Morita homotopy theory of categories
Abstract.
In this article we establish the foundations of the Morita homotopy theory of categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by ) on the category of small unital categories. The weak equivalences are the Morita equivalences and the cofibrations are the functors which are injective on objects. As an application, we obtain an elegant description of BrownGreenRieffel’s Picard group in the associated homotopy category . We then prove that is semiadditive. By group completing the induced abelian monoid structure at each Homset we obtain an additive category and a composite functor which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes corepresented in by the tensor unit object.
Key words and phrases:
categories, Model categories, Morita equivalence, Grothendieck group, Picard group.2010 Mathematics Subject Classification:
46L05, 46M99, 55U35, 16D90.1. Introduction
The theory of categories, first developed by Ghez, Lima and Roberts [glr] in the mideighties, has found several useful applications during the last decades. Most notably, it has been used by Doplicher and Roberts [doplicherroberts:new_duality, doplicherroberts:endo] in the development of a duality theory for compact groups with important applications in algebraic quantum field theory and by Davis and Lück [davislueck] in order to include the BaumConnes conjecture into their influential unified treatment of the theoretic isomorphism conjectures. Several authors – see [mitchener:symm, mitchener:Cstar_cats, mitchner:KKth, mitchener:gpd] [joachim:KC*] [kandelaki:KK_K, kandelaki:multiplier, kandelaki:karoubi_villamayor, kandelaki:fredholm] [vasselli:bundles, vasselli:bundlesII] [zito:2C*]…– have subsequently picked up these strands of ideas and employed categories in various operatortheoretic contexts, often in relation to Kasparov’s theory. In most of the above situations categories are only to be considered up to Morita equivalence, the natural extension of the classical notion of Morita(Rieffel) equivalence between algebras. Hence it is of key importance to development the foundations of a Morita theory of categories.
A functor between categories is called a Morita equivalence if it induces an equivalence on the completions of and under finite direct sums and retracts. This operation , named saturation, can easily be performed without leaving the world of categories; see §LABEL:subsec:saturated.
The first named author has initiated in [ivo:unitary] the study of categories via homotopytheoretic methods, in particular by constructing the unitary model structure, where the weak equivalences are the unitary equivalences of categories. In the present article we take a leap forward in the same direction by establishing the foundations of the Morita homotopy theory of categories. Our first main result, which summarizes Theorem LABEL:thm:morita_model, Propositions LABEL:prop:monoidal, LABEL:prop:simplicial and LABEL:prop:bousfield, and Corollary LABEL:cor:fibrant_replacement, is the following:
Theorem 1.1.
The category of (small unital) categories and (identity preserving) functors admits a Quillen model structure whose weak equivalences are the Morita equivalences and whose cofibrations are the functors which are injective on objects. Moreover, this model structure is cofibrantly generated, symmetric monoidal, simplicial, and is endowed with a functorial fibrant replacement given by the saturation functor. Furthermore, it is a left Bousfield localization of the unitary model structure.
We have named this Quillen model the Morita model category of categories (and denoted it by ) since two unital algebras become isomorphic in the associated homotopy category if and only if they are Morita equivalent ( MoritaRieffel equivalent) in the usual sense; see Proposition LABEL:prop:Morita_agreement. The Morita homotopy category becomes then the natural setting where to formalize and study all “up to Morita equivalence” phenomena. As an example we obtain an elegant conceptual description of the Picard group (see §LABEL:sec:Pic).
Proposition 1.2.
For every unital algebra there is a canonical isomorphism
(1.3) 
between its automorphism group in the Morita homotopy category and its Picard group , as originally defined by BrownGreenRieffel in [browngreenrieffel] using imprimitivity bimodules.
As a consequence, the lefthandside of (1.3) furnishes us with a simple Morita invariant definition of the Picard group of any category. Our second main result, which summarizes Theorems LABEL:thm:semiadditive and LABEL:thm:map_sum, Proposition LABEL:prop:morphisms_Morita, and Corollary LABEL:cor:new, is the following:
Theorem 1.4.
The homotopy category is semiadditive, i.e. it has a zero object, finite products, finite coproducts, and the canonical map from the coproduct to the product is an isomorphism. Its Homsets admit the following description
where denotes the category of functors from to the saturation of and the equivalence relation on objects is unitary isomorphism. Moreover, the canonical abelian monoid structure thereby obtained on each Homset is induced by the direct sum operation on .
Intuitively speaking, Theorem 1.4 shows us that by forcing Morita invariance we obtain a local abelian monoid structure. By group completing each Hom monoid we obtain then an additive category and hence a composed functor
consult §LABEL:sec:K0 for details. Our third main result (see Theorem LABEL:thm:characterization) is the following:
Theorem 1.5.
The canonical functor takes values in an additive category, inverts Morita equivalences, preserves all finite products, and is universal among all functors having these three properties.
Our last main result, collecting Theorem LABEL:thm:corepresentability and Proposition LABEL:prop:ring_comm, provides a precise link between our theory and the theory of algebras. Note that, since is symmetric monoidal, its tensor structure descends to and then extends easily to the group completion .
Theorem 1.6.
For every unital algebra there is a canonical isomorphism
(1.7) 
of abelian groups, where the righthandside denotes the classical Grothendieck group of . When is moreover commutative, the usual ring structure on (induced by the tensor product of vector bundles) coincides with the to one obtained on the lefthandside by considering as a ring object in the symmetric monoidal category .
As a consequence the lefthandside of (1.7) provides us with an elegant Morita invariant definition of the Grothendieck group of any category. Note that by Theorem 1.5 this definition is completely characterized by a simple universal property. In Remark LABEL:remark:Ktheory we compare our approach with those of other authors.
Conventions
We use the symbol to denote the base field, which is fixed and is either or . Except when stated otherwise, all categories are small (they have a set – as opposed to a class – of objects) and unital (they have an identity arrow for each object ). Similarly, all functors are unital (they preserve the identity maps ). We will generally follow the notations from [ivo:unitary].
2. Direct sums and idempotents
In this section we consider, in the context of categories, the additive hull and the idempotent completion constructions. Both will play a central role in the sequel.
2.1. categories, categories, Banach categories, and categories
For the reader’s convenience, we start by recalling some standard definitions and facts; consult [ivo:unitary]*§1 for more details and examples. Recall that an category is a category enriched over vector spaces (see [kelly:enriched_book]); concretely, each Homset carries an vector space structure for which composition is bilinear. A category is an category which comes equipped with an involution on arrows. More precisely, the involution is a conjugatelinear contravariant endofunctor on which is the identity on objects and which is its own inverse. The arrow is called the adjoint of . A Banach category is an category where moreover each Homspace is a Banach space in such a way that for all composable arrows and for all identity arrows. A category is simultaneously a Banach category and a category, where moreover the norm is a norm, i.e. for every arrow we require that:

the equality holds;

the arrow is a positive element of the endomorphism algebra , i.e. its operatortheoretic spectrum is contained in .
A functor is a functor preserving the linear structure and the involution. If are categories, will automatically be normreducing on each Homspace, and if is moreover faithful (i.e. injective on arrows) it will automatically be isometric, i.e. norm preserving: (the converse being obvious).
Notation 2.1.
The category of all (small) categories and all (identity preserving) functors will be denoted by .
Example 2.2.
Every (unital) algebra can be identified with the (small) category with precisely one object and whose endomorphisms algebra is given by . Then a functor between unital algebras is the same as a unital homomorphism. The collection of all Hilbert spaces and all bounded linear operators between them (together with the operator norm and the usual adjoint operators) is an example of a (large) category .
The axioms of a category are designed so that the following basic standard result holds: every small category admits a concrete realization as a subcategory of . This is essentially the GNS construction; see [glr]*Prop. 1.14. The converse is also clear: every normclosed closed subcategory of inherits the structure of a category.
Notation 2.3.
Given a category (or a category, Banach category,…), we denote by its underlying category that one obtains by simply forgetting some of its structure. This defines a faithful functor from categories to ordinary (small) categories and ordinary functors. Similarly, we have a forgetful functor to (small) categories and linear functors between them.
2.2. Some categorical notions
In the context of categories, or more generally categories, it is natural to require all categorical properties and constructions to be compatible with the involution. The following notions concern objects, morphisms, and more generally diagrams inside a given category and the terminology is inspired by the example .
A unitary morphism (or isomorphism) is an invertible morphism such that . We say that two objects are unitarily isomorphic if there exists a unitary morphism between them. An isometry (or mono) is an arrow such that . We call a retract of (or retract); whenever we say that is a retract of , we will assume that an isometry has been specified. Dually, a coisometry (or epi) is a morphism such that ; note that is an isometry if and only if is a coisometry.
A projection (or idempotent) is a selfadjoint idempotent morphism . If is an isometry, then is a projection on . In this case we say that has range object , or that is the range projection of . If and are two retracts of the same object , it follows that and are range objects for the same projection (i.e. ) if and only if there exists a unitary isomorphism such that . Therefore, if the range object of a projection exists then it is uniquely determined up to a unique unitary isomorphism.
Remark 2.4.
Note that identity maps are both projections and unitary isomorphisms. Moreover, in a category all unitaries, isometries and projections automatically have norm one or zero; indeed, this is wellknown for bounded operators between Hilbert spaces, and as we have recalled every category is isomorphic to a category of such operators. (Generally speaking, this reasoning is a quick way to gain some intuition on categories for those familiar with Hilbert spaces.)
A direct sum (or biproduct) of finitely many objects is an object together with isometries such that the following equations hold:
where is the evident Kronecker delta: if or otherwise.
Remark 2.5.
By definition, the direct sum is also a biproduct in the underlying category, and thus both a product and a coproduct in the underlying category . Moreover, in analogy with retracts, a direct sum is uniquely determined up to a unique unitary isomorphism.
Remark 2.6.
A functor between categories preserves each one of the above notions.
Definition 2.7.
Let and be two categories. A unitary equivalence (or equivalence) between and is a functor for which there exist a functor and natural unitary isomorphisms and . Equivalently, is fully faithful and unitarily essentially surjective, i.e. for every there exists an and a unitary isomorphism in .
It can be shown that any two objects in a category are isomorphic if and only if they are unitarily isomorphic; see [ivo:unitary]*Prop. 1.6. In particular, a functor between categories is unitarily essentially surjective if and only if it is essentially surjective in the usual sense. Therefore, it is a unitary equivalence if and only if it is an equivalence of the underlying categories (that is, if and only if is an equivalence of categories).
2.3. Adding direct sums
Let be a category. We say that is additive if its underlying category is additive. This amounts to requiring that admits biproducts, or equivalently that admits a zero object, all finite products and coproducts, and that the canonical maps comparing coproducts with products are isomorphisms (cf. §LABEL:sec:semi_add). As it will become apparent in what follows, this is the same as requiring that admits all finite (compatible) direct sums.
Definition 2.8 (Additive hull ; see [mitchener:symm]*Def. 2.12).
The additive hull of is the category defined as follows: the objects are the formal words on the set and the Homspaces are the spaces of matrices, written as follows:
Composition is the usual matrix multiplication, , and adjoints are given by the conjugatetranspose . There exists a unique norm on making the canonical fully faithful functor
isometric. Moreover, is complete for this norm, i.e. it is a Banach category and so in fact a category. (For a quick proof of these facts choose a faithful, and hence isometric, representation . Since the functor has an evident extension to , one can now argue with bounded operators). Given a functor , we define a functor by setting
for all objects and arrows . We obtain in this way a welldefined additive hull functor .
Remark 2.9.
Note that the object , together with the evident matrices
is a canonical choice for the direct sum in of the objects . In particular, the empty word provides a zero object . Hence admits all finite direct sums and thus it is additive (cf. Remark 2.5). Note also that admits all finite direct sums if and only if is a unitary equivalence. For this use the fact that is fully faithful and that direct sums are unique up to a unitary isomorphism.
Notation 2.10.
In the following, whenever we write in some additive hull , we mean the canonical direct sum with the above matrix isometries. Similarly, by we will always mean the empty word.
Remark 2.11.
If we ignore norms and adjoints, the same precise construction as in Definition 2.8 provides an additive hull for any linear category . Hence is additive (i.e. admits finite biproducts) if and only if is an equivalence. Note that in the case of a category we have the equality .
The additive hull can be characterized by the following 2universal property.
Lemma 2.12.
Let be an additive category. Then the induced functor
is a unitary equivalence.
Here, denotes the internal Hom functor, which for two categories yields the category of functors and bounded natural transformations first introduced in [glr]*Prop. 1.11; see also [ivo:unitary].
Proof.
Every functor extends along by the formula , which requires the choice of direct sums in . Nonetheless, the extension is unique up to unitary isomorphism of functors. Every bounded natural transformation extends diagonally to a bounded natural transformation and the extension is unique since every (bounded) natural transformation must be diagonal: if and then