In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in M iff it is in M'. In particular some aspects related to "internal" (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T, whose objects should be considered as possibly "infinitesimal" and suitably "regular" topological spaces. While in C the classes M and M' play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that X\to 1 is in M (resp. M') iff it is a discrete (resp. compact) space. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x \to X and the coslice projection x\X \to X, obtained as the second factors of x:1 \to X according to (E,M) and (E',M') in C, correspond in T to the "infinitesimal" neighborhood of x \in X and to the closure of x. Furthermore, the open-closed complementation (generalized to reciprocal stability) becomes the key tool to internally treat, in a coherent way, some categorical concepts (such as (co)limits of presheaves) which are classically related by duality.
Deep Dive into Balanced Category Theory II.
In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category endowed with two reciprocally stable factorization systems such that X \to 1 is in M iff it is in M’. In particular some aspects related to “internal” (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T, whose objects should be considered as possibly “infinitesimal” and suitably “regular” topological spaces. While in C the classes M and M’ play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that X\to 1 is in M (resp. M’) iff it is a discrete (resp. compact) space. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x \to X and the coslice projection x\X \
In [Pisani, 2008] we argued that a good deal of basic category theory can be carried out in any strong "balanced factorization category" (bfc). Recall that a finitely complete category C is a bfc if it is endowed with two factorization systems (E, M) and (E โฒ , M โฒ ) which are reciprocally stable: the pullback of a map in E (resp. E โฒ ) along a map in M โฒ (resp. M) is itself in E (resp. E โฒ ). We say that C is a "strong" bfc if, furthermore, M/1 = M โฒ /1 (the category S of "internal sets"). We refer to "weak" bfc's when we wish to emphasize that this condition is not required to hold. The motivating example of a strong bfc is Cat, with the comprehensive factorization systems: M and M โฒ are the classes of discrete fibrations and opfibrations, while E and E โฒ are the classes of final and initial functors, so that M/1 = M โฒ /1 โ Set (while E/1 = E โฒ /1 are the connected categories).
In the first part of the present paper, we review and further develop some aspects of balanced category theory. In particular, we consider the bifunctors โX : C/X ร C/X โ S and their restrictions โ X : M โฒ /X ร M/X โ S, where n โ X m := ฯ 0 (n ร X m) is the internal set of components of (the total of) the product over X (and reduces in Cat to the tensor product of the corresponding set-functors). Now, while the bifibrations associated to the factorization systems of the bfc C are summarized, in terms of indexed categories, by the adjunctions
for any f : X โ Y in C, and the reflections
for any X โ C (and in particular ฯ 0 := โ 1 = โ 1 โฃ i : S โ C), the reciprocal stability axiom allows us to obtain also the following “coadjunction” laws:
natural in m โ M/X (or M/Y ), n โ M โฒ /Y (or M โฒ /X) and p, q โ C/X. With this toolkit, we are in a position to straghtforwardly prove familiar properties of colimits of “internal-set-valued” maps m โ M/X or n โ M โฒ /X, and also that, for any
x : 1 โ X in C, there is a bicartesian arrow โ X x โ โ X x of the bimodule ten X : (M โฒ /X) op โ M/X, obtained by composing โ X with the points functor S(1, -) : S โ Set. Thus the subcategories X of “slices (projections)” โ X x : X/x โ X in M/X and X โฒ of “coslices (projections)” โ X x : x\X โ X in M โฒ /X are dual. Furthermore, under a “Nullstellensatz” hypothesis, we prove that the bicartesian arrows of ten X correspond to the retracts of slices in M/X (or coslices in M โฒ /X), so offering an alternative perspective on Cauchy completion also in the classical case C = Cat. It is also shown how these retracts may arise as reflections of figures P โ X whose shape P is an “atom” (such as the monoid with an idempotent non-identity arrow for C = Cat).
In the second part, most of which can be read indipendently from the first one, we sketch how some relevant aspects of topology can be developed in a bfc too. While perfect maps are known to form the second factor of a factorization system on the category Top of topological spaces, we intend to show that, by replacing Top with a suitable category T , it is reasonable to assume that the same is true for local homeomorphisms and that reciprocal stability holds therein.
The existence of a reflection ฯ 0 : T โ M/1 in “sets” suggests that the spaces X โ T are “locally connected”, and in fact the neighborhoods X/x are connected that is, the map ! X/x : X/x โ 1 is in E. Some homotopical properties of spaces can be studied through"finite coverings" that is, maps in B = M โฉ M โฒ ; for instance, a space is “simply connected” if ! * X : B/1 โ B/X is an equivalence. By the reciprocal stability law, spaces in T are also locally simply connected, so that finite coverings are in fact locally trivial (Corollary 13.8).
Thus we maintain that (weak) bfc’s form a common kernel shared by category theory and topology, and that both the subjects are enlighted by this point of view. For example, the reciprocal stability law allows us, on the topological side, to extend (via exponentiation) the classical complementarity between open and closed parts to local homeomorphisms and perfect maps in T , with evident conceptual advantages; on the other side, it provides a sort of internal duality for categorical concepts (as sketched above) which often turns out to be more effective than an “obvious” duality functor.
1.1. Outline.
After three preliminary “technical” sections on bimodules, factorization systems and balanced factorization categories, and after recalling some concepts of balanced category theory, we emphasize in sections 7, 8 and 9 the central role of the reciprocal stability law in treating “internal aspects” of (balanced) category theory. Namely, we study (co)limits of internal presheaves in M/X or M โฒ /X, and the role of the retracts of the representable ones (that is, (co)slice projections). In the last three sections we sketch the idea of balanced topology; in particular, we present some “evidences” of the fact that the reciprocal stability law should hold in an appropriate “topological” category T , in which local homeomorp
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