Restriction categories provide a categorical framework for partiality. In this paper, we introduce three new categorical theories for partiality: local categories, partial categories, and inclusion categories. The objects of a local category are partially accessible resources, and morphisms are processes between these resources. In a partial category, partiality is addressed via two operators, restriction and contraction, which control the domain of definition of a morphism. Finally, an inclusion category is a category equipped with a family of monics which axiomatize the inclusions between sets. The main result of this paper shows that restriction categories are $2$-equivalent to local categories, that partial categories are $2$-equivalent to inclusion categories, and that both restriction/local categories are $2$-equivalent to bounded partial/inclusion categories. Our result offers four equivalent ways to describe partiality: on morphisms, via restriction categories; on objects, with local categories; operationally, with partial categories; and via inclusions, with inclusion categories. We also translate several key concepts from restriction category theory to the local category context, which allows us to show that various special kinds of restriction categories, such as inverse categories, are $2$-equivalent to their analogous kind of local categories. In particular, the equivalence between inverse (restriction) categories and inverse local categories is a generalization of the celebrated Ehresmann-Schein-Nambooripad theorem for inverse semigroups.
We define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. In addition to proving several general results for constructing examples of locally bounded enriched categories and locally bounded closed categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly's reflectivity a…
In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories…
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable.
A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. In this article we first determine an abstract characterization of the categories which are semi-localizations of an exact Mal'tsev category, by specializing a result due to S. Mantovani. We then turn our attention to semi-abelian categories, where a special type of semi-localizations are known to coincide with torsion-free subcategories. A new characterisation of protomodular categories in terms of binary relations is obtained, inspired by the one discovered in the point…
In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the objective of establishing the free compact closed category on one generator as a fibrant replacement of the free symmetric monoidal category on one generator, in our localized model categories. We go on to show that the fibrant objects in our model categories are compact closed categories.
We consider the quotient of an exact or one-sided exact category $\mathcal{E}$ by a so-called percolating subcategory $\mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $\mathcal{E}$ at a suitable class $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ of morphisms. The localization $\mathcal{E}[S_\mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $\mathcal{E}{/\mkern-6mu/} \mathcal{A}$ of $\mathcal{E}[S_\mathcal{A}^{-1}]$ and shows that this satisfies th…
Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.
We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.
We examine the analogues for the respective categories of locales of two well-known results about regularity and effectiveness of some categories of spaces. We show that the category of compact regular locales is effective regular (=Barr-exact). We also show that the category of compactly generated Hausdorff locales is regular, provided that it is coreflective within Hausdorff locales. We do not appeal to the existence of points (which would render the two results trivial) but rely on the treatment of the subject by methods that are valid in the internal logic of a topos. On the course to the …
In this paper, we investigate the property (P) that finite products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal'tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre $Pt_{\mathbb{C}}(X)$ of the fibration of points $π: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ …
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.
We investigate several versions of the telescope conjecture on localized categories of spectra, and implications between them. Generalizing the "finite localization" construction, we show that on such categories, localizing away from a set of strongly dualizable objects is smashing. We classify all smashing localizations on the harmonic category, HFp-local category and I-local category, where I is the Brown-Comenetz dual of the sphere spectrum; all are localizations away from strongly dualizable objects, although these categories have no nonzero compact objects. The Bousfield lattices of the h…
In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let $\mathcal{A}$ be a percolating subcategory of an exact category $\mathcal{E}$, the quotient $\mathcal{E} {/\mkern-6mu/} \mathcal{A}$ is constructed in two steps. In the first step, we associate a set $S_\mathcal{A} \subseteq \operatorname{Mor}(\mathcal{E})$ to $\mathcal{A}$ and consider the localization $\mathcal{E}[S^{-1}_\mathcal{A}]$. In general, $\mathcal{E}[S_\mathcal{A}^{-1}…
In this article we introduce the notion of a square structure on a model category, that generalises cubical model categories. We then show that under some homotopical conditions on this square structure the induced cubical category is a localisation of the model category.
We show that the category of finite-dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2-Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.
Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Pi-types, Sigma-types and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou-Hofmann interpretation of Martin-Löf type theory in locally cartesian closed categories yields a biequivalence…
Given Gray-categories $P$ and $L$, there is a Gray-category $\mathrm{Tricat}_{\mathrm{ls}}(P,L)$ of locally strict trihomomorphisms with domain $P$ and codomain $L$, tritransformations, trimodifications, and perturbations. If the domain $P$ is small and the codomain $L$ is cocomplete, we show that this Gray-category is isomorphic as a Gray-category to the Gray-category $\mathrm{Ps}$-$T$-$\mathrm{Alg}$ of pseudo algebras, pseudo functors, transformations, and modifications for a Gray-monad $T$ derived from left Kan extension. Inspired by a similar situation in two-dimensional monad theory, we a…
We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith's theorem for accessible model categories.
Nakaoka-Ogawa-Sakai considered the localization of an extriangulated category. This construction unified the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Recently, Herschend-Liu-Nakaoka defined $n$-exangulated categories as a higher dimensional analogue of extriangulated categories. Let $\mathcal C$ be an $n$-exangulated category and $\mathcal{F}$ be a multiplicative system satisfying mild assumption. In this article, we give a necessary and sufficient condition for the localization of $\mathcal C$ be an $n$-exangulated category. This way gives a ne…
We introduce Artin-Wraith glueing and locally closed inclusions in double categories. Examples include locales, toposes, topological spaces, categories, and posets. With appropriate assumptions, we show that locally closed inclusions are exponentiable, and the exponentials are constructed via Artin-Wraith glueing. Thus, we obtain a single theorem establishing the exponentiability of locally closed inclusions in these five cases.
We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly un…
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The definition of the new class of model categories is based on the corresponding extension of the theory of locally presentable and accessible categories developed in the companion paper [arXiv:1110.0605], where we introduced the concepts of locally class-presentable and class-accessible categories. In this work we prove that the category of weak equivalences of …
We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.
We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categor…
Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ in terms of the section functors. Finally we study abstract local cohomology functor on the derived category $\mathcal{D}^+(\mathcal{A})$.
We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives and we consider here the case of mixed motives over a regular base scheme.
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by func…
We prove that the tensor category of quasi-coherent modules $\mathsf{Qcoh}(X \times_S Y)$ on a fiber product of quasi-compact quasi-separated schemes is the bicategorical pushout of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ over $\mathsf{Qcoh}(S)$ in the $2$-category of cocomplete linear tensor categories. In particular, $\mathsf{Qcoh}(X \times Y)$ is the bicategorical coproduct of $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$. For this we introduce idals, which can be seen as non-embedded ideals, and use them to study localizations of cocomplete tensor categories in general.
New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.
We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.
It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a topologically enriched locally contractible small category is Quillen equivalent to the standard Quillen model structure of topological spaces. We give a geometric interpretation of this fact in directed homotopy.
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produc…
The invertibility hypothesis for a monoidal model category S asks that localizing an S-enriched category with respect to an equivalence results in an weakly equivalent enriched category. This is the most technical among the axioms for S to be an excellent model category in the sense of Lurie, who showed that the category of S-enriched categories then has a model structure with characterizable fibrant objects. We use a universal property of cubical sets, as a monoidal model category, to show that the invertibility hypothesis is consequence of the other axioms.
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan correspondence. We study homotopy coherent coalgebras associated to a monoidal monoidal category. We show examples when these coalgebras cannot be rigi…
We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local" geometric realization functor $Cat_{sS} \to Cat_\infty$, and we prove that their homotopy theory presents the $\infty$-category of $\infty$-categories, i.e. that this functor induces an equivalence $Cat_{sS} [[ W_{DK}^{-1} ]] \xrightarrow{\sim} Cat_\infty$ from a localization of the $\infty$-category of $sS$-enriched $\infty$-categories. Following Dwyer--Ka…
We formulate a general abstract criterion for verifying the local-to-global principle for a rigidly-compactly generated tensor triangulated category. Our approach is based upon an inductive construction using dimension functions. Using our criterion we give a new proof of the theorem that the local-to-global principle holds for such categories when they have a model and the spectrum of the compacts is noetherian. As further applications we give a new set of conditions on the spectrum of the compacts that guarantee the local-to-global principle holds and use this to classify localising subcateg…
We examine various triangulated quotients of the module category of a finite group. We demonstrate that these are not compactly generated by the simple modules and present a modification of Rickard's Idempotent Module construction that accounts for this. When the localizing subcategories are sufficiently nice we give an explicit description of the objects in the Bousfield triangles for modules that are direct limits of sequences of finite dimensional modules in terms of homotopy colimits.
** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete $C^{*}$-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal $*$-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand…
The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors -- a natural generalization of inverse semigroups -- and extend the ESN theorem to an equivalence between this category and the category of top-heavy locally inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally…
Local cohomology functors are constructed for the category of cohomological functors on an essentially small triangulated category T equipped with an action of a commutative noetherian ring. This is used to establish a local-global principle and to develop a notion of stratification, for T and the cohomological functors on it, analogous to such concepts for compactly generated triangulated categories.
We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids.
We construct a category of fibrant objects $\mathbb{C}\langle P\rangle$ in the sense of K. Brown from any indexed frame (a kind of indexed poset generalizing triposes) $P$, and show that its homotopy category is the Barr-exact category $\mathbb{C}[P]$ of partial equivalence relations and compatible functional relations. In particular this gives a presentation of realizability toposes as homotopy categories. We give criteria for the existence of left and right derived functors to functors $\mathbb{C}\langleΦ\rangle : \mathbb{C}\langle P\rangle\to \mathbb{C}\langle Q\rangle$ induced by finite-me…
In a Brown category of cofibrant objects, there is a model for the mapping spaces of the hammock localization in terms of zig-zags of length 2. In this paper we show how to assemble these spaces into a Segal category that models the infinity-categorical localization of the Brown category.
Cofibration categories are a formalization of homotopy theory useful for dealing with homotopy colimits that exist on the level of models as colimits of cofibrant diagrams. In this paper, we deal with their enriched version. Our main result claims that the category $[\mathcal{C},\mathcal{M}]$ of enriched diagrams equipped with the projective structure inherits a structure of a cofibration category whenever $\mathcal{C}$ is locally cofibrant (or, more generally, locally flat).
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints t…
In 1996, Franke constructed a purely algebraic category that is equivalent as a triangulated category to the E(n)-local stable homotopy category for n^2+n < 2p-2. The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n=1 Franke's functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke's systems of triangulated diagram categories and …
We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R.
