assumptions are called ‘simple’ spaces. Denote
the full subcategory of ‘simple’ spaces in hTop by SS. For a ‘simple’ space X, the construction of the l−localised
space Xl shows that it is a ‘simple’ space. Let Ul be the inclusion of the full subcategory of l−local ‘simple’
spaces SSl in SS. Then, by theorem [7, §.IV.1.Th.2], there exists an adjunction
Ll ∶ SS ⇄ SSl ∶ Ul
with Ll (X) = Xl , ∀X ∈ SS, i.e. the l−localisation is a reflector of SS in SSl ; and the components of the unit of
the adjunction η ∶ idSS → Ul Ll are isomorphic to ϕ X for X ∈ SS. Since Zl ⊗ Zl ≅ Zl then the adjunction monad
Ul Ll is 2−idempotent.
Notice that a morphism f ∶ X → Y in SS induces an isomorphism between l−localised reduced homology
groups if and only if fl ∶= Ll (f ) ∶ Xl → Yl induces an isomorphism between reduced homology groups, or
equivalently be a homotopy equivalence, i.e. an isomorphism in SS, and hence in SSl . That is due to ϕ X
inducing isomorphisms on l−localised reduced homology groups for every X ∈ SS, chasing the unit diagram for
a morphism f , and applying theorem [11, Th.2.1].
Recall that for every integer d there exists an endomorphism fn,d ∶ S n → S n in SS that is of degree d for
every integer n ≥ 0, see [5, Ex.2.31]. For every prime p ∉ l and integer n ≥ 0, the degree map fn,p induces an
isomorphism of localised reduced cohomologies
̃∗ (S n ) ⊗ Zl Ð→ H
̃∗ (S n ) ⊗ Zl .
fn,d ∗ ∶ H
Then, by [11, Th.2.1], Ll (fn,d ) are isomorphisms in SSl . In fact, Ll is a 2−universal functor sending the degree
maps f2,d to isomorphisms for every p ∉ l, see [2, §.1.E.2]. Call a category under SS l−S n−local if the structure
functor sends the degree maps fn,d to isomorphisms for every p ∉ l, then in particular Ll ∶ SS → SSl is l−S 2−local.
Under the above assumption on SS, a category under SS is l−S 2−local if and only if it is l−S 1−local.
The above notations and results are based on the cellular construction of localisation introduced in Sullivan’s
lecture notes. Since our aim from recalling localisation of topological types here is to motivate the concept of
localising categories, we wont continue further with this review. However, it is worth mentioning that different
notions of localisation of more general topological types have been developed, and those notions are not always
equivalent outside nilpotent spaces, see for instance [8, Part.2].
LOCALISATION AND QUOTIENT CATEGORIES
5
Localisation of Groups. l−localising an abelian group G, considered as Z−module, away from a multiplicative
subset of Z inverts degree maps for prime degrees that are not in l. The degree maps for abelian additive groups
as just a special case of the power maps for multiplicative groups. Then, similarly, for a set of prime numbers
l, one calls a multiplicative group (not necessary abelian) l−local when the p−power maps are isomorphisms for
p ∉ l, and call a homomorphism of groups f ∶ G → H l−equivalence if the pre-composition with f induces a
bijection
f ∗ ∶ Grp(H, Z) Ð→ Grp(G, Z)
for every l−local group. Then, the l−localisation of group G away from l is defined to be a universal morphisms
Ll ∶ G → Gl
from G to the inclusion of the full subcategory of l−local groups in Grp, if exist. Then, in particular, Ll ∶ G → Gl
is an l−equivalence of groups.
Also, for a set of prime numbers l, a categories under Grp is called l−local if its structure functor sends
p−power maps −p ∶ G → G to isomorphisms for every group G and p ∉ l. Then, the l−localisation functor for
groups is defined to be the 2−universal l−local category under Grp, if exists.
Localisation of Categories. The localisation of categories generalises the above notions of localisations. The
main idea underlying the above arguments is being able to universally invert a class of morphisms in a category
associated to the objects in consideration. The eager reader can find the definition in 1.1. Localisation of
categories can be given in the language of categories (1−categories). However, in order to see how the definition
fits into the bigger picture, and to see how it honestly generalise the above mentioned notions of localisations,
one needs to view it as 2−universal 1−morphisms.
We start with an argument that ‘naturally’ motivate the definition in terms of 2−universal 1−morphisms,
Def.1.1. Then, we expand the definition and recover the more common definition in terms of 1−categories.
Let C be a category, S a class of morphisms in C . A functor F ∶ C → D is called S−local if it sends morphisms
in S to isomorphisms in D. Then, one priorly wants to define the localisation of C with respect to S to be the
initial universal S−local functor
LS ∶ C Ð→ C [S −1 ],
if exists. Then the ‘category’ C [S −1 ] is then isomorphic to the category of fractions of C with respect to S,
recalled below in 1.1. The construction of the category of fractions has several disadvantages. They are mainly
due to its size and to the nature of its morphisms, that it might be big enough not to be a category in given
foundations, and its morphisms are formal and hard to work with compared to those of C . On the other
hand, one rarely obtains isomorphisms of categories in every day mathematics, and it is rather equivalence of
categories what one usually in interested in. Since universal morphisms are defined up-to isomorphisms, and
strict 2−universal 1−morphisms are defined up-to 1−equivalences. Then, in order to relax the above suggested
notion of localisation, and obtain a notion up-to equivalence of categories, on can define the localisation functor
LS ∶ C Ð→ C [S −1 ] using to be the initial strict 2−universal S−local functor, if exists.
To be precise, strict 2−universal morphisms are defined from objects in a strict 2−category to a strict 2−functor,
and hence one needs to realise the property of functors being S−local in terms of the existence of a 1−morphisms
to an suitable strict 2−functor.
Let CAT2 be the very large strict 2−category of categories, functors, and natural transformations; let iS ∶
S ↪ C be the subcategory of C generated by S, and observe that a functor F ∶ C → D sends morphisms in S
- and hence morphisms in S - to isomorphisms if and only if there exists a strict 2−commutative squares
S
SqF ∶
−op

iS
e
x
S op
∼
j
/C
F
/D
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ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
in CAT2 , i.e. if there exists a natural isomorphism e ∶ F iS ⇒ j−op . That, on the one hand, the existence of
such natural isomorphism induces a commutative square
F (X)
eX
/ j(X op )
O
j(sop )
F (s)
F (Y )
eY
/ j(Y op )
in D whose horizontal morphisms are isomorphisms, for every morphism s ∶ X → Y in S , in particular for
op
those in S. That in turn implies that F (s) is an isomorphism with an inverse e−1
X j(s )eY , for every morphism
s ∶ X → Y in S . On the other hand, assume that F sends morphisms in S to isomorphisms and define the
−1
functor j ∶ S op → D by j(X op ) = F (X) for X op ∈ S op and j(sop ) = F (s) for sop in S op . Then, there exists
op
a natural isomorphism e ∶ F iS ⇒ j− given by eX = idX for every X ∈ S , and hence the strict 2−commutative
squares SqF exists. It is evident that j in SqF is unique up-to natural isomorphism.
Let ●←●→● be the span small category, and consider the very large strict 2−category CAT2●←●→● of strict 2−functors
from ●←●→● to CAT2 , their pseudo-natural transformations, and the modifications of the latter; i.e. objects of
●←●→● to CAT2 are spans of functors of large categories
U o
G
T
H
/V ,
its 1−morphisms between objects U ← T → V and U ′ ← T ′ → V ′ are triples of functors (U ∶ U → U ′ , T ∶ T →
∼
∼
T ′ , V ∶ V → V ′ ) and pairs of natural isomorphisms (φ ∶ U G ⇒ G′ T, ψ ∶ V H ⇒ H ′ T )
U o
U
U′ o
G
∼
T
φ
/V
ψ
T
G′
H
% y
T′
∼
H′
V
/ V ′,
its 2−morphisms between 1−morphisms ((U, T, V ), (φ, ψ)) and ((U ′ , T ′ , V ′ ), (φ′ , ψ ′ )) are triples of natural transformations (u ∶ U ⇒ U ′ , t ∶ T ⇒ T ′ , v ∶ V ⇒ V ′ ) that respects evident composition of natural transformations,
i.e.
−1
−1
H ′ t = ψ ⋅ vH ⋅ ψ ′
, and
G′ t = φ ⋅ uG ⋅ φ′
U o
T
G
/V
H
′
∼φ
U′
U o
V′
T
G
/V
H
ψ′
U
u
t
U′ o
G′
∼φ
U′ o
G′
T′
∼
ψ
V
∼
$ z
T′
H′
v
/V′
T
$ z
T′
H′
/ V ′.
There is a diagonal strict 2−functor
∆ ∶ CAT2 Ð→ CAT●←●→●
2
sending a category D to the span Did←D Did→D D, a functor F ∶ D → E to the triple of functors (F, F, F ) and pair
of identity natural isomorphisms (idF , idF ), and a natural transformation e ∶ F ⇒ G ∶ D → E to the triple of
natural transformations (e, e, e).
LOCALISATION AND QUOTIENT CATEGORIES
7
Notice that a functor F ∶ C → D fits into a strict 2−commutative square SqF if and only if it fits into a
op
1−morphism ((G, H, F ), (φ, ψ)) from the span S op −← S i→S C to ∆(D), and hence F inverts morphisms of S if and
only if fits into such 1−morphism. Notice that the choice of G and H in the above 1−morphism to ∆(D), if
exists, is unique up-to 2−isomorphisms. However, the choice of φ, ψ is not necessary unique. To ease notion,
when ψ = φ = idF iS , and hence H = F iS and G = F iS −op , we refer to the above 1−morphism by F .
Definition 1.1 (Localisation of Categories). Let C be a category, S a class of morphisms in C . Then, the
localisation of C with respect to S, if exists, is defined to be a strict 2−universal 1−morphism
LS ∶ C → C [S −1 ]
from the span
−op iS
S op ← S → C
to the strict 2−functor
∆ ∶ CAT2 Ð→ CAT●←●→●
2
where iS ∶ S ↪ C is the subcategory of C generated by S.
Since MapCAT2 (U , V ) ∶= Funct(U , V ), then a functor LS ∶ C → C [S −1 ] is the localisation of C with respect
to S if and only if the pre-composition with LS induces an equivalence of categories
(S op −←
L∗S ∶ Funct(C [S −1 ], D) ≅ MapCAT●←●→●
2
op
given by
L∗S (G)
iS
S →C
, ∆(D)) ,
= GLS , being for every category D.
The above definition of the localisation LS means in particular that it fits into a strict 2−push-out square of
op
the span S op −← S i→S C in the very large strict 2−category CAT2 . Strict 2−colimits are unique up-to 1−equivalence,
if exists, and hence the localisation is unique up-to equivalence of categories, if exists.
Recall that L∗S is an equivalence of categories if and only if it is essentially surjective and fully faithful.
The functor L∗S is essentially surjective if and only if every functor F ∶ C → D that sends morphisms of S to
isomorphisms 2−factorise through LS , i.e. factorises up-to natural isomorphism. On the other hand, L∗S is fully
faithful if and only if the pre-composition with natural isomorphism idL induces a bijection of sets
L∗S G,H ∶ Funct(C [S −1 , D)(G, H) ≅ MapCAT2●←●→● (S op −←
op
iS
S →C
, ∆(D)) (GLS , HLS ) ≅ Funct(C , D)(GLS , HLS )
for every category D and for every pair of functors G, H ∶ C [S −1 ] → D. That in turn is equivalent to the
pre-composition with L,
L∗S ∶ Funct(C [S −1 ], D) → Funct(C , D),
being fully faithful for every category D.
Therefore, the above definition is equivalent to definition one induces from proposition [3, §.1.3], and to the
definition [6, Def.7.1.1]. That is the localisation of C with respect to S, if exists, is a functor LS ∶ C → C [S −1 ]
such that
● For every functor F ∶ C → D that sends morphisms of S to isomorphisms there exists a functor
GF ∶ C [S −1 ] → D and a natural isomorphism φ ∶ GF LS ≅ F .
● The functor given by pre-composition with L,
L∗S ∶ Funct(C [S −1 ], D) → Funct(C , D),
is fully faithful for every category D.
If the localisation LS ∶ C → C [S −1 ] exists, the by definitions it sends morphisms of S to isomorphisms.
However, it may as well send morphisms that are not in S to isomorphisms. We define the saturation of S to be
the multiplicative kernel of LS , if exists, i.e. the subcategory of C whose morphisms are precisely morphisms
of C sent to isomorphisms by LS . In particular the saturation of S contains all isomorphisms of C .
Recall that the class of isomorphisms satisfy the two-out-of-six-property in any category, i.e. given three
f
g
h
composable morphisms X → Y → Z → W in the category, if gf, hg are in the class then so is f, g, h, and hgf . It
is evident that the saturation of S satisfies the two-out-of-six-property in C . In fact, it is easy to see that the
saturation of S is the smallest subcategory of C that contains S and satisfies the two-out-of-six-property in C .
Lemma 1.2 (Reflective Localisations). Let C be a category, S be a class of morphisms in C , and LS ∶ C Ð→
C [S −1 ] be a localisation of C with respect to S. If LS admits a right adjoint US , then the following conditions
are equivalent
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ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
● US is fully faithful.
● The adjunction counit is a natural isomorphism.
Proof. See [3, §.I.Prop.1.3].
Localisations that satisfy either of the equivalent conditions of the above lemma are said to be reflective,
as they are reflectors from C [S −1 ] to a subcategory of C . Then, in particular one has C [S −1 ] fully faithfully
embedded in C . Not all localisations are reflective in general. However, several interesting localisations are, as
seen below.
Example 1.3. The localisation of left R−modules away from a multiplicative subset S of R and the localisation
of ‘simple’ spaces at a set of prime numbers are reflective, as seen above. Also, the localisations of the category
of CW−complexes with respect to weak equivalences, resulting in the homotopy category of CW−complexes, and
the localisation of the complexes of abelian category with respect to quasi-isomorphisms are also reflective.
Example 1.4. The localisation of homotopy categories derived from Bousfield localisation of model categories
is reflective, see [1, Model Categories, Def. 2.8.40] .
1.1. Category of Fractions. Let C be a category, S be a class of morphisms in C . Then, one can always
consider the (priorly very large) category of fractions of C with respect S. When it exist, i.e. when its homobjects being sets as opposed to proper classes, it gives a localisation of C with respect S.
Caution 1.5. Recall the construction of the category of fractions of C with respect S. It given by the quotient
category of the free category F G(C , S −1 ) with respect to the minimal relation R that satisfies:
● ∀X ∈ C , ()X RX,X (idX ).
● (f, g) RX,Z (g ○ f ) for every pair of composable morphisms, f ∶ X → Y and g ∶ Y → Z.
● ∀s ∶ X → Y in S, ()X RX,X (s, s−1 ), and ()Y RY,Y (s−1 , s).
It has the same class of object as of C , whereas its morphisms are equivalence classes of zigzags1 of morphisms
in C with the components directed backwards being from S, modulo the evident equivalence relation induced
from R, see [3, §.I.1.1] for the concrete construction. This construction results in several difficulties:
● The category of fractions of C with respect S might not exist, in the given foundations e.g. the universe
C lives in. That, hom-objects do not have to be small sets, and in general they may be proper classes.
● The nature of the morphisms of the category of fractions becomes formal and obscure compared to
those of C , even if C is a ‘nice’ well-understood category.
The above difficulties can be remedied when the class we are localising with respect to is a set and when
the given category admit suable structure, like left or right calculus of fractions, or a model category structure.
We review below calculus of fractions, and we devote another file for the study of model structures and their
localisation, see [1, Model Categories, §.2] .
1.2. Calculus of Fractions. Recall that, in reflective localisation, the localised category is fully faithfully
embedded in the original category. That guarantees its existence in the given foundations (that admit the axiom
of choice) and makes it more accessible. Analysing the reflective localisation, isolating the axioms resulting in
a such good behaved localised category, as done in [3, §.I.2], one distinguished the below desired properties of
the class one is localisation with respect to.
Definition 1.6 (Calculus of Fractions). Let C be a category, S a set of morphisms of C . Then, C is said to
admit calculus of left fractions with respect to S, and S is said to be a left multiplicative system in C if the
following axioms hold:
(1) S is a subcategory of C that contains all its objects, i.e. the identity of every object of C is in S and
the composition of morphisms of S is in S.
(2) (Weak Push-outs) For every solid diagram
X
f
s
X′
/Y
s′
f′
/ Y′
in C with s ∈ S, there exists doted morphisms in C completing it to a commutative square with s′ ∈ S.
1Strings from the joint class of morphisms of C and formal inverses of morphisms of S.
LOCALISATION AND QUOTIENT CATEGORIES
9
(3) (Annihilation) For every commutative solid diagram
X′
s
/X
f
g
// Y
s′
/ Y′
in C with s ∈ S there exists a doted morphism s′ ∈ S making the whole diagram commutative.
C is said to admit calculus of right fractions with respect to S, and S is said to be a right multiplicative system
in C if the dual axioms hold. Moreover, C is said to admit calculus of fractions with respect to S, and S is said
to be a multiplicative system in C if C admits both calculus of left and right fractions with respect to S.
When C admits a calculus of left fractions, a construct of the category of left fractions S −1 C can be given
by the data:
● The class of objects Ob(S −1 C ) = Ob(C ).
● For every pair of objects X, Y ∈ Ob(S −1 C ), S −1 C (X, Y ) is the set of equivalence classes of S−left roofs
(u, α)
′
7Y g α
u
Y
X
for some Y ′ ∈ C , u ∈ C (X, Y ′ ), and α ∈ S, with the equivalence relation being generated by having two
S−left roofs (u1 , α1 ) ∶ X → Y1′ ← Y and (u2 , α2 ) ∶ X → Y2′ ← Y equivalent if and only if they admit a
corefinement, i.e. if there exists an S−left roof (u, α) ∶ X → Y ′ ← Y and morphisms s1 ∶ Y1′ → Y ′ and
s2 ∶ Y2′ → Y ′ in S making the below diagram commutative
4 Y1′ j
u1
α1
s1
X
u
/ Y′ o
O
α
Y.
s2
u2
α2
* Y′ t
2
This relation is indeed an equivalence relation due to S being a left multiplicative system in C , especially
due to the weak push-outs and annihilation properties, by lemma [4, §.III.2.Lem.8]. In fact, one can
view S −1 C (X, Y ) as a filtered colimit of S−left roofs, see [10, Tag 04VB].
● For every triple of objects X, Y, Z ∈ Ob(S −1 C ) and S−left roofs (v, β) ∶ Y → Z, and (u, α) ∶ X → Y ,
applying the axiom of choice, one fix a commutative diagram induced by the weak push-out axiom with
α′ ∈ S
′′
6 Z h α′
v′
u
′
7Y h
α
v
′
6Z g
β
X
Y
Z.
Let S −roof (X, Y ) be the class of roofs from X to Y . Then, there exist a class function
○ ∶ S −roof (Y, Z) × S −roof (X, Y ) Ð→ S −roof (X, Z)
given by
(v, β) ○ (u, α) ∶= (v ′ u, α′ β)
for morphisms fitting in the above fixed diagrams. This function extends readily to a well-defined
function on equivalence classes, i.e. to a composition function
○ ∶ S −1 C (Y, Z) × S −1 C (X, Y ) Ð→ S −1 C (X, Z).
● For every object X ∈ Ob(S −1 C ), let idX ∶ ∗ → EndS −1 C (X) be given by (idX , idX ).
One sees readily that S −1 C is a category, see the proof of [4, §.III.2.Lem.8]. The construction of S −1 C mimics the
construction of the ring of left fraction of a ring away from a multiplicative subset of the given ring. Obviously,
the two coincides when C is additive and has one object. One think of the class of S−left roof (u, α) as [α−1 u].
Dual constructions defines the category of right fractions C S −1 .
There is an evident functor [−]S ∶ C → S −1 C fixing objects and sending morphisms f ∶ X → Y to classes of
S−left roofs [f, idY ].
10
ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
Theorem 1.7. Let C be a category, S be a set of morphisms of C , and assume that C admits calculus of left
fractions with respect to S. Then, [−]S ∶ C → S −1 C is a localisation of C with respect to S.
Proof. See [3, §.I.2.4] and [4, §.III.2.Lem.8].
Recall the main problem with the category of fractions is that its morphisms are classes of zigzags
−1
−1
−1
s−1
0 f0 s1 f1 s2 f2 ⋯sn fn
of arbitrary length, with s−1
i being a formal inverse of si ∈ S for 0 ≤ i ≤ n. In general one do not have cancellation
or commutativity relations that allow better understanding of these formal zigzags in terms of morphisms of the
original category. However, the main advantage one obtains when C admits a calculus of left or right fractions
with respect to S is that the formal inverses of morphisms of S, appearing in the above morphism, can be always
gathered into a single formal inverse on the left or the right, respectively, turning all such zigzags, priorly of
arbitrary length, into the relatively better-understood roofs, see [4, §.III.2.Rem.7].
2. Quotient Category
One is sometimes interested in steadying mathematical objects up-to some equivalent relation that respect
their structure, in such case it is useful to consider the quotient of such objects with respect to the given
equivalence relation, when such quotient is defined, as in the case of subsets, normal groups, ideals of rings,
submodules, etc. The same principle applies to category theory, for instance, the classical homotopy category
of topological spaces hTop = πTop is defined to be the quotient category of topological spaces Top with respect
to homotopical equivalence. Hereby, we recall hereby the notions of congruence relation and quotient category.
Definition 2.1 (Congruence Relation). Let C be a category. A relation on C is a (class) function R that
assigns to each pair of objects X, Y ∈ C a binary relation RX,Y on C (X, Y ) that respects the composition in
C , i.e. if f, f ′ ∈ C (X, Y ) and g, g ′ ∈ C (Y, Z) such that f RX,Y f ′ and gRY,Z g ′ , then gf RX,Z g ′ f ′ . A relation R
on C is called a congruence relation if all binary relations RX,Y are equivalence relations for every X, Y ∈ C .
Let C be a category, R a relation on C , then the argument in [7, p.52] shows that there is a minimal
congruence relation R̄ on C with R ⊆ R̄. In particular, when R is a congruence relation, then R̄ = R.
Having the notion of homotopy and homotopy equivalence for topological spaces in mind, on generalises these
notion as follows:
Definition 2.2. Let C be a category, and R a relation on C . Let f, g ∶ X → Y be morphisms in C , f is
said to be R−related to g if f RX,Y g, for short one writes f ∼R g. Also, for objects X, Y ∈ C , X is said to be
R−equivalence to Y if there exist morphisms f ∶ X → Y and g ∶ Y → X such that gf ∼R idX and idY ∼R f g.
Notice that when R is a congruence relation then R−equivalence is an equivalence relation.
Definition 2.3. Let C be a category, R be a relation on C . Call a category D under C R−congruent if the
structure functor F ∶ C → D identifies R−related parallel morphisms of C in D, i.e. for every X, Y ∈ C and
f, f ′ ∈ C (X, Y ) such that f RX,Y f ′ one has F (f ) = F (f ′ ). Then, the quotient category of C with respect to R,
if exists, is defined to be to be a universal morphism
QR ∶ C Ð→ C /R
from idC to the inclusion of the very large full subcategory of S−congruent categories under C in the very large
category C ↓ CAT.
That is equivalent to QR ∶ C Ð→ C /R being R−congruent category under C , and QR being R−congruent
equivalence, where a functor F ∶ D → E between categories under C is called R−congruent equivalence if the
functor
F ∗ ∶ C ↓ CAT(E , Z ) Ð→ C ↓ CAT(D, Z )
is an isomorphism between functor categories, for every R−congruent category Z under C .
One notices that the above definition is strict, i.e. taken in the very large 1−category CAT. Of course, it
makes sense to define its weaker version as it is the case for localisation. However, for an arbitrary category C
and a relation R on C , proposition [7, Prop.§.II.8.1] implies that the quotient QR ∶ C Ð→ C /R exists, in the
strict sense, and hence one does not need to weaken the definition. The argument on page 52 loc.cit. shows
that C /R ≅ C /R̄ is isomorphic to the category C /R that is given by
● A class of objects Ob(C /R) = Ob(C ).
LOCALISATION AND QUOTIENT CATEGORIES
11
● For each pair of objects X, Y ∈ Ob(C /R), the set of morphisms
C /R(X, Y
) ∶= C (X, Y )/R̄X,Y .
is the set-theoretical quotient with respect to the minimal equivalence relations RX,Y ⊆ R̄X,Y .
● For each triple of objects X, Y, Z ∈ Ob(C /R) the composition maps of C in the first row induces a unique
composition map in C /R in the second row that R respects the composition of C
/ C (X, Z)
C (Y, Z) × C (X, Y )
○C ∶
/ C /R(X, Z).
○C/R ∶
C /R(Y, Z) × C /R(X, Y
)
● For each object X ∈ Ob(C /R), the equivalence class [idX ] of idX in C gives the identity of X in C /R, i.e.
the composition of the identity element in C and the quotient morphism gives the identity at X in C /R
idX
∗
/ C (X, X)
/ / C /R(X, X).
Example 2.4. Recall that a commutative ring R can be though of as a pre-additive monoid R with one object
∗ and a commutative ring of endomorphisms EndR (∗) = R. Let I ⊂ R be an ideal in R, then the relation on
∼I on R defined x ∼I y if and only if x − y ∈ I defines a congruence relation I on R. The quotient category R/I
is isomorphic to the the pre-additive monoid R′ with one object ∗ and a commutative ring of endomorphisms
EndR′ (∗) = R/I. On the other hand, let I be a congruence relation I on R that respect its pre-additive structure,
then I ∶= {x ∈ R∣x ∼I 0} is an ideal in R, and the quotient category R/I is isomorphic to the the pre-additive
monoid R′ with one object ∗ and a commutative ring of endomorphisms EndR′ (∗) = R/I.
Notice that a relation R on a category C for which idX R idX for every X ∈ C can be realised as a subcategory
R
(i1 ,i2 )
↪ C × C with f ∼R g if and only if (f, G) is in R. Then,
● R is reflexive if and only if there exists a section functor s ∶ C → R for both of i1 and i2 , i.e. (i1 , i2 )s =
(idX , idX ).
● R is symmetric if and only if there exists a symmetric braiding functor ψ ∶ R → R that interchanges i1
and i2 , i.e. (i1 , i2 )ψ = (i2 , i1 ).
● R is transitive if and only if there exists a ‘multiplication’ functor µ ∶ R ×C R → R, where the fibre
product is given by the Cartesian square
p1
R ×C R
p2
R
/R
I
i2
i1
/ C,
such that (i1 , i2 )µ = (i1 p1 , i2 p2 ).
The first two equivalences are obvious. For transitivity, on the one hand let f ∼R g and g ∼R h in C , then
((f, g), (g, h)) is a morphism in R×C R that i2 ((f, g)) = i1 ((g, h)). Assume the existence of such ‘multiplication’
functor µ, then (f, h) = µ ((f, g), (g, h)) in R, and hence f ∼R h. The other direction is evident.
Moreover, the quotient category C /R is given by the coequalizer of the solid diagram
R
 (i1 ,i2 ) /
C ×C
π1
π1
// C
QR
/ C /R.
in the very large category CAT. Then, the the quotient category C /R is in particular a quotient object in
CAT.
One can generalise the notion of a relation and define a relation on a category C to consider arbitrary
subcategories (i1 , i2 ) ∶ R ⊂ C × C , and define categorical congruence relation on C to be a categorical relation
R ⊂ C × C that is reflexive, symmetric, and transitive in the above sense. Then, for a categorical relation R on
C , define the categorical quotient of C with respect to R to be the coequalizer of the solid diagram
R
 (i1 ,i2 ) /
C ×C
π1
π1
// C
QR
/ C /R.
12
ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
in the very large category CAT. Since the functor (i1 , i2 ) is a mono, then the functors s, ψ, and µ for which
(i1 , i2 ) is reflexive, symmetric, and transitive in the above sense are unique, if exists. Hence, a congruency is a
property on categorical relation, as opposed to a structure.
The above notion of ‘categorical’ relations can be generalised further to any category C with products and
(i1 ,i2 )
fibre product. That an internal relation on an object X ∈ C is defined to be a subobject R ↪ X × X. An
internal relation R on X is called an internal congruence relation if it is reflexive, symmetric, and transitive,
where
● R is reflexive if there exists a section morphism s ∶ X → R in C for both of i1 and i2 , i.e. (i1 , i2 )s =
(idX , idX ).
● R is symmetric if there exists a symmetric braiding morphism ψ ∶ R → R in C that interchanges i1 and
i2 , i.e. (i1 , i2 )ψ = (i2 , i1 ).
● R is transitive if there exists a ‘multiplication’ morphism µ ∶ R ×X R → R in C , where the fibre product
being given by the Cartesian square
p1
R ×X R
p2
I
R
/R
i2
i1
/ X,
such that (i1 , i2 )µ = (i1 p1 , i2 p2 ).
Then, in particular the above ‘categorical’ congruence relation is an internal congruence relation in the very
large category CAT.
3. Relation between Localisation and Quotienting
Although the notions of localisation and quotienting of categories are priorly different, they are pertained.
Their relation fades in some occasions and embellishes in others.
The main similarity between the two construction is that both constructions result in inverting a class of
morphisms, and both of them are ‘colimits’. Localisations inverts morphisms of a given class, and by doing so
it identifies the domain and codomain of morphisms of S, of course up-to isomorphism. Whereas quotienting
is a ‘finer’ construction, that also invert the saturated class R−equivalences, however its effect is not restrict to
that.
Quotient Categories We start with the more comprehensible case of quotient categories. The main three
advantage of the construction of quotient categories are:
● They always exist.
● Their morphisms are represented by morphisms of the original category, and hence the nature of the
resulting morphisms is well understood, at least in comparison to morphisms of the original category.
● They admit unique functors induced from functors on the original categories, that respects their relations.
Let C be a category R a relation on C , and QR ∶ C → C /R be the quotient functor of C with respect to R.
As a result of identifying R̄−related morphisms in C /R, the quotient functor QR takes R̄−equivalences in C to
isomorphisms in C /R, and hence the quotient functor with respect to R 2−factorises in the very large 2−category
CAT through the localisation functor of C with respect to the class of R̄−equivalences, if the localisation exists.
Denote the class of R̄−equivalences in C by WR , when the localisation LWR ∶ C → C [WR−1 ] exists, there exist
a functor FR ∶ C [WR−1 ] → C /R, unique only up-to natural isomorphism, and hence there exists a triangle of
functors (not necessary commutative)
4/ C /R,
QR
C
LWR
and a unique natural isomorphism FR LWR ≅ QR .
*
C [WR−1 ]
FR
LOCALISATION AND QUOTIENT CATEGORIES
13
(C , WR ) is a category with weak equivalences. In particular, the transitivity of R̄ implies that WR is closed
under composition, its reflexivity implies that WR contains isomorphisms. One direction of the two other cases
of the two-out-of-three property are evident, whereas the rest results from R̄ being symmetric and transitive.
In fact, a morphism in C is an R̄−equivalence if and only if it is sent to isomorphism in C /R, that equalities
between morphisms in C /R accounts to being R̄−related in C . The natural isomorphism FR LR ≅ QR implies
that morphisms in C that are sent to isomorphism in C [WR−1 ] are also sent to isomorphisms in C /R. Hence
a morphism in C is an R̄−equivalence if and only if it is sent to isomorphism in C [WR−1 ], i.e. the class WR is
saturated.
In some occasions the functor FR is an equivalence of categories, as it is the case for quotient category of
classical homotopy category of CW−complexes and the localisation of CW−complexes with respect to homotopy
equivalences. However, in general the functor FR is not even fully faithful, as we will see in the below.
Since the two notions of localisation and quotienting of categories generalise their ring-theoretical counterparts, it is useful to consider examples arising from commutative algebra.
Let R be a commutative ring, I ⊂ R be an an ideal in R, R and R/I be the pre-additive monoids corresponding
to the rings R and R/I, where I is the congruence relation on R given by f ∼I g if and only if f − g ∈ I, as in
example 2.4. Then, a morphism f ∶ ∗ → ∗ in R, i.e. f ∈ R, is an I−equivalence if and only if there exists a g ∈ R
such that f g − 1 ∈ I (i.e. if [f ] is invertible in R/I). The set SI ⊂ R of I−equivalences forms a multiplicative
system in R and one has a factorisation of ring morphisms and pre-additive 1−monoids, respectively
4/ R/I
QI
R
LSI
*
R[SI−1 ]
FI
, and
4/ R/I.
QI
R
LI
*
R[SI−1 ]
FI
Notice that when I is a maximal ideal in R then SI = R ∖ I, and hence R[SI−1 ] ≅ RI . Then, the the morphism
FI is not an isomorphism in general as it is the case for F(2) ∶ Z(2) ↠ Z/2, and hence FI is not an equivalence
of categories in general.
Avoid the confusion, in general R ∖ I is not necessary a multiplicative system, and even when it is, i.e. when
I is a prime ideal, SI and R ∖ I do not have to coincide, and R/I does not have to factorise through RI . For
example, let k be a filed, R = k[x, y], and I = (x) ⊂ R be the prime ideal generated by x. y is invertible in RI ,
but is not invertible in R/I, hence the quotient morphism QI ∶ R → R/I does not factorises through RI .
Localisation of Categories On the contrary to quotient categories, localisation of categories is defined
as a strict 2−colimit up-to equivalence of categories and has the following three disadvantages:
● They do not always exist.
● Their morphisms are represented by zigzags of morphisms of the original category, and hence the nature
of the resulting morphisms is very formal compared to morphisms of the original category.
● They do not always induce canonical functors from functors on the original categories that respects the
localised classes.
One approach to remedy some of these disadvantages is given by calculus of fractions, when it is admitted
by the localised class, see 1.2, where zigzags can be simplified and be replaced by roofs.
Another approach follows from Quillen’s development of homotopical algebra. For a category C and a class
of morphisms W ⊂ C , one looks for extra structure M on C , namely model structure [1, Model Categories,
Def. 2.1.23] , which guarantees the existence of the localisation C [W −1 ] ≅ Ho(C , M ). This extra structure
induces a congruence relation H on a full subcategory Cf c ⊂ C (namely, the full subcategory of fibrant-cofibrant
objects of (C , M )). Moreover, the existence of such structure implies that the quotient of Cf c /H is equivalent
to the desired localisation C [W −1 ]. Hence, for such categories, called model categories, the localisation category
C [W −1 ] becomes more accessible through quotient categories Cf c /H.
In general, for a category C and a class of morphisms W ⊂ B there may not exist a model structure M
on C for which C [W −1 ] ≅ Ho(C , M ), and the localised category C [W −1 ] does not have to be equivalent
to a quotient of C or its full subcategories with respect to any congruence relation. Model categories are
(co)complete by definition, hence when C is not (co)complete one may use the Yoneda embedding to embed C
14
ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
in the (co)complete category of pre-sheaves on C , or a suitable subcategory of the latter, and then look for a
model structure for the ambient category.
Counter Example 3.1. Using the notation of example 2.4 for R = Z, S = Z ∖ (2) is a multiplicative system
in R, but Z(2) is not isomorphic to the quotient of R with respect to any ideal of Z, and hence R[S −1 ] is not
equivalent to R/I for any ideal I ⊂ R. However, that also can be remedied easily when on consider the faithful
(but not full) embedding R ↪ R′ , where EndR (∗) = R and EndR′ (∗) = R[x], then for the multiplicative system
S = {an ∣n ∈ N} ⊂ R, for a non-zero divisor a ∈ R, one has
R[S −1 ] ≅ R′ /I
where I is the congruence relation defined on R′ by f (x) ∼I g(x) if and only if f (x) − g(x) ∈ (ax − 1).
4. Symmetric Monoidal Categories
Symmetric monoidal structures on categories generalises tensor product on vector spaces to arbitrary categories. Roughly speaking, it is a bifunctor on a category satisfying the usual associativity, unitary, and
commutativity axioms upto isomorphism, see [7, Ch.VII§.1 and §.2] for the definition and basic properties.
4.1. Internalisation. Traditional structures (not necessary first-order structures) like group, R−module, ring,
topological space structures are set-theoretical, i.e. defined for a set theory. Since the representable functors
embed a given (small) category in the category of (small) sets, then one can use these functors to ‘internalise’
the traditional structure into the given category. Let S be the category whose objects are sets with a traditional
structure S, and whose morphisms are structure preserving maps, then there is a forgetful functor U ∶ S → Set.
Let C be a category, and X ∈ C . Then, the object X is said to be an S−object, or to admit an internalised
S−structure if the representable functor hX ∶ C op → Set factorises through S .
Example 4.1 (Group Objects). A group object in a category C is an object X for which the representable
functor hX factorises through Grp,
5/ Set.
hX
C op
h′X
(
Grp
U
Since h′X (Y ) admits traditional group structure for every Y ∈ C , h′X is a functor, and that hX (Y ) × hX (Y ) ≅
hX×X (Y ) when C has finite products, then it readily seen that X is a group object in a category C with finite
products if and only if there exist morphisms
µX ∶ X × X → X
,
ιX ∶ X → X
, and
ηX ∶ ∗ → X
for which µX , ιX , and ηX satisfy the traditional group structure axioms, see [6, §.8.1]. Similar argument applies
for monoid, commutative group, ring, or R−module objects, etc.
For instance, all groups are group objects, or admit an internalised group structure, in the category of
groups. A (commutative) monoid object in the category of abelian groups is a (commutative) unitary ring, and
a (commutative) monoid object in the category of R−modules is a (commutative) R−algebra.
When all objects of a category admit an internalised stricture S, then it is natural to look for such categories
where composition maps respect the traditional structure S. For instance, when S is the structure of abelian
groups, then the categories whose all objects admit an internalised stricture S and whose composition maps
respect the traditional structure S are called pre-additive categories, and they are recalled in [1, Homological
Algebra in Abelian Categories, Section 1] .
4.1.1. Internalisation of Algebraic Theories. Recall that algebraic objects like (commutative) group, ring, R−module
objects can given in categories with finite Cartesian products by the existence of morphisms between suitable
Cartesian powers satisfying the traditional structures axioms. Cartesian product is in particular a symmetric
monoidal structure, and the conditions the algebraic objects can by given for any symmetric monoidal structure
by the existence of morphisms between suitable symmetric monoidal powers satisfying the traditional structures
axioms. Notice that algebraic objects given for arbitrary symmetric monoidal structure do not have to coincide
with the algebraic objects arisen from the above definition.
LOCALISATION AND QUOTIENT CATEGORIES
15
4.2. Enriched Categories. The sets of morphisms between objects may admit a natural mathematical structure, usually arising from the structure of the objects of the given category, when they are set-theoretical. For
instant, in the category of groups Grp, for every pair of groups G, H ∈ Grp, the set Grp(G, H) has a natural
group structure. The same applies for the category of vector spaces Vectk over a field k, and left R−modules
R − Mod, for a ring R. Also, in topological spaces, if we restrict ourself to compactly generated Hausdorff
spaces, then for any such pair of spaces X, Y , the set CGHaus(X, Y ) admits a canonical topology, making it
into topological space, namely the open-compact topology. In all these examples the composition maps
○ ∶ C (Y, Z) × C (X, Y ) → C (X, Z)
are compatible with the induced structure, i.e. they are morphisms in the same category. This observation
gives rise to enriched categories, where one generalises the notion of category replacing hom-sets with more
interesting hom-objects.
Definition 4.2 (Enriched Categories). Let (M , ⊗, 1, α, l, r) be a (strict/lax) monoidal category. Then, a small
M−enriched category C is given by
● A class Ob(C ), called the class of objects in C .
● For each pair (X, Y ) ∈ Ob(C ) × Ob(C ) of objects of C , an object C (X, Y ) ∈ M , called the hom-object
from X to Y .
● For each triple (X, Y, Z) of objects in C , a morphism
○X,Y,Z ∶ C (Y, Z) ⊗ C (X, Y ) → C (X, Z)
in M , called the composition morphism.
● For each object X ∈ Ob(C ), a morphism idX ∶ 1 → C (X, X), called the identity ‘element’ of X.
such that composition in C is (strict/lax) associative, and (strict/lax) unital. Then, C is said to be an
M−category.
Example 4.3. Any category is a Set−enriched category.
A comparison between the definition of categories and enriched categories shows the naturality of the generalisation, and highlights the need to the restriction on M to be (strict/lax) monoidal.
Example 4.4. The category of compactly generated Hausdorff spaces is Top−-enriched category.
Remark 4.5. When M is concrete, i.e. its objects are sets with a traditional structure S, it admits a forgetful
functor U ∶ M → Set, and then all objects of C has an internalised structure S. This is the case in the above
mentioned examples of enriched categories. In such case, the enriched categories are in particular categories,
that there hom-objects are sets equipped with some traditional structure. However, that does not have to
hold in general. When the enriched category C happens also to be a category, we distinguish hom-sets from
hom-objects, where we denote the hom-set morphisms from X to Y in C by C (X, Y ), and the hom-object from
X to Y by M apC (X, Y ), or M ap(X, Y ) if no confusion arises.
Definition 4.6 (Enriched Functors). A (strict/lax) enriched functor F ∶ C → D between (strict/lax) M−enriched
categories is given by
● A class function F ∶ Ob(C ) → Ob(D).
● For each pair (X, Y ) ∈ Ob(C ) a morphism FX,Y ∶ Map(X, Y ) → Map(F (X), F (Y )) in M that
(strictly/weakly) commutes with the composition and the identity.
4.2.1. Strict 2−Categories. Recall that the category of small categories Cat is a strict monoidal category. Enriched categories and functors over the strict category Cat are called strict 2−categories and strict 2−functors,
respectively. They can be given by classes of objects (0−morphisms), 1−morphisms that satisfy the strict associativity and identity axioms, and 2−morphisms that satisfy the interchange law. The large category Cat of
small categories gives rise to the large strict 2−category Cat2 whose objects are small categories, 1−morphisms
are functors between small categories, and 2−morphisms are natural transformations between the latter. Also,
the very large category CAT of large categories similarly gives rise to the very large strict 2−category CAT2 of
large categories, functors between them, and natural transformations between the latter.
The localisation of categorises is given by an (initial) 2−universal morphism in the very large strict 2−category
CAT2 . Therefore, we recall (initial) 2−universal morphisms in strict 2−categories.
16
ALAMEDDIN, ANWAR
UNIVERSITY OF LIVERPOOL, UK
(Initial) 2−Universal Morphisms in Strict 2−Categories To motivate the definition, we first recall
(initial) universal morphisms in categories, and study its generalisation to 2−categories.
Let F ∶ C → D be a functor between categories, d ∈ D, then an (initial) universal morphism from d to F is
defined to be the initial object in the comma category d ↓ F , if exists. If it exists, it is a morphism ηd ∶ d → F (cd )
for cd ∈ C such that for every morphism f ∶ d → F (c) for c ∈ C there exists a unique morphism gf ∶ cd → c in C
such that the below diagram strictly commutes
d
ηd
/ F (cd )
f
! F (c)
F (gf )
cd
F
←
gf
∃!
c.
In other words, it is a morphism ηd ∶ d → F (cd ) that induces a bijection of sets
ηd∗ ∶ C (cd , c) → D(d, F (c))
for every c ∈ C . The surjectivity of ηd∗ is equivalent to the existence of the factorisation, whereas its injectivity
is equivalent to the uniqueness of the factorisation when it exists. The (initial) universal morphism ηd is unique
up-to isomorphism, if exists, and the above factorisation of f is unique for the choice of the universal morphism.
Let F2 ∶ C2 → D2 be a strict 2−functor between strict 2−categories, d ∈ D, then an (initial) 2−universal
1−morphism from d to F2 is a 1−morphism ηd ∶ d → F2 (cd ) for cd ∈ C that induces an equivalence of categories
ηd∗ ∶ MapC (cd , c) → MapD (d, F2 (c))
for every c ∈ C . Since a functor in part of equivalence if and only if it is essentially surjective and fully faithful,
then ηd is a 2−universal 1−morphism from d to F2 if and only if it satisfies the below two conditions:
Es. surjective For every 1−morphism f ∶ d → F (c) with c ∈ C there exists (by essential surjectivity) 1−morphism
gf ∶ cd → c in C and a 2−isomorphism
θf ∶ F2 (gf )ηd ≅ f ′
Fully faithful For every two parallel 1−morphisms f, f ′ ∶ d → F (c) with c ∈ C and a 2−morphism ψ ∶ f ⇒ f ′ there
exists (by fullness) a unique (by faithfulness) 2−morphism ξψ ∶ gf ⇒ gf ′ , for the choice of (gf , φf ) and
(gf ′ , φf ′ ), such that
ψ = φf ′ ⋅ F2 (ξψ )ηd ⋅ φ−1
f .
That, in particular, implies that the choice of (gf , φf ) is unique up-to 2−isomorphism (by faithfulness).
d
/ F2 (cd )
ηd
∼φ
f
d
f
{
ηd
cd
/ F2 (cd )
cd
F2 (gf )
gf ∃! up-to 2−isomorphism
F2 (ξψ )
F2
ψ
f
∼φ
′
{
←
f′
F2 (g f ′ )
+ F2 (c)
+ F2 (c)
ξψ ∃!
gf ′
c
c
Notice that the (initial) 2−universal 1morphism ηd is unique up-to 1−equivalences, if exists, and the above
factorisation (gf , φf ) of 1−morphism f is unique up-to 2−isomorphisms, for the choice of the universal morphism
ηd , whereas the above factorisation ξψ of 2−morphism ψ is unique, for the choice of the factorisation (gf , φf )
and (gf ′ , φf ′ ) of the involved 1−morphism f and f ′ .
LOCALISATION AND QUOTIENT CATEGORIES
17
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Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England, UK
E-mail address: anwar.alameddin@liv.ac.uk