Geach and Relative Identity

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The Theory of Relative Identity is a logical innovation due to Peter Thomas Geach  (P.T. Geach-Logic Matters) motivated by the same sort of mathematical examples as Frege’s definition by abstraction. Like Frege Geach seeks to give a logical sense to mathematical talk “up to” a given equivalence E through replacing E by identity but unlike Frege he purports, in doing so, to avoid the introduction of new abstract objects (which in his view causes unnecessary ontological inflation). The price for the ontological parsimony is Geach’s repudiation of Frege’s principle of a unique and absolute identity for the objects in the domain over which quantified variables range. According to Geach things can be same in one way while differing in others. For example two printed letters aa are same as a type but different as tokens. In Geach’s view this distinction does not commit us to a-tokens and a-types as entities but presents two different ways of describing the same reality. The unspecified (or “absolute” in Geach’s terminology) notion of identity so important for Frege is in Geach’s view is incoherent.

Geach’s proposal appears to account better for the way the notion of identity is employed in mathematics since it does not invoke “directions” or other mathematically redundant concepts. It captures particularly well the way the notion of identity is understood in Category theory. According to Baez & Dolan

In a category, two objects can be “the same in a way” while still being different.

So in Category theory the notion of identity is relative in exactly Geach’s sense. But from the logical point of view the notion of relative identity remains highly controversial. Let x,y be identical in one way but not in another, or in symbols: Id(x,y) & ¬Id'(x,y). The intended interpretation assumes that x in the left part of the formula and x in the right part have the same referent, where this last same apparently expresses absolute not relative identity. So talk of relative identity arguably smuggles in the usual absolute notion of identity anyway. If so, there seems good reason to take a standard line and reserve the term “identity” for absolute identity.

We see that Plato, Frege and Geach propose three different views of identity in mathematics. Plato notes that the sense of “the same” as applied to mathematical objects and to the ideas is different: properly speaking, sameness (identity) applies only to ideas while in mathematics sameness means equality or some other equivalence relation. Although Plato certainly recognizes essential links between mathematical objects and Ideas (recall the “ideal numbers”) he keeps the two domains apart. Unlike Plato Frege supposes that identity is a purely logical and domain-independent notion, which mathematicians must rely upon in order to talk about the sameness or difference of mathematical objects, or any other kind at all. Geach’s proposal has the opposite aim: to provide a logical justification for the way of thinking about the (relativized) notions of sameness and difference which he takes to be usual in mathematical contexts and then extend it to contexts outside mathematics (As Geach says):

Any equivalence relation … can be used to specify a criterion of relative identity. The procedure is common enough in mathematics: e.g. there is a certain equivalence relation between ordered pairs of integers by virtue of which we may say that x and y though distinct ordered pairs, are one and the same rational number. The absolute identity theorist regards this procedure as unrigorous but on a relative identity view it is fully rigorous.

Conjuncted: Of Topos and Torsors

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The condition that each stalk Fx be equivalent to a classifying space BG can be summarized by saying that F is a gerbe on X: more precisely, it is a gerbe banded by the constant sheaf G associated to G.

For larger values of n, even the language of stacks is not sufficient to describe the nature of the sheaf F associated to the fibration X~ → X. To address the situation, Grothendieck proposed (in his infamous letter to Quillen; see [35]) that there should be a theory of n-stacks on X, for every integer n ≥ 0. Moreover, for every sheaf of abelian groups G on X, the cohomology group Hn+1sheaf(X;G) should have an interpreation as sheaf classifying a special type of n-stack: namely, the class of n-gerbes banded by G. In the special case where the space X is a point (and where we restrict our attention to n-stacks in groupoids), the theory of n-stacks on X should recover the classical homotopy theory of n-types: that is, CW complexes Z such that the homotopy groups πi(Z, z) vanish for i > n (and every base point z ∈ Z). More generally, we should think of an n-stack (in groupoids) on a general space X as a “sheaf of n-types” on X.

When n = 0, an n-stack on a topological space X simply means a sheaf of sets on X. The collection of all such sheaves can be organized into a category ShvSet(X), and this category is a prototypical example of a Grothendieck topos. The main goal of this book is to obtain an analogous understanding of the situation for n > 0. More precisely, we would like answers to the following questions:

(Q1) Given a topological space X, what should we mean by a “sheaf of n-types” on X?

(Q2)  Let Shv≤n(X) denote the collection of all sheaves of n-types on X. What sort of a mathematical object is Shv≤n(X)?

(Q3)  What special features (if any) does Shv≤n(X) possess?

Our answers to questions (Q2) and (Q3) may be summarized as follows:

(A2)  The collection Shv≤n(X) has the structure of an ∞-category.

(A3)  The ∞-category Shv≤n(X) is an example of an (n+1)-topos: that is, an ∞-category which satisfies higher categorical analogues of Giraud’s axioms for Grothendieck topoi.

Grothendieck’s vision has been realized in various ways, thanks to the work of a number of mathematicians (most notably Jardine), and their work can also be used to provide answers to questions (Q1) and (Q2). Question (Q3) has also been addressed (at least in limiting case n = ∞) by Toën and Vezzosi

To provide more complete versions of the answers (A2) and (A3), we will need to develop the language of higher category theory. This is generally regarded as a technical and forbidding subject. More precisely, we will need a theory of (∞, 1)-categories: higher categories C for which the k-morphisms of C are required to be invertible for k > 1.

Classically, category theory is a useful tool not so much because of the light it sheds on any particular mathematical discipline, but instead because categories are so ubiquitous: mathematical objects in many different settings (sets, groups, smooth manifolds, etc.) can be organized into categories. Moreover, many elementary mathematical concepts can be described in purely categorical terms, and therefore make sense in each of these settings. For example, we can form products of sets, groups, and smooth manifolds: each of these notions can simply be described as a Cartesian product in the relevant category. Cartesian products are a special case of the more general notion of limit, which plays a central role in classical category theory. 

Badiou, Heyting Algebras cross the Grothendieck Topoi. Note Quote.

Let us commence by introducing the local formalism that constitutes the basis of Badiou’s own, ‘calculated phenomenology’. Badiou is unwilling to give up his thesis that the history of thinking of being (ontology) is the history of mathematics and, as he reads it, that of set theory. It is then no accident that set theory is the regulatory framework under which topos theory is being expressed. He does not refer to topoi explicitly but rather to the so called complete Heyting algebras which are their procedural equivalents. However, he fails to mention that there are both ‘internal’ and ‘external’ Heyting algebras, the latter group of which refers to local topos theory, while it appears that he only discusses the latter—a reduction that guarantees that indeed that the categorical insight may give nothing new.

Indeed, the external complete Heyting algebras T then form a category of the so called T-sets, which are the basic objects in the ‘world’ of the Logics of Worlds. They local topoi or the so called ‘locales’ that are also ‘sets’ in the traditional sense of set theory. This ‘constitution’ of his worlds thus relies only upon Badiou’s own decision to work on this particular regime of objects, even if that regime then becomes pivotal to his argument which seeks to denounce the relevance of category theory.

This problematic is particularly visible in the designation of the world m (mathematically a topos) as a ‘complete’ (presentative) situation of being of ‘universe [which is] the (empty) concept of a being of the Whole’ He recognises the ’impostrous’ nature of such a ‘whole’ in terms of Russell’s paradox, but in actual mathematical practice the ’whole’ m becomes to signify the category of Sets – or any similar topos that localizable in terms of set theory. The vocabulary is somewhat confusing, however, because sometimes T is called the ‘transcendental of the world’, as if m were defined only as a particular locale, while elsewhere m refers to the category of all locales (Loc).

An external Heyting algebra is a set T with a partial order relation <, a minimal element μ ∈ T , a maximal element M ∈ T . It further has a ‘conjunction’ operator ∧ : T × T → T so that p ∧ q ≤ p and p ∧ q = q ∧ p. Furthermore, there is a proposition entailing the equivalence p ≤ q iff p ∧ q = p. Furthermore p ∧ M = p and μ ∧ p = μ for any p ∈ T .

In the ‘diagrammatic’ language that pertains to categorical topoi, by contrast, the minimal and maximal elements of the lattice Ω can only be presented as diagrams, not as sets. The internal order relation ≤ Ω can then be defined as the so called equaliser of the conjunction ∧ and projection-map

≤Ω →e Ω x Ω →π1 L

The symmetry can be expressed diagrammatically by saying that

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is a pull-back and commutes. The minimal and maximal elements, in categorical language, refer to the elements evoked by the so-called initial and terminal objects 0 and 1.

In the case of local Grothendieck-topoi – Grothendieck-topoi that support generators – the external Heyting algebra T emerges as a push-forward of the internal algebra Ω, the logic of the external algebra T := γ ∗ (Ω) is an analogous push-forward of the internal logic of Ω but this is not the case in general.

What Badiou further requires of this ‘transcendental algebra’ T is that it is complete as a Heyting algebra.

A complete external Heyting algebra T is an external Heyting algebra together with a function Σ : PT → T (the least upper boundary) which is distributive with respect to ∧. Formally this means that ΣA ∧ b = Σ{a ∧ b | a ∈ A}.

In terms of the subobject classifier Ω, the envelope can be defined as the map Ωt : ΩΩ → Ω1 ≅ Ω, which is internally left adjoint to the map ↓ seg : Ω → ΩΩ that takes p ∈ Ω to the characteristic map of ↓ (p) = {q ∈ Ω | q ≤ p}27.

The importance the external complete Heyting algebra plays in the intuitionist logic relates to the fact that one may now define precisely such an intuitionist logic on the basis of the operations defined above.

The dependence relation ⇒ is an operator satisfying

p ⇒ q = Σ{t | p ∩ t ≤ q}.

(Negation). A negation ¬ : T → T is a function so that

¬p =∑ {q | p ∩ q = μ},

and it then satisfies p ∧ ¬p = μ.

Unlike in what Badiou calls a ‘classical world’ (usually called a Boolean topos, where ¬¬ = 1Ω), the negation ¬ does not have to be reversible in general. In the domain of local topoi, this is only the case when the so called internal axiom of choice is valid, that is, when epimorphisms split – for example in the case of set theory. However, one always has p ≤ ¬¬p. On the other hand, all Grothendieck-topoi – topoi still materially presentable over Sets – are possible to represent as parts of a Boolean topos.

The Ubiquity of Self-Predicative Universality of Adjoint Functors. Note Quote.

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One of the most important and beautiful notions in category theory is the notion of a pair of adjoint functors. The developers of category theory, Saunders MacLane and Samuel Eilenberg, famously said that categories were defined in order to define functors, and functors were defined in order to define natural transformations. Adjoints were defined more than a decade later by Daniel Kan but the realization of their ubiquity (“Adjoint functors arise everywhere” (MacLane) and their foundational importance has steadily increased over time (Lawvere). Now it would perhaps not be too much of an exaggeration to see categories, functors, and natural transformations as the prelude to defining adjoint functors. The notion of adjoint functors includes all the instances of self-predicative universal mapping properties discussed above. As Steven Awodey (179) put it:

The notion of adjoint functor applies everything that we have learned up to now to unify and subsume all the different universal mapping properties that we have encountered, from free groups to limits to exponentials. But more importantly, it also captures an important mathematical phenomenon that is invisible without the lens of category theory. Indeed, I will make the admittedly provocative claim that adjointness is a concept of fundamental logical and mathematical importance that is not captured elsewhere in mathematics.

“The isolation and explication of the notion of adjointness is perhaps the most profound contribution that category theory has made to the history of general mathematical ideas.” (Goldblatt)

How do the ubiquitous and important adjoint functors relate to theme of self- predicative universals? MacLane and Birkhoff succinctly state the idea of the self-predicative universals of category theory and note that adjunctions can be analyzed in terms of those universals. The construction of a new algebraic object will often solve a specific problem in a universal way, in the sense that every other solution of the given problem is obtained from this one by a unique homomorphism. The basic idea of an adjoint functor arises from the analysis of such universals. (MacLane and Birkhoff)

We will use a specific novel treatment of adjunctions (Ellerman) that shows they arise by gluing together in a certain way two universal constructions or self-predicative universals (“semi-adjunctions”). But for illustration, we will stay within the methodological restriction of using examples from partial orders (where adjunctions are called “Galois connections”).

We have been working within the inclusion partial order on the set of subsets ζ(U) of a universe set U. Consider the set of all ordered pairs of subsets <a,b> from the Cartesian product ζ(U) x ζ(U) where the partial order (using the same symbol) is defined by pairwise inclusion. That is, given the two ordered pairs <a’, b’> and <a,b>, we define

<a’,b’> ⊆ <a,b> if a ⊆’  a and b ⊆’  b.

Order-preserving maps can be defined each way between these two partial orders. From ζ(U) to ζ(U) x ζ(U), there is the diagonal map Δ(x) = <x,x>, and from ζ(U) x ζ(U) to ζ(U), there is the meet map ∩(<a,b>)  = a ∩ b. Consider now the following “adjointness relation” between the two partial orders:

Δ(c) ⊆ <a,b> iff c ⊆ ∩ (<a,b>) Adjointness Equivalence

for sets a, b, and c in ζ(U). It has a certain symmetry that can be exploited. If we fix <a,b>, then we have the previous universality condition for the meet of a and b: for any c in ζ(U), c ⊆ a ∩ b iff Δ(c) ⊆ <a,b> Universality Condition for Meet of Sets a and b.

The defining property on elements c of ζ(U) is that Δ(c) ⊆ <a,b>. But using the symmetry, we could fix c and have another universality condition using the reverse inclusion in ζ(U) x ζ(U) as the participation relation: for any <a,b> in ζ(U) x ζ(U), <a,b> ⊇ Δ(c) iff c ⊆ a ∩  b. Universality Condition for Δ(c). Here the defining property on elements <a,b> of ζ(U) x ζ(U) is that “the meet of a and b is a superset of the given set c.” The self-predicative universal for that property is the image of c under the diagonal map Δ(c) = <c,c>, just as the self-predicative universal for the other property defined given <a,b> was the image of <a,b> under the meet map ∩(<a,b>) = a ∩ b.

Thus in this adjoint situation between the two categories ζ(U) and ζ(U) x ζ(U), we have a pair of maps (“adjoint functors”) going each way between the categories such that each element in a category defines a certain property in the other category and the map carries the element to the self-predicative universal for that property.

Δ: ζ(U) → ζ(U) x ζ(U) and ∩: ζ(U) x ζ(U) → ζ(U) Example of Adjoint Functors Between Partial Orders

The notion of a pair of adjoint functors is ubiquitous; it is one of the main tools that highlights self-predicative universals throughout modern mathematics.

Conjuncted: The Prerogative of Category Theory Over Set Theory in Physics. Note Quote.

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When it comes to deal with structures, in particular in abstract branches of mathematics – abstract in comparison to number theory, analysis and the geometry of figures, curves and planes -, such as algebraic topology, homology and homotopy theory, universal algebra, and what have you, a vast majority of mathematicians considers Category-Theory (CT) vastly superior to set-theory. CT also is the only rival to ZFC (Zermelo–Fraenkel Choice set theory) in providing a general theory of mathematical structure and in founding the whole of mathematics. The language of CT is two-sorted: it contains object-variables and arrow-variables. An arrow sends objects to objects; an identity-arrow sends an object to itself. Simply put, structures are categories, and a category is something that has objects and arrows, such that the arrows can be composed so as to form a composition monoid, which means that: (i) every object has an identity-arrow, and (ii) arrow-composition is associative. The languages of CT (L↑) and ZFC (L∈) are inter-translatable. In CT there is the specific category Set, whose objects can be identified with sets and whose arrows are maps. In ZFC one can identify objects with sets and arrows with ordered pair-sets of type ⟨f, C⟩, consisting of a mapping f and a co-domain C.

In spite of the fact that some mathematical physicists have applied categories to physics, not a single structural realist on record has advocated replacing ZFC with CT. One of the very few critics of the use of set-theory for Structural Realism is E.M. Landry, who has argued that the set-theoretical framework does not always do the work it has been suggested to do; but even she does not openly advocate CT as the superior framework for StrR, although she does advocate it for mathematical structuralism.

The objects of CT are more general than the Ur-elements one can introduce in ZFC, because whereas primordial elements are not sets, the objects of CT can be anything, arrows, sets, functors and categories included. Similar to ZFCU is that CT does not have axioms that somehow restrict the interpretation of ‘object’. A CT-object is anything that can be sent around by an arrow, similar to the fact that a set-theoretical Ur-element is anything that can be put in a set. CT-objects obtain an ‘identity’, a ‘nature’, from the category they are in: different category, different identity. Outside categories, these objects lose whatever properties and relations they had in the category they came from and they become essentially indiscernible.

One great advantage of CT is that structures, i.e. categories, are not accompanied by all these sets that arise by iterated applications of the power-set and union-set operation. Nevertheless, the grim story we have been telling for Structural Realism in the framework of ZFC, can be repeated in the framework of CT, of course with a few appropriate adjustments. 

Object as Category-Theoretic or Object as Ontological: The Inadequacy of Implicitly Quantifying Over Elements. (2)

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It will be convenient to use the term ‘object’ in two senses. First, as an object of a category, i.e. in a purely mathematical sense. We shall call this a C- object (‘C’ for category-theoretic). Second, in the sense commonly used in structural realist debates, and which was already introduced above, viz. an object is a physical entity which is a relatum in physical relations. We shall call this an O-object (‘O’ for ‘ontological’).

We will also need to clarify our use of the term ‘element’. We use ‘element’ to mean an element of a set, or as it is also often called, a ‘point’ of a set (indeed it will be natural for us to switch to the language of points when discussing manifolds, i.e. spacetimes.) This familiar use of element should be distinguished from the category-theoretic concepts of ‘global element’ and ‘generalized element’, which is introduced below.

Jonathan Bain’s first strategy for defending (Objectless) draws on the following idea: the usual set-theoretic representations of O-objects and relations can be translated into category-theoretic terms, whence these objects can be eliminated. In fact, the argument can be seen as consisting of two different parts.

In the first part, Bain attempts to give a highly general argument, in the sense that it turns only on the notion of universal properties and the translatability of statements about certain mathematical representations (i.e. elements of sets) of O-objects into statements about morphisms between C-objects. As Bain himself notes, the general argument fails, and he thus introduces a more specific argument, which is what he wishes to endorse. The specific argument turns on the idea of obtaining a translation scheme from a ‘categorical equivalence’ between a geometric category and an algebraic category, which in turn allows one to generalize the original C-objects. The argument is ‘specific’ because such equivalences only hold between rather special sorts of categories.

The details of Bain’s general argument can be reconstructed as follows:

G1: Physical objects and the structures they bear are typically identified with the elements of a set X and relations on X respectively.

G2: The set-theoretic entities of G1 are to be represented in category-theoretic language by considering the category whose objects are the relevant structured sets, and whose morphisms are functions that preserve ‘structure’.

G3: Set-theoretic statements about an object of a category (of the type in G2) can often be expressed without making reference to the elements of that object. For instance:

1. In any category with a terminal object any element of an object X can be expressed as a morphism from the terminal object to X. (So for instance, since the singleton {∗} is the terminal object in the category Set, an element of a set X can be described by a morphism {∗} → X.)

2. In a category with some universal property, this property can be described purely in terms of morphisms, i.e. without making any reference to elements of an object.

To sum up, G1 links O-objects with a standard mathematical representation, viz. elements of a set. And G2 and G3 are meant to establish the possibility that, in certain cases, category theory allows us to translate statements about elements of sets into statements about the structure of morphisms between C-objects.

Thus, Bain takes G1–G3 to suggest that: 

C: Category theory allows for the possibility of coherently describing physical structures without making any reference to physical objects.

Indeed, Bain thinks the argument suggests that the mathematical representatives of O- objects, i.e. the elements of sets, are surplus, and that category theory succeeds in removing this surplus structure. Note that even if there is surplus structure here, it is not of the same kind as, e.g. gauge-equivalent descriptions of fields in Yang-Mills theory. The latter has to do with various equivalent ways in which one can describe the dynamical objects of a theory, viz. field. By contrast, Bain’s strategy involves various equivalent descriptions of the entire theory.

Bain himself thinks that the inference from G1–G3 to C fails, but he does give it serious consideration, and it is easy to see why: its premises based on the most natural and general translation scheme in category theory, viz. redescribing the properties of C-objects in terms of morphisms, and indeed – if one is lucky – in terms of universal properties. 

First, the premise G1. Structural realist doctrines are typically formalized by modeling O-objects as elements of a set and structures as relations on that set. However, this is seldom the result of reasoned deliberation about whether standard set theory is the best expressive resource from some class of such resources, but rather the product of a deeply entrenched set-theoretic viewpoint within philosophy. Were philosophers familiar with an alternative to set theory that was at least as powerful, e.g. category theory, then O-objects and structures might well have been modeled directly in the alternative formalism. Of course, it is also a reasonable viewpoint to say that it is most ‘natural’ to do the philosophy/foundations of physics in terms of set theory – what is ‘natural’ depends on how one conceives of such foundational investigations.

So we maintain that there is no reason for the defender of O-objects to accept G1. For instance, he might try to construct a category such that O-objects are modeled by C-objects and structures are modeled by morphisms. For example, there are examples of categories whose C-objects might coincide with the mathematical representatives of O-objects. For instance, in a path homotopy category, the C-objects are just points of the relevant space, and one might in turn take the points of a space to be O-objects, as Bain does in his example of general relativity and Einstein algebras. Or he might take as his starting point a non-concrete category, whose objects have no underlying set and thus cannot be expressed in the terms of G1.

The premise G2, on the other hand, is ambiguous—it is unclear exactly how Bain wants us to understand ‘structure’ and thus ‘structure-preserving maps’. First, note that when mathematicians talk about ‘structure-preserving maps’ they usually have in mind morphisms that do not preserve all the features of a C-object, but rather the characteristic (albeit partial) features of that C-object. For instance, with respect to a group, a structure-preserving map is a homomorphism and not an isomorphism. Bain’s example of the category Set is of this type, because its morphisms are arbitrary functions (and not bijective functions).

However, Bain wants to introduce a different notion of ‘structure’ that contrasts with this standard usage, for he says:

(Structure) …the intuitions of the ontic structural realist may be preserved by defining “structure” in this context to be “object in a category”.

If we take this claim seriously, then a structure-preserving map will turn out to be an isomorphism in the relevant category – for only isomorphisms preserve the complete ‘structural essence’ of a structured set. For instance, Bain’s example of the category whose objects are smooth manifolds and whose morphisms are diffeomorphisms is of this type. If this is really what Bain has in mind, then one inevitably ends up with a very limited and dull class of categories. But even if one relaxes this notion of ‘structure’ to mean ‘the structure that is preserved by the morphisms of the category, whatever they happen to be’, one still runs into trouble with G3.

We now turn to the premise G3. First, note that G3 (i) is false, as we now explain. It will be convenient to introduce a piece of standard terminology: a morphism from a terminal object to some object X is called a global element of X. And the question of whether an element of X can be expressed as a global element in the relevant category turns on the structure of the category in question. For instance, in the category Man with smooth manifolds as objects and smooth maps as morphisms, this question receives a positive answer: global elements are indeed in bijective correspondence with elements of a manifold. This is because the terminal object is the 0-dimensional manifold {0}, and so an element of a manifold M is a morphism {0} → M. But in many other categories, e.g. the category Grp, the answer is negative. As an example, consider that Grp has the trivial group 1 as its terminal object and so a morphism from 1 to a group G only picks out its identity and not its other elements. In order to obtain the other elements, one has to introduce the notion of a generalized element of X, viz. a morphism from some ‘standard object’ U into X. For instance, in Grp, one takes Z as the standard object U, and the generalized elements Z → G allow us to recover the ordinary elements of a group G.

Second, while G3 (ii) is certainly true, i.e. universal properties can be expressed purely in terms of morphisms, it is a further – and significant – question for the scope and applicability of this premise whether all (or even most) physical properties can be articulated as universal properties.

Hence we have seen that the categorically-informed opponent of (Objectless) need not accept these premises – there is a lot of room for debate about how exactly one should use category theory to conceptualize the notion of physical structure. But supposing that one does: is there a valid inference from G1–G3 to C? Bain himself notes that the plausibility of this inference trades on an ambiguity in what one means by ‘reference’ in C. If one merely means that such constructions eliminate explicit but not implicit reference to objects, then the argument is indeed valid. On the other hand, a defense of OSR requires the elimination of implicit reference to objects, and this is what the general argument fails to offer – it merely provides a translation scheme from statements involving elements (of sets) to statements involving morphisms between C-objects. So, the defender of objects can maintain that one is still implicitly quantifying over elements. 

Category Theory as Structuralist. Part Metaphysic, Part Mathematic. (1)

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What are categories good for? Elementary category theory is mostly concerned with universal properties. These define certain patterns of morphisms that uniquely characterize (up to isomorphism) a certain mathematical structure. An example that we will be concerned with is the notion of a ‘terminal object’. Given a category C, a terminal object is an object I such that, for any object A in C, there is a unique morphism of type f : A → I. So for instance, on Set the singleton {∗} is the terminal object, and so we obtain a characterization of the singleton set in terms of the morphisms in Set. Other standard constructions, e.g. the cartesian product, disjoint union etc. can be characterized as universal.

Just as morphisms in a category preserve the structure of the objects, we can also define maps between categories that preserve the composition law. Let C and D be categories. A functor F : C → D is a mapping that:

(i) assigns an object F (A) in D to each object A in C; and

(ii) assigns a morphism F(f) : F(A) → F(B) to each morphism f : A → B in C, subject to the conditions F(g ◦ f) = F(g) ◦ F(f) and F(1A) = 1F(A) for all A in C.

Examples abound: we can define a powerset functor P : Set → Set that assigns the powerset P (X ) to each set X , and assigns the function P(f)::X →f[X] to each function f :X →Y, where f[X] ⊆ Y is the image of f.

Functors ‘compare’ categories, and we can once again increase the level of abstraction: we can compare functors as follows. Let F : C → D and G : C → D be a pair of functors. A natural transformation η : F ⇒ G is a family of functions {ηA : F (A) → G(A)} A ∈|C| indexed by the objects in C, such that for all morphisms f : A → B in C we have ηB ◦ F (f ) = G(f ) ◦ ηA.

Category theory can be thought of as ‘structuralist’ in the following simple sense: it de-emphasizes the role played by the objects of a category, and tries to spell out as many statements as possible in terms of the morphisms between those objects. These formal features of category theory have developed into a vision of how to do mathematics. This has for instance been explicitly articulated by Awodey, who says that a category-theoretic ‘structuralist’ perspective of mathematics, is based on specifying :

…for a given theorem or theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or “determination” of the objects involved.

Awodey presents one reasonable methodological sense of ‘category-theoretic struc- turalism’: a view about how to do mathematics that is guided by the features of category theory.

Let us now contrast Awodey’s sense of structuralism with a position in the philosophy of science known as Ontic Structural Realism (OSR). Roughly speaking, OSR is the view that the ontology of the theory under consideration is given only by structures and not by objects (where ‘object’ is here being used in a metaphysical, and not a purely mathematical sense). Indeed, some OSR-ers would claim that:

(Objectless) It is coherent to have an ontology of (physical) relations without admitting an ontology of (physical) relata between which these rela- tions hold.

On the face of it, the ‘simple structuralism’ that is evident in the practice of category theory is very different from that envisaged by OSR; and in particular, it is hardly obvious how this simple structuralism could be applied to yield (Objectless). On the other hand, one might venture that applying (some form of) this simple structuralism to physical theories will serve the purposes of OSR.

Let us consider which forms of OSR have an interest in such an category-theoretic argument for (Objectless). According to Frigg and Votsis’ detailed taxonomy of structural realist positions, the most radical form of OSR insists on an extensional (in the logical sense of being ‘uninterpreted’) treatment of physical relations, i.e. physical relations are nothing but relations defined as sets of ordered tuples on appropriate formal objects. This view is faced not only with the problem of defending (Objectless) but with the further implausibility of implying that the concrete physical world is nothing but a structured set.

More plausible is a slightly weaker form of ontic structural realism, which Frigg calls Eliminative OSR (EOSR). Like OSR, EOSR maintains that relations are ontologically fundamental, but unlike OSR, it allows for relations that have intensions. Defenders of EOSR have typically responded to the charge of (Objectless)’s incoherence in various ways. For example, some claim that our ontology is ‘structure all the way down’ without a fundamental level (Ladyman and Ross)

, or that the EOSR position should be interpreted as reconceptualizing objects as bundles of relations.

Category-less Category Theory. Note Quote.

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Definition:

Let us axiomatically define a theory which we shall call an objectless or object free category theory. In this theory, the only primitive concepts (besides the usual logical concepts and the equality concept) are:

(I) α is a morphism,
(II) the composition αβ is defined and is equal to γ, The following axioms are assumed:

1. Associativity of compositions: Let α, β, γ be morphisms. If the compositions βα and γβ exist, then

• the compositions γ(βα) and (γβ)α exist and are equal;
• if γ(βα) exists, then γβ exists, and if (γβ)α exists then βα exists.

2. Existence of identities: For every morphism α there exist morphisms ι and ι′, called identities, such that

• βι = β whenever βι is defined (and analogously for ι′),

• ιγ = γ whenever ιγ is defined (and analogously for ι′).

• αι and ι′α are defined.

Lemma:

Identities ι and ι′ of axiom (2) are uniquely determined by the morphism α.

Proof:

Let us prove the uniqueness for ι (for ι′ the proof goes analogously). Let ι1 and ι2 be identities, and αι1 and αι2 exist. Then αι1 = α and (αι12 = αι2. From axiom (1) it follows that ι1ι2 is determined. But ι1ι2 exists if an only if ι1 = ι2. Indeed, let us assume that ι1ι2 exist then ι1 = ι1ι2 = ι2. And vice versa, assume that ι1 = ι2. Then from axiom (2) it follows that there exists an identity ι such that ιι1 exists, and hence is equal to ι (because ι1 is an identity). This, in turn, means that (ιι12 exists, because (ιι12 = ιι2 = ιι1 = ι. Therefore, ι1ι2 exists by Axiom 1.

Let us denote by d(α) and c(α) identities that are uniquely determined by a morphism α, i.e. such that the compositions αd(α) and c(α)α exist (letters d and c come from “domain” and “codomain”, respectively).

Lemma 2.2 The composition βα exists if and only if c(α) = d(β), and consequently,

d(βα) = d(α) and c(βα) = c(β).

Proof. Let c(α) = d(β) = ι, then βι and ια exist. From axiom (1) it follows that there exists the composition (βι)α = βα. Let us now assume that βα exists, and let us put ι = c(α). Then ια exists which implies that βα = β(ια) = (βι)α. Since βι exists then d(β) = ι.

Definition:

If for any two identities ι1 and ι2 the class ⟨ι12⟩ = {α : d(α) = ι1, c(α) = ι2},

is a set then objectless category theory is called small.

Definition:

Let us choose a class C of morphisms of the objectless category theory (i.e. C is a model of the objectless category theory), and let C0 denote the class of all identities of C. If ι123 ∈ C0, we define the composition

mC0ι123 : ⟨ι1, ι2⟩ × ⟨ι2, ι3⟩ → ⟨ι1, ι3

by mC0 (α, β) = βα. Class C is called objectless category.

Proposition:

The objectless category definition is equivalent to the standard definition of category.

Proof:

To prove the theorem it is enough to reformulate the standard category definition in the following way. A category C consists of

(I) a collection C0 of objects,
(II) for each A,B ∈ C0,
a collection ⟨A,B⟩ C0 of morphisms from A to B,

(III) for each A,B,C ∈ C0, if α ∈ ⟨A,B⟩ C0 and β ∈ ⟨B,C⟩ C0, the composition

mC0 : ⟨A,B⟩ C0 × ⟨B,C⟩ C0 → ⟨A,C⟩ C0

is defined by mC0A,B,C (α, β). The following axioms are assumed

1. Associativity: If α ∈ ⟨A,B⟩C0, β ∈ ⟨B,C⟩C0 , γ ∈ ⟨C,D⟩C0 then γ(βα) = (γβ)α.

2. Identities: For every B ∈ C0 there exists a morphism ιB ∈ ⟨B,B⟩C0 such that

A∈C0α∈⟨A,B⟩C0 ιBα = α, ∀C∈C0β∈⟨B,C⟩C0 βιB = β.

To see the equivalence of the two definitions it is enough to suitably replace in the above definition objects by their corresponding identities.

This theorem creates three possibilities to look at the category theory: (1) the standard way, in terms of objects and morphisms, (2) the objectless way, in terms of morphisms only, (3) the hybrid way in which we take into account the existence of objects but, if necessary or useful, we regard them as identity morphisms.

RAPL (Right Adjoint Preserve Limits) Theorem. Part 8b/End Part.

Fix a small category I and a diagram D ∶ I → C of shape I. Then the limit of D, if it exists, consists of an object l ∈ C and a natural isomorphism

Cone ∶ HomC(−,l) ≅ HomCI((−)I,D) ∶ Uni

in the category SetCop.

This is intuitively plausible if we recall the definition of limits. Recall that a cone under D consists of an object l ∈ C and a natural transformation Λ ∶ lI ⇒ D. We say that the cone (l,Λ) is the the limit of D if, for any other cone Φ ∶ cI ⇒ D, there exists a unique arrow υ ∶ c → l making the following diagram in CI commute:

img_20170214_201951_hdr

The map sending the cone Φ ∶ cI ⇒ D to the unique arrow υ ∶ c → l is the desired function HomCI (cI,D) → HomC(c,l). Furthermore, it’s clear that this function is a bijection since we can pull back any arrow α ∶ c → l to the cone Λ ○ αI ∶ cI ⇒ D. The main difficulty is to show that the data of naturality for these bijections is equivalent to the data of the canonical cone Λ ∶ lI ⇒ D.

Proof: First assume that the limit of D exists and is given by the cone (limID,Λ). In this case we want to define a family of bijections

Unic ∶HomcI(cI,D) →~ Hom (c,limID)

that is natural in c ∈ Cop. (Then the inverse Cone ∶= Uni−1 is automatically natural. So consider any element Φ ∈ HomCI(cI,D), i.e., any cone Φ ∶ cI ⇒ D. By the definition of limits we know that there exists a unique arrow υ ∶ c → limID making the following diagram commute:

img_20170214_203133_hdr

Therefore the assignment Unic(Φ) ∶= υ defines an injective function (recall that the functor (−)I is faithful, so that υ1I = υ2I implies υ1 = υ2). To see that Unic is surjective, consider any arrow α ∶ c → limID in C. We want to define a cone Φα ∶ cI ⇒ D with the property that Unicα) = α. By definition of Unic this means that we must have Φα ∶= Λ ○ αI — in other words, we must have α)i ∶= Λi ○ α indices i ∈ I. And note that this does define a natural transformation Φα ∶ cI ⇒ D since for all arrows δ ∶ i ∈ j in I we have

D(δ) ○ (Φα)i =D(δ) ○ (Λi ○ α)

= (D(δ) ○ Λi) ○ α

= Λj ○ α (Naturality of Λ)

= (Φα)j

We conclude that Unic is a bijection. To see that Unic is natural in c ∈ Cop, consider any arrow γ ∶ c1 → c2 in C (i.e., any arrow γ ∶ c2 → c1 in C). We want to show that the following diagram commutes:

img_20170214_204617

And to see this, consider any cone Φ ∶ cI1 ⇒ D. By composing with the natural transformation γI ∶ cI2 ⇒ cI1 we obtain the following commutative diagram in CI:

img_20170214_205052_hdr

Since the diagonal embedding (−)I ∶ C → CI is a functor, the bottom arrow is given by

(Unic1 (Φ))I ○ γI = (Unic1 (Φ) ○ γ)I

But by the definition of the function Unic2 this arrow also equals (Unic2(Φ ○ γI))I

Then since (−)I is a faithful functor we conclude that

Unic2 (Φ ○ γI) = Unic1 (Φ) ○ γ

and hence the desired square commutes. Conversely, consider an object l ∈ C and suppose that we have a bijection

Conec ∶HomC(c,l) ←→ HomCI(cI,D) ∶ Unic

that is natural in c ∈ Cop. In other words, suppose that for each arrow γ ∶ c1 → c2 in Cop (i.e., for each arrow γ ∶ c2 → c1 in C) we have a commutative square:

img_20170214_210335_hdr

We want to show that this determines a unique cone Λ ∶ lI ⇒ D such that (l, Λ) is the limit of D. The only possible choice is to define Λ ∶= Conel(idl). Now given any cone Φ ∶ cI ⇒ D we want to show that there exists a unique arrow υ ∶ c → l with the property Λ ○ υI = Φ.

So suppose that there exists some arrow υ ∶ c → l with the property Λ ○ υI = Φ. By substituting γ ∶= υ into the above diagram we obtain a commutative square:

img_20170214_210951_hdr

Then following the arrow idl ∈ HomC(l, l) around the square in two different ways gives

idl ○ υ = Unic(Conel(idl) ○ υI)

υ = Unic(Λ ○ υI)

υ = Unic(Φ)

Thus there exists at most one such arrow υ. To show that there exists at least one such arrow, we must check that the arrow Unic(Φ) actually does satisfy Λ ○ (Unic(Φ))I = Φ. Indeed, by substituting υ ∶= Unic(Φ) into the above diagram we obtain a commutative square:

img_20170214_211902_hdr

Then following the arrow idl ∈ HomC(l,l) around the

Conel(idl) ○ (Unic(Φ))I) = Conec(idl ○ Unic(Φ)) Λ ○ (Unic(Φ))I

= Conec(Unic(Φ))

Λ ○ (Unic(Φ))I = Φ

square in two ways gives as desired.

[Remark: We have proved that the limit of a diagram D ∶ I → C, if it exists, consists of an object limID ∈ C and a natural isomorphism

HomC(−, limID) ≅ HomCI((−)I,D) of functors Cop → Set. It turns out that if all limits of shape I exist in C then there is a unique way to extend this to a natural isomorphism

HomC(−,limI−) ≅ HomCI((−)I,−)

of functors Cop × CI → Set, and hence that we have an adjunction (−)I ∶ C ⇄ CI ∶ limI. However, we don’t need this result right now so we won’t prove it. Dually, the colimit of D, if it exists, consists of an object colimID ∈ C and a natural isomorphism HomC(colimID, −) ≅ HomCI(D, (−)I) of functors C → Set. If all colimits of shape I exist in C then this extends uniquely to an adjunction colimI ∶ CI ⇄ C ∶ (−)I. This explains the title of the previous lemma.]

Theorem (RAPL):

Let L ∶ C ⇄ D ∶ R be an adjunction and consider a diagram D ∶ I → D of shape I in D. If the diagram D ∶ I → D has a limit cone Λ ∶ lI ⇒ D then the composite diagram RI(D) ∶ I → C also has a limit cone, which is given by RI(Λ) ∶ R(l)I ⇒ RI(D).

Proof:

In this proof we will write limID ∶= l ∈ D, and we will just assume that the limit object limI RI(D) ∈ C exists. Now we want to show that the following objects are isomorphic in C : R(limID) ≅ limI RI (D). (We will ignore the data of the limit cone.)

So assume that L ∶ C ⇄ D ∶ R is an adjunction. Then we have the following sequence of bijections, each of which is natural in c ∈ Cop:

Homc(c, R(limID)) →~ Homc(L(c), limID) (L ⊣ R)

~ HomDI(L(c),D) (Diagonal ⊣ Limit)

~ HomDI (LI (cI),D)

~ HomCI(c,RI(D))

~ Homc(c,limIRI(D)) (Diagonal ⊣ Limit)

By composing these we obtain a family of bijections

Homc(c,R(limID)) →~ Homc(c,limI RI(D))

that is natural in c ∈ Cop. In other words, we obtain an isomorphism of hom functors HR(limI(D)) ≅ HlimI RI(D) in the category SetCop. Then since the Yoneda embedding H(−) : C → SetCop is essentially injective (from the Embedding Lemma), we obtain an isomorphism of objects R(limID) ≅ limI RI(D) in the category C.