typical contexts
A quality type is a key concept in W. Lawvere‘s axiomatic approach to cohesion which permits to analyze a space understood as a domain of quantitative variation via its qualitative aspects. This can be viewed as an axiomatisation of the commonly encountered situation in geometry or dynamics that problems permit (only) a qualitative analysis.
A primary example for that kind of deferred analysis is the study of topological spaces via the homotopy category (hence via the homotopy types which they represent). So in a very broad sense, quality types are intended as ingredients to a synthetic homotopy theory^{1} where the (homotopy) contraction of a space (the collapsing of a cylinder/two idempotents) is extracted as essence of the concept of a spatial ‘attribute’.
Technically, a quality type amounts to a special sort of essential localization and is therefore called a quintessential localization in (Johnstone 1996).
In typical (topos) cases these come in the form of adjoint cylinders $\Pi\dashv\Delta\dashv\Gamma$ (with $\Delta$ fully faithful) such that the connected components and the global sections functor coincide (=“collapse”): $\Pi\cong\Gamma$ (cf. points-to-pieces transform). This implies for spaces in the domain of the functors that every connected component contains exactly one point. Spaces with this property are called infinitesimal spaces^{2} in (Lawvere 2008).
Hence from a more geometrical point of view, an object in a quality type is a particular simple kind of space with ‘degenerate’ components, or, if you prefer, a space with ‘thick’ or ‘coarse’ points which in turn can be viewed as a minimal vestige of cohesion: when a set is a space with no cohesion, an object in a quality type is a space with almost no cohesion.^{3} (For an elaboration of this perspective in the context of cohesive toposes see at infinitesimal cohesion.)
Quality types together with the continuity axiom are an essential ingredient to Lawvere’s (2007 axioms) for geometry:
That Continuity axiom (preservation of infinite products by $\pi_0$) was introduced in order to obtain homotopy types that are “qualities” in an intuitive sense^{4} (as they should be automatically in the continuous case). Lawvere (message to catlist oct 26th 2011)
(Lawvere 1998, Lawvere 2008, Marmolejo-Menni 2016) also point to the role of continuity in insuring that the Hurewicz homotopy category becomes a quality type. The definition of quality type yields important contrasts with pure variation and sufficient cohesion permitting to attend to the fine structure of the petit-gros landscape. So we can say with only slight exaggeration that the 2007 axioms revolve around the concept of a quality type and that cohesive spaces are those that admit qualitative (homotopical) analysis!
In the following we discuss the case for ordinary categories, for the higher order generalization see infinitesimal cohesive (infinity,1)-topos.
Let $\mathcal{S},\mathcal{F}$ be extensive categories. A fully faithful functor $q^*:\mathcal{S}\to\mathcal{F}$ is said to exhibit $\mathcal{F}$ as a quality type over $\mathcal{S}$ if $q^*$ has both a left and a right adjoint which moreover coincide: $q_!\dashv q^*\dashv q_*$ and $q_!\cong q_*$.
Let $\mathcal{E}$ be a (pre)cohesive category over $\mathcal{S}$ with adjoint string $p_!\dashv p^* \dashv p_* \dashv p^!:\mathcal{S}\to\mathcal{E}$ and $q^*:\mathcal{S}\to\mathcal{F}$ a quality type over $\mathcal{S}$. A functor $s:\mathcal{E}\to\mathcal{F}$ is called an intensive quality on $\mathcal{E}$ if $s$ preserves finite products and finite coproducts and $q_*\circ s =p_*$.
Dually, a finite coproduct preserving functor $h:\mathcal{E}\to\mathcal{F}$ with $q_!\circ h=p_!$ is called an extensive quality on $\mathcal{E}$.
Intuitively, an intensive quality is compatible with the points of its domain spaces and an extensive quality with the connected components.
Note that a quality type yields itself a (pre)cohesive category with (degenerate) adjoint string $q_!\dashv q^*\dashv q_*\dashv q^*$. This situations obtains when the induced transformation $p_*\to p_!$ of the Nullstellensatz is not only an epimorphism but a natural isomorphism (cf. Lawvere 2007).
If we divest smooth spaces of all global cohesion, keeping only the jets (on which the Thom-Mather singularities depend), we obtain a category in which every connected component of any object has exactly one point, so that the natural map between those functors is an isomorphism. Lawvere (2004, p.108)
Kan complexes over $Set$: The role of Kan complexes in this context is discussed in Marmolejo-Menni (2016).
Galois theory - presheaves on the opposite of the category of finite-dimensional local k-algebras (Lawvere 2004).
The properties of (pre)cohesive categories are fine-tuned to yield a canonical extensive quality via the Hurewicz construction $\pi (X^Y)$. In the blueprint for this, namely the combinatorial homotopy theory in Gabriel-Zisman (1967, chap. III), the category of spaces has objects Kelley spaces. In a sense, the concept of continuity can be viewed as a mean to bypass the construction of the category of fractions necessary in this case.
Lawvere’s work on the petit-gros topos distinction (Lawvere 1976) started with the examination of two toposes of graphs (Lawvere 1986). The first one of “irreflexive” graphs is given by the presheaf topos of actions of the small diagram category $E\rightrightarrows V$ on $Set$ and is an étendue hence petit.
Whereas the second one of reflexive graphs is given by the actions of the graphic monoid $\Delta _1=\{1,\partial_0,\partial_1\}$ with $\partial_i\partial_j=\partial_i$ for $i,j=0,1$. $\Delta _1$ is Morita equivalent to the diagram category $E\stackrel{\leftrightarrows}{\leftarrow} V$ and consists entirely of idempotents. Its topos of actions is gros and when taken over $FinSet$ satisfies the axioms for a (sufficiently) cohesive topos. The ‘abelianization’ of $\Delta _1$ is $\mathbb{F}_1=\{1,e\}$, the multiplicative monoid with a generic (nontrivial) idempotent $e$ which incidentally is also the monoid underlying the blueprint for the field with one element.
As observed in (Lawvere 1989, p.277) $\mathcal{S}^{\mathbb{F}_1^{op}}$ is a quality type over $\mathcal{S}$. It was probably the first one to arise and its status of being neither petit nor gros being commented on. Some of the details are spelled out in the following section.
The surjective homomorphism $q:\Delta_1\to\mathbb{F}_1$ induces an adjoint string $q_!\dashv q^*\dashv q_*:\mathcal{S}^{\Delta_1^{op}}\to\mathcal{S}^{\mathbb{F}_1^{op}}$ with $q_*$ discarding all non-loops in a reflexive graph.
For details how the zeta function arises via the Burnside ring of $\mathcal{S}^{\mathbb{F}_1^{op}}$ in this context see (Lawvere 1989, pp.291-292). $\mathbb{F}_1$-torsors are discussed in Johnstone (2002, p.380).
Properties of quality types were first studied by P. Johnstone (1996) using the language of localizations.
Let $\mathcal{C}$ be a finitely complete category. An essential localization $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is called quintessential if $l$ is naturally isomorphic to $i$.
A trivial example of a quintessential localization is provided by $id_\mathcal{C}$.
Another simple example of a quintessential localization is given by the category $\mathcal{C}$ with objects pairs $(X, e)$ where $X$ is a set and $e=e^2$ an idempotent map $X\to X$. A morphism $f:(X_1, e_1)\to (X_2, e_2)$ is a function $f:X_1\to X_2$ with $f\cdot e_1=e_2\cdot f$. These equivariant morphisms are bound to preserve fixpoints: when $e_1(x)=x$ then $f(e_1(x))=f(x)=e_2(f(x))$. Then the fixpoint set functor $r:\mathcal{C}\to Set$ with $r(X, e)=\{x\in X | e(x)=x \}$ is left as well as right adjoint to $i(X)=(X, id_X)$ since an equivariant morphism $f:(X,e)\to (Y,id_Y)$ is uniquely determined by its restriction to the fixpoints of $e$ and its values are given by $f(e(x))$. The adjoint modalities $i\cdot r\dashv i\cdot r:\mathcal{C}\to\mathcal{C}$ corresponding to $i\dashv r\dashv i:Set\to\mathcal{C}$ map $(X,e)$ to $(r(X),id_{r(X)})$. Of course, $\mathcal{C}$ is up to equivalence just the category $\mathcal{S}^{\mathbb{F}_1^{op}}$ from above!
Consider the monoid $(\overline{\mathbb{N}}, \cdot , 0)$ consisting of the natural numbers in their usual order with a point $\infty$ attached at infinity such that $\forall n\in \mathbb{N}: n$<$\infty$ with multiplication $m\cdot n=max\{m,n\}$. $\overline{\mathbb{N}}$ is commutative and satisfies the graphic identity $x\cdot y\cdot x = x\cdot y$. Then $i:Set\hookrightarrow Set^{\overline{\mathbb{N}}}$ with $i(X)=(X, \pi_2:\overline{\mathbb{N}}\times X\to X)$ exhibits $Set^{\overline{\mathbb{N}}}$ as a quality type over $Set$ (Cf. Freyd et al. (1999) and O’Hearn et al. (1995)).
This can be viewed as a more elaborate version of the preceding example where $\infty$ corresponds to $e$ upon identifying $\mathcal{C}$ as the category $\mathcal{S}^{\mathbb{F}_1^{op}}$.
(Caveat: this needs checking!)
To say that $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is a quintessential localization amounts to say that $i:\mathcal{L}\to\mathcal{C}$ exhibits $\mathcal{C}$ as a quality type over $\mathcal{L}$ with $r$ providing the right adjoint to $i\simeq l$ (provided $\mathcal{L}$, $\mathcal{C}$ are extensive).
Note that a quintessential subtopos is dense since $i$ is up to natural isomorphism a left adjoint whence preserves all colimits and the initial object in particular!
Since the adjoint modalities $l\cdot r\dashv i\cdot r$ corresponding to a quintessential subtopos as a level coincide up to natural isomorphism, one sees that quintessential levels are their own Aufhebung.
Let $\mathcal{C}$ be a finitely complete category. A localization $r\dashv i:\mathcal{L}\to\mathcal{C}$ is called persistent if $\mathcal{L}$ is closed under subobjects in $\mathcal{C}$.
Quintessential localizations of toposes are persistent (Johnstone 1996, p.94). Since separated objects for a Lawvere-Tierney topology $j$ are precisely the subobjects of $j-sheaves$ it follows that a persistent (and, in particular, a quintessential) subtopos coincides with its quasitopos of $j$-separated objects. Conversely, a subtopos with the property that it coincides with the corresponding quasitopos is persistent.
Let $\mathcal{E}$ be a Cauchy- complete category. The quintessential localizations of $\mathcal{E}$ correspond precisely to idempotent endomorphisms of the identity functor on $\mathcal{E}$. Moreover, quintessential subcategories of $\mathcal{E}$ form a semilattice under intersection.
For the proof cf. (Johnstone 1996, p.93).
The definition of quality types occurs in (Lawvere 2007) with important additional remarks in (Lawvere 2008). The first occurrence of a quality type is in the work of the 1980s in the context of the gros topos of graphs: a discussion of this example is in (Lawvere 1989, p.277) where already the contrast to sufficient cohesion is observed. (Lawvere 1991, pp.9-10) has a short but suggestive discussion in the context of synthetic differential geometry. The short abstract (Lawvere 1998) anticipates the 2007 axioms as ‘fine-tuning’ to accomodate the homotopy category as a quality type. Quality types resurface without an explicit definition in the 2004 paper on data types.
Under the name of quintessential localization they are the focus of (Johnstone 1996) where the characterization via central idempotents is given. Some additional results occur in the context of the Nullstellensatz in (Johnstone 2011). (Menni 2014) attends to the contrast between quality types and sufficient cohesion. A variant is studied as the concept of a bireflective subcategory in Freyd et al. (1999).
P. J. Freyd, P. W. O’Hearn, A. J. Power, M. Takeyama, R. Street, R. D. Tennent, Bireflectivity , Theor. Comp. Sci. 228 (1999) pp.49-76.
P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory , Springer Heidelberg 1967.
P. Johnstone, Remarks on Quintessential and Persistent Localizations , TAC 2 no.8 (1996) pp.90-99. (pdf)
P. Johnstone, Remarks on Punctual Local Connectedness , TAC 25 no.3 (2011) pp.51-63. (pdf)
P. Johnstone, Sketches of an Elephant vol. 1 , Cambridge UP 2002. (pp.202,380)
M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.
F. W. Lawvere, Variable quantities and variable structures in topoi , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.
F. W. Lawvere, Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs, Revista Colombiana de Matemáticas XX (1986) pp.179-186. Reprinted with commentary in TAC 9 (2005) pp.1-7. (pdf)
F. W. Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs , Cont. Math. 92 (1989) pp.261-299.
F. W. Lawvere, Some Thoughts on the Future of Category Theory , pp.1-13 in Springer LNM 1488 (1991).
F. W. Lawvere, Categories of Space and Quantity, pp.14-30 in: J. Echeverria et al (eds.), The Space of Mathematics , de Gruyter Berlin 1992.
F. W. Lawvere, Unity and Identity of Opposites in Calculus and Physics , App. Cat. Struc 4 (1996) pp.167-174.
F. W. Lawvere, Are Homotopy Types the same as Infinitesimal Skeleta ? , abstract (1998). (link)
F. W. Lawvere, Kinship and Mathematical Categories , pp.411-425 in: R. Jackendoff, P. Bloom, K. Wynn (eds), Language, Logic, and Concepts - Essays in Memory of John Macnamara, MIT Press 1999.
F. W. Lawvere, Left and right adjoint operations on spaces and data types , Theor. Comp. Sci. 316 (2004) pp.105-111.
F. W. Lawvere, Axiomatic cohesion , TAC 19 no.3 (2007) pp.41–49. (pdf)
F. W. Lawvere, Cohesive Toposes: Combinatorial and Infinitesimal Cases, Como Ms. 2008. (pdf)
F. Marmolejo, M. Menni, On the relation between continuous and combinatorial , arXiv:1602.02826 (2016). (abstract)
M. Menni, Sufficient Cohesion over Atomic Toposes , Cah. Top. Géom. Diff. Cat. LV (2014). (preprint)
M. Menni, Continuous Cohesion over Sets , TAC 29 no.20 (2014) pp.542-568. (pdf)
P. W. O’Hearn, A. J. Power, M. Takeyama, R. D. Tennent, Syntactic Control of Interference Revisited , Electronic Notes in Theor. Comp. Sci. 1 (1995) pp.1-40.
R. Paré, R. Rosebrugh, R. J. Wood, Idempotents in Bicategories , Bull. Austr. Math. Soc. 39 (1989) pp.421-434.
C. Ruiz S., R. Ruiz, Conditions for a Realization Functor to Commute with Finite Products , Revista Colombiana de Matemáticas XV (1981) pp.113-146. (gdz)
The term synthetic homotopy theory is slightly misleading here as Lawvere’s ideas don’t provide a fully fledged homotopy theory but are rather observations on the role of the cylinder configuration in the analysis of space and spatial dimension. Important facets of his views are documented in Lawvere 1992, 1994, 1999 and a more thorough discussion of the homotopical apects of the concept appears in Marmolejo-Menni (2016). In a complementary direction, homotopy type theory provides a synthetic formulation of homotopy theory proper, not just of the homotopy category, and cohesive homotopy type theory implements cohesion in that natively homotopy-theoretic context. ↩
A motivation for this terminology from SDG can be found in a remark in (Lawvere 1991, pp.9-10). ↩
From another perspective, one could view an object in a quality type as a set with germs of cohesion. ↩
The claim that quality types intend to model the philosophical concept of quality reoccurs at several places in Lawvere’s writings though the concrete connection still needs to be spelled out. Some motivation is provided in Lawvere (1992) where it is linked to the negation of quantity as a logical category of being that is indifferent to non-being (cf. Hegel’s Science of Logic) e.g. whereas the temperature varies continuously or “indifferently” below and above zero degree, the same transition makes a crucial difference for the phase or “qualitative being” of water. Note that Lawvere sticks here to the traditional view that quantitative being comes before qualitative being whereas Hegel in his Logic reversed the traditional hierarchy of the categories. ↩