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Electronic Notes in Theoretical Computer Science 230 (2009) 39-69

www.elsevier.com/locate/entcs

A Framework for Component Categories

Emmanuel Haucourt 1,2

PPS/LSL Paris 7/CEA Saclay Paris, France

Abstract

This paper provides further developments in the study of the component categories which have been introduced in [7]. In particular, the component category functor is seen as a left adjoint hence preserves the pushouts. This property is applied to prove a Van Kampen like theorem for component categories. This last point is very important to make effective calculations. The original purpose of component categories is to suitably reduce the size of the fundamental categories which are the directed counterpart of classical fundamental groupoids (see [15]). In concrete examples, the fundamental category is as "big" as R while the component category is "finitely generated". We take advantage of this fact to define the cohomology of a directed geometrical shape as the cohomology of its component category. The cohomology of small categories is defined in [2] and [1]. Still, in the recent paper [18], the homology of small categories is defined in a very similar way and applied to the study of asynchronous transition systems.

Keywords: algebraic topology, directed algebraic topology, partially ordered spaces, pospaces, local pospaces, Yoneda inversible morphisms, inessential morphisms, weak equivalence subcategory, loop-free category, fundamental category, component category, Van Kampen theorem, fundamental monoid, directed (co)homology

1 Introduction

Given a small category C and a subcategory S of C, we define the quotient category C/e applying the results developed in [3]. Indeed, the size of X(XX)/e decreases as the one of S increases. As one can expect, if S = tt!(XX) then tv{(XX)/e is {*}. Then the component category of a pospace XX is defined as tt!(XX)/e where S is the greatest weak-equivalences subcategory of C and tTi (XX) the fundamental category of XX. We have in mind that S is made of the dipaths 4 of X 5 along which "no choice is made" so we do not lose information removing them 6 .

1 Thanks to everyone who should be thanked

2 Email: haucourt@cea.fr

3 Email: couserid@codept.coinst.coedu

4 or execution traces from a computer scientist point of view.

5 seen as the space of states of a computer on which a program runs.

6 precisely, they are not removed but turned into identities.

1571-0661/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.entcs.2009.02.016

The previous construction can be done in a category whose objects are taken in the class of pairs (C, £), such a pair is called a system over C, where C is an object of a subcategory of CAT and £ CC. The idea is to equip the objects C of a sub-category of CAT with a sub-poset of the poset of all subcategories of C. Then we define the quotient functor sending (C, £) on C/s. The component category is obtained when £ is optimal i.e. when the size of C/s is minimal without loss of relevant information. Several examples are given, involving different subcategories of CAT, and we define component categories of pospaces, local pospaces and d-spaces. Some proofs of technical points are skipped and the paper is organized in the following way:

(i) Pospaces, local pospaces and d-spaces are defined. Concrete but informal examples are given to make the reader understand what component categories should be.

(ii) Generalized congruences and some related tools are described.

(iii) A general theorem describes a framework in which the component category functor can be defined. As we shall see, this theorem makes the component category functor a left adjoint.

(iv) The previous theorem is applied to define the component category of pospaces, local pospaces and d-spaces. We check that we have obtained what was expected.

(v) Preservation of colimits by component category functor is applied to prove Van Kampen like theorems for component category (instead of fundamental category). Examples are given.

(vi) A form of directed cohomology is defined as the cohomology of the component category.

2 Geometrical intuition of component categories through examples

Component categories first appear in [7] in order to reduce the size of the fundamental category. Pospaces are certainly the simplest model of directed topology one may find.

Definition 2.1 [Pospaces] A pospace is a triple (X,tx, <x) where (X,tx) is a topological space, (X, <x) is a poset and <x is a closed subset of (X, tx) x (X, tx). A dimap from a pospace (X,tx, <x) to a pospace (Y,ty, <y) is a set-theoretic function f from X to Y inducing a continuous map from (X, tx) to (Y, Ty) and an increasing map from (X, <x ) to (Y, <y). The collection of po-spaces together with dimaps between them form a category denoted POSPC. Isomorphisms of POSPC are called dihomeomorphisms and are bijective (one-to-one, onto) dimaps whose inverse is also a dimap. Monomorphisms are one-to-one dimaps.

Note that the caracterization of epimorphisms is much more complicated. The unit segment [0, l] with classical topology and order is a pospace as well as all its

Fig. 1. Square with centered hole and Swiss flag

products with product topology and order. [0,1] is in fact the "standard" example in the sense that it is the cogenerator 7 of the category of compact pospaces 8 . The examples of figure 1 are built up from the unit square with classical topology and order in which "holes" have been dug. In each case the underlying space is divided into "components" which give the set of objects of the component category, their borders are drawn with the dashed lines. Two components sharing a frontier are "neighbours" and we put a unique "prime arrow" between neighbour components, the source component being the left most bottom most one. The morphisms of the component category are "generated" by those "prime arrows". In the first example, the component category is free, in the "swiss flag" example (figure 1) it is not the case any more because we have BD> o AB> = GD> o AG> .

The two examples of figure 2 are not dihomeomorphic since their component categories are clearly not isomorphic, it suffices to compare how many morphisms go from the left most bottom most object to the right most upper most one.

Before going further in the study of examples, let me emphasize the fact that we "read" the dipaths of the pospace in its component category. In mathematical terms, we have a lifting property which says that any morphism of the fundamental category is represented by a unique morphism of the component category, conversely, any morphism of the component category represents a morphism of the fundamental one. This property can be found in [7], it is also given for free provided we define the component category by means of generalized congruences, see [3] and the description of the component category given in the rest of the paper. Next examples are 3[ dimensional, the unit cube (with classical topology and order) with a centered hole is shown on the left side of figure 3. The right side picture depicts its components, whose border are represented by "walls":

In figure 5, the blue parallelepipeds are holes and the red cube is a deadlock

7 see [4] for the definition.

8 i.e. the underlying topological space is compact.

Fig. 3. The cube with a centered hole

The point at the center of the left side figure is represented by a 3—morphism (the grey filling on the right side figure) in a suitable 3—category built up from the component category.

Fig. 4. Dimensional duality

area, i.e. any dipath entering in it will not go beyond the deep right upper corner of the red cube. On the right side, the corresponding component category is depicted, but the conventions of representation are different, vertices are components, edges are elementary arrows and faces represent relations between morphisms. By the way, this convention of representation induces a "dimensional duality", components are 3—dimensional subpospaces of the cube and they are represented by points, which are 0—dimensional. Faces of the components are 2—dimensional subpospaces and they are represented by "elementary arrows" (hence 1 — dimensional) from component to the neighbour it shares the face with. A segment shared by four faces

between the four "elementary arrows" representing the four faces. This relation can be seen as a 2 — dimensional arrow provided we turn the component category into a 2—category adding a trivial groupoid between 3 o a and 5 o 7. One can even go further with a point of the pospace shared by six segments all of them being shared by four faces, which makes us reach 3—categories, see figure 4. This "duality" property has been practically applied by Eric Goubault to write a program which provides a 3-dimensional "view" of the component category of the 3-dimensional pospaces. A detailed description of the method is available in [11]. The right side picture of figure 5 has been produced by this program.

3 Generalized congruences

is a 1— dimensional subpospace and is represented by a relation

This section is devoted to generalized congruences which have been formalized in

Fig. 5. Three philosophers diner

Definition 3.1 [Generalized Congruences [3]] A generalized congruence on a small category C, is an equivalence relation on Ob(C) and a partial equivalence relation on Mo(C)+ (the set of all non-empty finite sequences of morphisms of C) satisfying the following conditions (• is the usual concatenation, the a's, P's and 7's range over Mo(C)):

• (pn,...,p0) • (ap,...,ao) (7q,...,70) ^ tgt(ap) src(Po)

• (Pn, ...,Po) (ap,..., ao) ^ tgt(Pn) tgt(ap) and src(Po) src(ao)

• x y ^ idx idy

• (Pn,...,Po) (ap,...,ao) and (Sg,...,So) (7r,...,7o) and tgt(Pn) src(So) ^

(Sg ,...,So) • (Pn,...,Po) (7r,..., 7o) • (ap, ...,ao)

• src(P) = tgt(a) ^ (P o a) (P, a)

Given a functor f E CAT[Ci, C2], in other words a morphism of CAT from Ci to C2, the generalized congruence ~f := (~o, ~m) is defined by x ~o y iff f (x) = f (y) and given ^o-composable sequences (pm,...,po) and (an,..., ao), (pm,...,po) (an,...,ao) iff f (Pm)o...o f (Po) = f (an)o...o f (ao). For any generalized congruence

(an,...,ao) is a ^o-composable sequence iff Vk E {0,...,n — 1}, src(ak+1) tgt(ak).

Theorem 3.2 (Quotient Category [3]) Given (~o, ~m) a generalized congruence on a small category C, we define the quotient category C/^ by

• Ob(C/-) := {[x]-«/x e Ob(C)}

• src([(7n,...,7o)]^m) = [src(70)]^o and tgt([(Yn, ..., Yo)]-m) = [tgt(Yn)]-0

• [(^n,..., ^o)]~m o [(aP,..., ao)]~m = [(^n,..., Po) • (ap,..., ao)]~m

Moreover, there is a quotient functor Q^ : C — C^, defined by Q^(x) = [x]-o and Q~(y) = [y]-m• Q~ enjoys the following universal property, for any functor f : C —>■ C2, if ^C^f then 3!g : C/^ — C2 making the following diagram commutes

Still, we have the following facts :

r\j f ^^r-u

• g is a monomorphism iff

• Q^ is an extremal epimorphism.

Lemma 3.3 ([3]) Generalized congruences on a given small category, ordered by componentwise inclusion form a complete lattice whose meets are componentwise intersections• The total relation which identifies all objects and all non-empty finite sequences of morphisms is a generalized congruence, precisely T of the lattice, while (=Ob(C), =Mo(C)+) is ^ Mo(C)+ is the set of non empty ~o-composable sequences•

Thus, for an arbitrary pair of relations Ro on Ob(C) and Rm on Mo(C)+, there is a least generalized congruence containing (Ro,Rm)•

4 The Component Category functor

4-1 Loop Free, One Way and Directed categories

Pureness first appears in [7] and is an unavoidable technical tool to study component categories, indeed, good properties of C/s directly depend on pureness of £. In ideas, if £ consists of execution paths 9 along which nothing happens then if 3oa £ £ it is expectable that 3,a £ £ too. It is also a convenient way to define loop free, one way and directed categories.

Definition 4.1 A sub-category B of C is pure in C iff Vf,g morphisms of C with src(g) = tgt(f), g o f £B^ f,g eB.

Pureness is a kind of generalization of convexity in poset framework, indeed, a subposet (A, <a) of a poset (X, <x) is convex iff A' is a pure subcategory of X', where A' and X' are the small categories coresponding to A and X.

9 it is a computer science point of view.

Definition 4.2 A loop free category is a category whose subcategory of endo-morphisms is pure and discrete io .

A one way category is a category whose subcategory of isomorphisms is pure and discrete. ii

A directed category or d-category is a category whose subcategory of isomorphisms is pure. Loop free, one way and directed small categories respectively form epi-reflective subcategories of CAT respectively denoted LFCAT, OWCAT, dCAT, see proposition 4.3.

The fundamental category of a pospace is obviously loop free, the one of a local pospace is one way, but it is much harder to prove, and I conjecture that the one of a d-space is directed, it is in fact the reason why I called them "directed", roughly speaking, it comes from the fact that dX is stable under direparametrization (see definition 7.2).

We say that A is a reflective subcategory of B when

• A is a full subcategory of B

• for all A E Ob(A) and B E Ob(B), if A and B are isomorphic in A then A E Ob(B).

• the inclusion functor A-^B has a left adjoint.

When the two first points are satisfied, we say that A is a replete subcategory of B. The left adjoint is called the reflection. Moreover, if all the elements of the counit of the adjunction are epimorphisms, we say that A is an epireflective subcategory of B.

Proposition 4.3

We have the inclusion functors

LFCAT-OWCAT-dCAT-CAT

and the domain of each inclusion depicted on the previous diagram is an epireflective subcategory of its source.

For further details about reflective subcategories, see [4].

Conjecture 4.4 We have the following commutative diagram.

POSPCc-^ LPOSPC-^dSPC

Jl Jl Jl

LFCAT-^ OWCAT-^dCAT

10i.e. for all diagram ^ ^ in C, a and ( are identities. Hence C has no "loops", whence the name.

11A one way category might have loops, but each loop is either clockwise or anticlockwise never both at the time.

4.2 Weak Equivalences Subcategory

Next materials are directly related to the choice of a £ such that C/s is the component category of C. As we shall see, all the rest of the subsection, in particular the existence of a non empty weak equivalences subcategory, holds for any directed category C. Then the component category of a pospace/local pospace/directed space, is defined as the component category of its fundamental category.

4.2.1 Yoneda inversible morphisms, Left/Right extension properties and Weak Equivalences Subcategories

Definition 4.5 [[7]] Let C be a category. A morphism a of C is said to be Yoneda revertible iff Vx E Ob(C), (C[x, src(a)] = 0 ^ 7 E C[x,src(a)] ^ a o 7 e C[x,tgt(a)]) is bijective and Vy E Ob(C), (C[tgt(a),y] = 0 ^ 7 E C[tgt(a),y] ^ 7 o a E C[src(a),y]) is bijective.

Definition 4.5 is closely related to representable functors of C and Yoneda's lemma (see [4]), however, the restriction Vx E Ob(C), (C[x, src(a)] = 0... and Vy E Ob(C), (C[tgt(a),y] = 0... cannot be removed, otherwise, a Yoneda inversible morphism would necessarily be an isomorphism which is silly for loop free and one way categories whose only isomorphisms are identities. From a computer science point of view, the subtle difference between Yoneda inversible morphisms and isomorphisms give a theoretical method for deadlock detection, but we will not develop this remark here. In all examples given in section 2, any dipath joining two points of the same component gives rise to a Yoneda inversible morphism of the fundamental category.

Lemma 4.6 Let C be any (small) category, x,y objects of C and a1,a2 E C[x,y] Yoneda inversible, 3!f1 ,f2 E Iso(C)[y, y],a2 = fi o a1,a1 = f2 o a2 and 3!g1,g2 E Iso(C)[x,x],a2 = a 1 o g1,a1 = a2 o g2.

Proof. C[y,y] = 0 hence by definition of Yoneda inversible applied to a1, 3!f1 e C[y,y] such that a2 = f1 o a1. Exchanging a1 and a2, 3!f2 e C[y,y] such that a1 = f2 o a2. In particular, a2 = f o (f2 o a2) = (f1 o f2) o a2 and a1 = f2 o (f1 o a1) = (f2 o f1) o a1, but, by definition of Yoneda inversible, idy is the only morphism of h E C[y, y] such that a2 = h o a2. It is also only morphism of h E C[y, y] such that a1 = h o a1. It follows that f1 o f2 = f2 o f1 = idy i.e. f1,f2 E Iso(Mo(C)). It works the same way for the g's. □

Corollary 4.7 Let C be a (small) category such that Iso(C) is dicrete, then given x,y objects of C, C[x,y] fl {Yoneda inversibles} is either 0 or a singleton.

Remark 4.8 Any isomorphism is Yoneda inversible morphism, and a composition of Yoneda inversible morphisms is a Yoneda inversible morphism. Moreover, if L is loop-free and a is a Yoneda inversible morphism of L then L[src(a),tgt(a)] = {a}.

To prove the last point, note that y £ L[src(a),src(a)] -—> a o 7 £ C[src(a),tgt(a)] is a bijection. Up to now, this definition has only proved its relevance in loop-free cases. First, we recall from [7] that the E-zigzag connected component of x in L denoted Cx is the subcategory of L whose objects are those connected to x by a zigzag of morphisms of E and satisfying for all objects y, z of Cx, Cx[y,z] = L[y,z] n E.

Definition 4.9 Right Extension Property

E has the right extension property with respect to C iff Vy : y' —* x', Va : x —► x' £ E, 3a' : y —► y' £ E, 3y' : y —> x such that a o y' = Y o a', i.e. the following diagram is commutative:

3a'e£ / \ 3j'

Left Extension Property is obtained "dualizing" definition 4.9

Definition 4.10 [Eric Goubault] 12 Let C be a small category, E C Mo(C) is a WE-subcategory iff (by definition) E is stable under composition (of C) and satisfies

1 Iso(C) C E C Yoneda(C)13

2 E is stable under pushouts and pullbacks (with any morphism in C). It means that E has both REP and LEP with respect to C and further the commutative squares provided by REP and LEP can be chosen in order to be respectively pullback and pushout squares in C.

Eric Goubault, in [12], has changed the definition of Weak Equivalences subcategory of [7] 14 replacing left and right extension axiom by pushout/pull back stability axiom, providing an extremely handy tool. Indeed, any WE-subcategory of any small category C is pure (it will be proved later) and has left and right extension properties (it is obvious). Moreover, if Iso(C) is pure in C (i.e. C is directed) then C has a C-biggest WE-subcategory.

4-2.2 Locale of the Weak Equivalences of a small category

We give several results which will be combined to prove that the collection of WE-subcategories of a small category C such that Iso(C) is pure in C forms a locale. We recall that a locale is a poset (L, <l) such that VU C L,U has a least upper bound and a greatest lower bound (it is a complete lattice) and V(bj)jej £ LJ Va £ L, a A

12 in directed categories framework, this definition is equivalent to Eric Goubault's one.

13Iso(C) and Yoneda(C) are subcategories of C respectively generated by isomorphisms and Yoneda inversible morphisms of C.

14definition of [7] was itself inspired by the notion of calculus of fractions, see [8] and [4].

{yjeJ j = jeJ(a A bJ) (see I5]' I19]' I9] OT I10])' Lemma 4.11 is due to Eric Goubault, it is the reason for definition 12. Indeed, in [7], we had to enforce the pureness of £ by an axiom, unfortunately, the resulting definition was not "stable" in the sense that the subcategory generated by two pure subcategory is not, in general, pure.

Lemma 4.11 Let C be a small category such that Iso(C) is pure in C. Then any WE-subcategory of C is pure in C.

Proof. Take a e £ and /i,/2 e Mo(C) such that a = /2 o /1. By 2nd point of definition 12, we have a a' e £ and /' which form a pushout square and a unique

g G Mo(C) making the push-out diagram 11 x x a

pushout

commutative. By pureness

of Iso(C) in C, /' and g are isomorphisms, hence by 1st point of definition 12, belongs to £. So, by stability under composition of £ (definition 12), /2 = g o a' e £. The same way, using the pull-back (instead of push-out) extension property, one proves that /1 e £. Thus £ is pure in C. □

Lemma 4.12 Let C be a small category. If Iso(C) is pure in C then Iso(C) is a WE-subcategory of C.

Proof. Stability under composition and 1st point of definition 12 are obviously satisfied since, any isomorphism is Yoneda inversible (remark 4.8) and isomorphisms

f 1 / \ id / \

compose. Let a G Iso(C) and f G Mo(C) be, then we have \ pushout ^ and

pullback ^ so the 2nd point is also satisfied.

Lemma 4.13 If (£j)jej is a non empty family of WE subcategories of a small category C then{~}jej £j is a WE-subcategory of C.

Proof. P| jj £j obviously enjoys stability under compose and the 1st point of definition 12. Suppose a e C\jeJ £j and / e Mo(C) with src(/) = src(a). Take

50 E. Haucourt / Electronic Notes in Theoretical Computer Science 230 (2009) 39-69

ji,j2 £ J, since a G Ejj we have a push out square

fi S ViV^

/ \ / \

^ pushout \ and also / pushout

because a G Ej2. By uniqueness (up to isomorphism) of the pushout, we have an isomorphism t from x2 to xi such that ai = t o a'2. By 1st point of definition 12 and for t is an isomorphism, t G Ej2 which is stable under composition (1st point), thus a'i = t o a'2 G Ej2. By the same argument, Vj G J, a\ G Ej i.e. a'i G C\jeJ Ej

fi - ienjeJs, / \

and we have \ pushout ^ . The same proof holds for pull-backs. □

-enjeJ sj\/f

Lemma 4.14 If (Ej)jej is a non empty family of WE subcategories of a small category C then 1+J jej Ej is a WE-subcategory of C. Where 1+J jej Ej is the least sub-category of C including all the Ej 's.

Proof. By definition, |+JjeJEj = {an ◦ ... o ai/for n G N* {ji,...,jn} C J and Vk G {1,..., n}ak G Ejk}, stable under composition and 1st point of definition 12 immediately follows since a composition of Yoneda inversible morphisms (respectively isomorphism) is Yoneda inversible (respectively isomorphism) (see remark 4.8). Take an0...oai G |+|jej Ej with n G N*, {ji,...,jn} CJ, Vk G {1,...,n}ak G Ejh

and f G Mo(C) with src(ai) = src(f ). We have

. With a finite

a 1 es, i

induction (apply consecutively the 2nd point of definition 12 for Ej 1,..., Ejn), we

eSji essn

have f, pushout fi fn-i pushout fn . Now, it is a general fact that a "composition" of

a 1 eEj ! Vn&Jri

a'nO...OG[ ^](1j Ej

push-out squares is a push-out square (see [4]) hence f

pushout fr .

It works analoguously for pull-backs, thus the 2rd point of definition 12 is satisfied. □

Lemma 4.15 Let C be a (small) category. If A is apure subcategory of C then for all families (Cj )jej of subcategories of C, An ^1+J jej Cj^J = l+J jej (AnCj )

Proof. The inclusion An ^1+JjejC^ 2 1+Jjej(AnCj) is always satisfied. Indeed, if f is an element of the right member, then one has n G N*, {ji,...,jn} C J,

yk e {1,..., n}ak eA fl £jk and / = an o ... o a1. Now A is a subcategory of C and in particular yk e {1, ...,n}ak e A, hence / e Mo(A). Conversely, suppose that we have n e N*, {j1, ...,jn} Q J, yk e {1, ...,n}ak e £jk and / = an o ... o a1 e Mo(A), by pureness of A, an,..., a1 e Mo(A). Then yk e {1,..., n}ak e Af£jk and / is an element of the left member. □

Remark 4.16 If C satisfies the following property: y71,72 e Mo(C),Y2oy1 = y2 ^ Y1 = id and y2 o y1 = y1 ^ Y2 = id, then the converse of lemma 4.15 is true.

Proof. Take Y2 o Y1 e Mo(A) where 72,71 e Mo(C). Set C1 := {71}, C2 := {72} 15 and apply the distributivity for the family {C1, C2}. If 71 $ Mo(A) and y2 $ Mo(A) then (A f C1) W (AfC2) = 0 while Af (C1 W C2) = {72 o 71}. If 71 £ Mo(A) and 72 e Mo(A) then (AfC1) W (AfC2) = {72} while Af (C1 W C2) = {72,72 o 71} and Y2 = y2 o 71 by the property of C, precisely, if we had y2 = y2 o 71, we would have

71 = idsrc(Yl) hence idsrc(Yl) e Mo(A) because A is a subcategory of C. □

The required property is true if C is a groupoid or a loop-free category. In fact, having Af ^1+J jej C^ = 1+J jej (AfCj) is equivalent to the existence of the right adjoint of the functor Af_: ({subcategories of C}, Q) —> ({subcategories of C}, Q), where the continuous lattice ({subcategories of C}, Q) is seen as a complete and co-complete small category. The equivalence directly comes from the special adjoint functor theorem . This equivalence is related to the link between locales and complete Heyting algebras, see [5] for further details.

Corollary 4.17 Let (£j)jej be a family of WE-subcategories of a (small) category C such that Iso(C) is pure in C and £ a WE-subcategory of C. Then £ f j j = j(£ f £j).

Proof. By lemma 4.11, £ is pure in C, the result follows by lemma 4.15. Note that the hypothesis that all the £j's are WE-subcategories is not used in the proof. □

Remark 4.18 P| and y are associative over the family of subcategories of a small category C.

Theorem 4.19 Let C be a small category such that Iso(C) is pure in C (i.e. C is directed). Then, the family of WE-subcategories of C is not empty and, together with Q it forms a locale whose l.u.b. operator is y and g.l.b operator is P|. Moreover, the least element of this locale ("bottom") is Iso(C).

Proof. Axioms of a locale are given by lemmas 4.13, 4.14 and corollary 4.17. □

As it is explained in [5] and [19], the notion of locale generalizes the notion of family of open subsets of a topological space, thus, theorem 4.19 gives us a kind of topology over C as soon as Iso(C) is pure in C. This pureness hypothesis is

15 an abuse of notation to say that C\ and C\ are subcategories of C respectively generated by 71 and 72

actually very "natural". Ideologically, if we want to consider an isomorphism of C as a path that can be run forward, which is the case when C is a fundamental category, it geometrically makes sense to expect that all its subpaths can also be run forward i.e. are isomorphisms. When this "geometrical" assumption is fulfilled by a small category C, roughly speaking, C describes the arc-wise connectedness of a "geometrical shape".

4.3 Quotient of a small category by one of its subcategory : C/s

Given £ a subcategory of a small category C, we can define C/s := C/^ where ~ is the least generalized congruence on C containing

(0, {(idtgt(a),a), srcsrc(a))/a e Mo(£)})

This definition holds by lemma 3.3.

Theorem 4.20 (Description and universal property of C/£)

Given a small category C and £ Q Mo(C), closed under composition (in fact, take £ a subcategory of C). Let (~0,s, s) be the least generalized congruence containing (0, {(idtgt(a) ,a), (a,idsrc{a))/a e £}). Then yx,y e Ob(C),x ~o,s iff there is a £-zig-zag between x and y. y(ftn,..., f30), (am, ...,a0) ~0ys-composable sequences (i.e. src(ai+1) ~0,s tgt(ai) and src(^i+1) ~0,s tgt(Pi)), we have (Pn,..,Po) (am,...,a0) iff there is a finite sequence of "elementary trans-

formation" from (am,..., a0) to (0n,..,Po), where an "elementary transformation" is either

• (an,...,ai+1,a,ai-1,...,a0) <s (an,...,ai+1,idsrc(a) or idtgt(a),ai-1,...,a0) if

a e £

• (an,...,ai+2,ai+1,ai,ai-1 ,...,ao) (an,...,ai+2,ai+1 o ai,ai-1,..., ao) if

src(a+) = tgt(ai).

C/£ is characterized by the following universal property, y/ eCAT[C,C'\, if ya e £,/(a) = id then 3!g eCAT[C,C/£] such that

Qt /' X g

/■commutes\

Moreover, if C1—C2 satisfies /(£1) Q £2 then 3! C1/£1—C2/£2 making the

following diagram commutes

Where Qs is the quotient functor (refer to theorem 3.2) associated to the generalized congruence induced by £. g is also denoted f/s, and in the same stream of notation h is denoted f/slts2.

Definition 4.21 (Definition of the Component Category) The component category of a directed category C is defined as C/twe(cC) where ~Twe(c) is the biggest weak equivalence subcategory of C.Given a pospace/local pospace/directed space X, the component category of X is defined as the component category of n1(X), the fundamental category of X.

It makes sense by theorem 4.19 and 4.3. Remark we have not the functoriality yet. Next theorem establishes a relation between connectedness 16 and component category of the fundamental groupoid of a topological space.

Theorem 4.22 Let G be a groupoid, then Mo(G) is the Q-biggest WE-subcategory of G. Moreover G/Mo(g) is (isomorphic to) the set (prcisely a discrete category seen as its set of objects) of zigzag connected components of G. If G := r1(X, tx) the fundamental groupoid of topological space (X,tx), then G / Mo(Q) is the set of arc-wise connected components of (X,tx).

Proof. Any morphism of Mo(G) is an isomorphism of Mo(G), thus, by 1st point of definition 12, if G has a WE-subcategory, it is necessarily Mo(G) which is stable under composition. By remark 4.8, each morphism of a groupoid is Yoneda inversible hence 1st point of definition 12 is satisfied. Finally, it is a general fact that if a is an isomorphism, then any morphism f such that src(f) = src(a) has a push-out along a and any morphism g such that tgt(g) = tgt(a) has a pull-back along a, thus we have the 2nd point of definition 12.

Then each morphism of G is identified with the identity of its source and target. Two objects x,y of G are identified iff there is a zigzag between them (note that, since G is a groupoid, it is equivalent to G[x,y] = 0). □

Remark 4.23 Any free category is obviously a one-way category, so we can always define the component category of a free category. For example, the component category of the monoid (N, +) seen as a small category is (N, +). However, (N, +) is not the component category of the free categories generated by the following graphs

16 in the classical algebraic topology sense.

... indeed, there are there own component category.

This is a rather severe drawback for the study of directed shapes with loops since, ideologically, we expected that they admit (N, +) as component category. The last section of the paper is devoted to some ways of fixing this up. Also note that S1 can be seen as a continuous generalization of the previous examples.17 .

Theorem 4.24 Let C be a small category and £ a wide subcategory of C. If C is loop-free and £ is a pure subcategory of Yoneda inversible morphisms admitting left and right extension properties then C/£ is loop-free. If £ is pure in C then C/£ is one-way.

Proof. Omitted.

Theorem 4.25 For any small category C, C/stoop is loop-free'.. Where £ioop is the wide subcategory of C generated by morphisms a such that G Mo(C) forming

a loop as follows

the above diagram is not required! to be commutative.

Proof. Omitted.

Note that £/oop is a pure subcategory of C

Definition 4.26 A category is thin iff its biggest weak equivalences subcategory is discrete. TLFCAT, TOWCAT and TdCAT are the full sub categories of thin loop-free categories of LFCAT, OWCAT and dCAT.

Conjecture 4.27 Let L be a small loop-free category and £l the biggest WE-subcategory of L. Then L/£l is thin (see definition 4.26).

4.4 Functoriality of component categories

Next theorem gives the general framework in which the notion of component category becomes functorial. As pointed out in the abstract, the idea is to equip any small category C in our scope of interest with a subcategory of distinguished mor-phisms (called "inessential" in [7]) which are unformally those along which "nothing happens".

Theorem 4.28 (General framework for component category functor)

Let K be a subcategory of CAT and $ be an "assignment" which gives to each C object of K a subposet of (Sb(C), Q) (which is the complete partial order of subcategories of C) with "top" and "bottom" elements. Then we define K$ the category whose objects

17Still, note that the fundamental category of ni(S ) is not free as described in section 2.

are pairs (C, £) where C is an object of K and £ G $(C) and K$[(Ci, £1), (C2, £2)] := {f £K[Ci, C2]/Va G £1, f (a) = id ^ f (a) G £2}.

(i) Vf £K[Ci, C2]Va G T$(Cl), f (a) = id ^ f (a) G T^)

(ii) Vf GK[Ci, C2]Va G ±*(ci), f (a) = id ^ f (a) G ±^2)

(iii) For all C object of K, V£ G $(C)

(a) ±$(C) C id(C)

(b) Qy : C —► C/£ is a morphism of K (hence C/£ is an object of K)

(c) Vf GK$[(C, £), (C, £')], f/Y : C/£ ^ C' and f /y,y' : C/£ C'/£' 18 are morphisms of K

Then we have

• (iiia) ^ (ii)

• If (i) is satisfied then R is well defined and U H R

• If (ii) is satisfied then L is well defined and L H U

• If (iii) is satisfied then CompQ is well defined and CompQ H L Where

U is the obvious forgetful functor.

Given C£K, L(C) := (C, ±$(C), R(C) := (C, T$(c)), Comp$(C, £) := C/e. Given f : Ci —> C2, R(f) is the induced morphism from (Ci, T$(Cl)) to (C2, T$(c2)) (i.e. U(R(f)) = f) L(f) is the induced morphism from (C1, ±$(c1)) to (C2, ±$(Ca)) (i.e. U(L(f)) = f) and for all f £K$[(Ci£i), (C2, £2)], Comp$(f) := fEi,E2.

Proof. (iiia) ^ (ii): Take f £K[C1,C2] and a £ ^$(c1) by (ii), a is an identity so necessarily f (a) is an identity.

(i) ^ U H R: R is well defined because the object part does not raise any problem and (i) is exactly the assumption we need to ensure that morphism of K from C1 to C2 induces a morphism of K$ from (C1, T$(Cl)) to (C2, T$(c2)). The unit of the ajunction is n(c,E) : (C, £) —> (C, T$(c)), which is a morphism of K$ since £ C T$(C). The co-unit is ec := idc. Given f : (C, £) —► (C, £'), put g := U(f), it is clearly the only morphism of K such that f = g o idc and f = R(g) o n(C,E). The naturality of n is obvious.

(ii) ^ L H U: L is well defined because the object part does not raise any problem

18see lemma 4.20 for notations //y, and //s,s/

and (ii) is exactly the assumption we need to ensure that morphism of K from C1 to C2 induces a morphism of K$ from (C1, to (C2, ±$(c2)). The unit of the

ajunction is nc = idc the co-unit is £(c,E) : (C, —* (C, £) which is a morphism

of K$ because T$(c) Q £. Given a morphism f GK[C1,C2], setting g := £(c,e)0L(f), we have f = nc o L(g) i.e. f = L(g).

(iii) ^ CC$ H L: The object part of CC$ is well defined by (iiib), the morphism

part of CC$ is well defined by (iiic) (f/E,E' : C/£ —► C'/£' is a morphism of K). The unit of the adjunction is the only morphism n(c,E) : (C, £) —> (C/£, ±$(c/e)) such that U(n(c,E)) = Qe, Qe is in K by (iiib), moreover Va G £, Qe(a) is an identity, hence by definition of K$, n(c,E) is in K$. Let f : (C, £) —► (C', ±$(c')) morphism of K$, it follows that Va G £, f (a) = id ^ f (a) G ±$(c'), however, by (iiia), -L§(c') Q id(C') then Va G £, f (a) is an identity. So we can apply lemma 4.20, f/E is the only morphism of small categories from C/£ to C' such that U(f) = f/EoQE. It follows that f/E is the only morphism of K (cf (iiic)) such that f = L(f/e) o n(c,E). Naturality of n(c,E) is a consequence of uniqueness property of lemma 4.20. □

Definition of the Component Category Functor by Means of Theorem 4.28

It suffices to set K:=LFCAT and $(C) := WE(C), (ii) and (iiia) are satisfied because ±$(c) := [idx/x G Ob(C)}. By theorem 4.24, V£ G WP(C), C/£ is a loop-free category, since LFCAT is a full sub-category of CAT, (iiib) and (iiic) are also satisfied. Note that (i) is not necessarily satisfied, hence we do not have, in general, the functor R.

We can do the same setting K:=OWCAT and $(C) := WE(C), (ii) and (iiia) are satisfied because ±$(c) := [idx/x G Ob(C)}. By theorem 4.24, V£ G WP(C), C/£ is a one-way category, since OWCAT is a full sub-category of CAT, (iiib) and (iiic) are also satisfied. Once again, (i) is not necessarily satisfied, hence we do not have, in general, the functor R.

For directed categories, things are slightly more intricate, the reason is that the least weak equivalences subcategory of a directed category C might contain isomorphisms which are not identities, hence (iiia) of theorem 4.28 is not necessarily satisfied. However, by theorem 4.3, OWCAT is a reflective subcategory of dCAT hence, if L H(dCAT^OWCAT) we define the compoent categroy functor as Compow o L where Compow is the component category functor defined in the case of one way categories. It is natural, isomorphisms are Yoneda inversible so they have to be turned into identities, the fact that we have to identify them before applying theorem 4.28 is just a technical twist which does not change the underlying philosophy of the method.

4.5 Comments and examples

4.5.1 Is there any relation with weak equivalences in model categories ?

In our context, morphisms of the weak eaquivalence subcategory of C are to be called weak equivalences. However, these weak equivalences are far from model categories ones. There is a slight analogy between them, due to the pushout/pullback stability property but it does not really go further. In fact, the main difference is that, in model categories, the weak equivalences are (almost) always given by an intrisic property of the morphisms, for example in SPC the category of topo-logical spaces, weak equivalences are continuous maps giving rise to isomorphisms between homotopy groups in all dimensions. This definition just depends on the map and its domains and codomains, in some sense, it is local. On the other hand, weak equivalences in our context are defined as part of a subcategory which is defined in a global way. Let us consider I* := {(x,y)/0 < x,y; x + y < 1} and ~C := {(x,y)/0 < x,y < 1} with classical topology and order. It is easy to check that in n1(T) as well as in n1 (C), all morphisms are Yoneda inversible. n1(C) clearly has all pushouts and pullbacks hence any morphism of n1((C) is a weak equivalence while the only weak equivalences of n1(C') are identities. The reason is that for any non identical morphism a of n1(C) one can find a morphism 7 so that the (right) extension property is not fulfilled. The last example emphasizes the global and geometric aspect of our weak equivalence definition.

4.5.2 Detailed calculation of the component category of the "L" pospace

The idea is to find morphisms that are "obviously" not weak equivalences and to check the remaining ones form a weak equivalence subcategory. Let L be the pospace depicted in figure 6 with classical topology and order. Given (x, y) < (x',y') there is, up to dihomotopy, a unique morphism from (x, y) to (x',y'), hence any morphism is Yoneda inversible. Now suppose that a morphism a crosses the vertical dotted segment, then take 7 a morphism which crosses the horizontal one. Clearly, the right extension property is not satisfied by a. Now it is easy to check that the subcategory made of the morphisms of n1 (L) which do not cross any dotted segments are weak equivalences. By the way, note that if a morphism has its source or target exactly on the dotted line, it is still a weak equivalence. This is due to topological properties of components which have been deeper studied in [7].

5 Tool for calculation of component categories

The presentation given above could let the reader think that theorem 4.28 is useless to define component categories, and forgeting the functoriality question, he is right! The point is that, in concrete case, we want to be able to calculate component categories and, in order to do so, we need efficient tools. One of the most classical results towards calculation of fundamental groups, groupoids and categories are Van

Fig. 6. The "L" pospace

Kampen theorem 19 . The idea of the theorem is as follows, given a geometrical shape X (classical or directed), instead of directly calculating the fundamental object of X, split X into two parts, say A and B whose fundamental objects are known (or at least easier to calculate) then "glue" the fundamental objects of A and B to have the fundamental object of X. If you see a geometrical shape as a program and its fundamental object as an abstract interpretation (see [6]) of this program, then Van Kampen theorem becomes a kind of "compositionality" result. Technical details of Van Kampen theorem are of out of the scope of this paper, so we just give an unformal statement.

In theorems 5.1 and 5.2, *SPC are *CAT are taken by pair according to the following

Table 19

★SPC ★CAT

POSPC LFCAT

LPOSPC OWCAT

dSPC dCAT

where *SPC is the domain of the fundamental category functor n and *CAT its codomain.

Theorem 5.1 (Van Kampen for fundamental category) Let ~X 1, ~X2 be sub-objects of X (object of *SPCJ such that the underlying topological space of ~X is the union of the interiors20 of the underlying topological spaces of X1 and X2 and let X0 := X1 nJt2. i1 : X0 x X 1,i2 : X0 x X2,31 : X1 xif and 32 : X2 xjt

19there are several versions depending on the framework : see [22] and [24] for groups, [15] for groupoids, [14] for categories.

20 with respect to the underlying topology of XX.

the inclusion maps. Then we have the following push-out squares

ni(jl)

nij'2)

C1 push-out ~X 2 n1(CX 1)

^l(il)

push-out (X" 2)

ni (i2 )

respectively in *SPC and *CAT.

n1(C 0)

Theorem 5.2 (Van Kampen for component category)

Let X1, XX2 be sub-objects of X3 (object of *SPC) such that the underlying topological space of X3 is the union of the interiors of the underlying topological spaces of x1 and X 2 and let X0 := XX1 n XX2. i1 : XX0 x XX1 ,i2 : X0 x XX2,^1 : XX1 x X3 and j2 : TX 2 x X3 the inclusion maps.

Moreover, we suppose that £1, £2 are WE-subcategories of n1(X 1), n1(XX2), ^1(j1)(£1^ n1(j2)(£2) (also denoted £3) is a WE-subcategory of n (X 3), n1(i1)(£0) C (£1) and n1(i2)(£0) C (£2) (i.e. n1(i1),n1(i2) are morphisms of ★CAT$ ).

Then i1,i2,j1 and j2 give rise to i'1,i'2,j'^ and j'2 morphisms of *CAT$ and we have

(n1(X3), £3)

(n1(X 1), £1) push out in (n1 (XX2), £2)

*CAT $

(n1(X0), £0)

Comp$(n1(X3), £3)

Comp$(n1(X 1), £1)

Comp$(ii

push out in *CAT

Comp$(n1(X0), £0)

Comp$(n1(XX2), £2)

yomp$(^2)

The proof of theorem 5.1 requires three cases, one for each line of table 19. POSPC/LFCAT case can be found in [13]. dSPC/dCAT is available in [14]. In all the cases one might define the fundamental category of a local pospace as the fundamental category of its corresponding directed space see theorem 4.3.

Proof. Theorem 5.1 gives us pushout squares in *SPC and *CAT:

X n1(X)

JX1 push-out JX 2 n\(jt l) push-out n\(Jt 2)

ni(jl) ni(j2)

/ \ . '

¿1^ ^¿2 ni(il)^ ^jri(i2)

XX 0 ^1(^0)

We have to prove that ^(XXo), niÇXt 1), n1(XX2) and respectively equiped

with £0, £1, £2 and £3 give rise to a pushout square in *CAT$. Given f1 : (n1(XX 1), £1) —► (L, £) and f1 : (n1(XX2), £2) —► (L, £) morphisms of *CAT$ such that f1 o i1 = f2 o i2, by hypothesis, 3!h : ■K1(XX3) —► L (morphism of *CAT) such that f1 = ^oj1 and f2 = hoj2. It remains to see that h gives rise to a morphism of *CAT$ i.e. h(£3) Ç £. By hypothesis, £3 = j1(E1^ j2(£2) so any element of £3 can be written j2(«2n+1) • j1(a2n) •... • j2(a1) • j\(a0) where Vk G {0, ...,n},a2k G £1 and a2k+1 G £2, so h(j2(a2n+1) • j1(a2n) •... • j2(a1) • j1(a0)) = (h o j2)(a2n+1) • (h o

j1 )(a2n) • ... • (h o j2)(a1) • (h o j1)(a0) = f2(a2n+1) • f 1 (a2n) • ... • f2(a1) • f1(a0) G E

since f 1, f2 are morphisms of *CAT$, hence h gives rise to a morphism of *CAT$ from

(ni(X3), £3) to (L, £). Thus we have a pushout square in *CAT$. Now by theorem 4.28, we know that Comp§ is a left adjoint hence 21 preserves colimits and, in particular, pushouts. □

Theorem 5.2 does not necessarily give the biggest WE-subcategory of n

(XX3), so

one has to guess what this biggest WE-subcategory is in order to choose appropriate £1 and £2, the choice of £0 is not as important, and once £1 and £2 are given, it might be possible to take £0 as the biggest WE-subcategory of n1(XX0) satisfying n1(i1)(£0) Ç (£1) and n1(i2)(£0) Ç (£2). A very simple application of theorem 5.2 to calculate the component category of the first example given in section 2.

Let us come back to the example of the rectangle with two holes:

21 general facts of category theory see [4].

which gives, by theorem 5.2

In this figure, rectangle filled with grey color are not commutative. The holes of the geometrical shape are represented by non-commutative squares in the component category.

Applying theorem 5.2 we can also prove that the component category of the cube with a centered cubical hole has 26 objects22 . It can be represented in R3 putting an object in the "center" of each vertex, edge and face (8 vertices +12 edges + 6 faces = 26 objects). Morphisms are generated by arrows from a point to its "closer neighbours in the future", for example those of (0,0, 0) are (0, 0, 2), (0,1, 0) and (1, 0, 0) while (1,1,1) has no such neighbours. In order to have the hypothesis of theorem 5.2 satisfied, we split the cube into two parts so that, following notation of theorem 5.1, Xo :=] 2 -e, 2 + e[x[0,1] x [0,1]. It is the analog of the previous example in three dimensions.

22geometrically, picture the Rubik's cube, the interior cube is the hole, all other cubes give an object.

62 E. Haucourt / Electronic Notes in Theoretical Computer Science 230 (2009) 39-69

6 Towards directed cohomology

In [2] and [1] a cohomology of small categories is presented by means of natural systems of factorization. The idea would be to define the cohomology of a directed geometrical object X as the cohomology of its fundamental category 23 . However, as we have already pointed it out, the fundamental category has often as many objects as R. Still, there is only few of them which is relevant, and finding them amounts to calculate the component category. Thus, the cohomology of XX could be defined as the cohomology of the component category of X. For example, with this definition, the fourth and fifth examples given in section 2 are distinguished by their first cohomology groups.

In this paragraph, "cubical" pospace means a disjoint union of unit cubes of dimension n in which finitely many parallelepipeds24 have been dug out. As we have remarked in the previous paragraph, the choice of a "good" natural system is influenced by "good" properties of the small category we want to calculate the cohomology groups.

Definition 6.1 • A morphism y is said prime iff for any morphisms Yn, ■ ■■, Yo such that y = Yn ◦ ■■■ ◦ Yo, 3!i G {1, ■■■, n},Yi = id.

• A category C is generated by primes iff any non trivial morphism of C can be written as a finite composition of prime morphisms.

• A category C is homogeneous iff C is generated by primes and for all composable sequences of prime morphisms (Yn, ■■■, Yo) and (y'n>, ■■■,Y'o) we have Yo ■■■ oy0) = (y'n' o ■■■ o y'o) ^ n = n'. n + 1 is the length of Yn ◦ ■ ■■ o y0.

• A category C is said strongly homogeneous iff C is generated by primes and Vx, y G Ob(C)3Nx,y such that for all composable sequences of prime morphisms (^^,■■■,^1^)) with src(Yo) = x and tgt(Yn) = y we have n = Nx,y. In this case, length depends only on src and tgt.

• A category C is said bounded iff the length of the composable sequences of C whose elements are not trivial are bounded, i.e. 3Nc G N such that for all composable sequences (Yn, ■■■, Yo) satisfying Yi = id, we have n < Nc.

• A category C is said weakly bounded iff Vy G Mo(C),Max({n G N/3(an, ■■■, a0) such that an o ■■■ o a0}) <

The relations existing between these properties are given in the following dia-

23 it is abusive to write "the" cohomology of a small category because, as far as I know, it depends on the natural system one has put on the small category one wants to calculate "the" cohomology. Hence, it becomes a part of the art to choose a good natural system. In partical cases, the component category of it has good properties which induce an "obviously" interesting natural system.

24with faces parallel to the faces of the unit cube.

Strongly Homogeneous = > Homogeneous

Bounded

Direct

generated by Primes

> Weakly Bounded

Loop-free

^Iso is discrete

Skeletal

Prime morphisms generalize prime numbers, indeed, the monoid (N, +) seen as a small category has prime morphisms which are exactly the prime numbers. In fact, it is homogeneous by the famous prime number decomposition theorem. In particular, n1(S ) is homogeneous. The notion of direct categories is related to model category theory, see [17] or [16] for further details.

Definition 6.2 A linear extension of a small category C is a functor f : C x X such that Vy £ Mo(C),f(Y) = id ^ Y = id and where X is an ordinal25 i.e. a poset whose any non empty subset has a minimum.

A direct category is a small category having a linear extension. An inverse category is a small category whose dual is direct.

Conjecture 6.3 The component category of a cubical pospace is homogeneous. Moreover, if its underlying space is connected, the component category is bounded.

In general it is not strongly homogeneous as shown by the right side of figure 2 nor bounded because it is always possible to have a infinite disjoint union of connected cubical pospaces C0U...UCnU... such that Vn £ N, Cn has a composable sequence of prime morphisms of length n. Being homogeneous induces a natural system as follows. Given a small category C, the category of factorizations of C (denoted FC) is given by Ob(FC) = Mo(C) and FC[a,3] is the collection of pairs (Y2,Y1) £ C\tgt(a),tgt(@)} x C[src(3),src(a)] such that 3 = Y2 ° a ° y1 26 . Given a small category C, a natural system (of abelian groups) 27 on C is a functor D : FC xAb, where Ab is the category of abelian groups and group morphisms between them.

25see [17] or [20] or any set theory textbook for the definition.

26 If C is small then so is FC. Moreover, if C is loop-free then so is FC.

27see [2] and [1] for further details.

Lemma 6.4 Let C be a homogeneous small category. We define a natural system on C setting D(y) := Zlength(Y and for

71 = 72

D(72,7i) : (xn,...,xi) G Zlength(a) ^ ( 0,............, 0 ,xn,..,xu 0,............, 0 ) G

* V * v

length(Y2) times length(Yi) times

Ziength(f}), with length(Y2) zeros on the left side of xn and length(Yi) zeros on the right side of x1.

Instead of a (boring and) formal proof that we actually have a functor, observe the following example, suppose length(^1) = 1, length(Y2) = 2, length(fî) = 6, then necesarily, length(a) = 3 and D (72,71) is an abelian group embedding pictured by the following diagram:

{0} x {0} x Z x Z x Z x {0}

Z x Z x Z x Z x Z x Z

It is important to notice that the image of a morphism of FC only depends on the length of 71,72 and a.

0 0 Î y

7 Dealing with loops: the fundamental monoid

As one can notice, the category POSPC does not contain any satifactory model of the directed circle " -1". Indeed, the only authorized paths of —1 are the clockwise ones 28 . The problem is to modelize this idea. What order relation should equip S1 in order to make it a pospaces whose dipaths are exactly the clockwise ones? Suppose that such a relation < exists, in particular, t £~t 1—> (cos(—2nt), sin(—2nt)) is clockwise, so we should have yt £ [0,1], (0,1) < (cos(—2nt),sin(—2nt)) < (0,1) hence, by antisymmetry, (cos(—2nt), sin(—2nt)) = (0,1) which is a contradiction. A naive solution consists of weakening the definition of a pospace asking < be a preorder instead of an order relation. But then, by transitivity, yt,t' £ [0,1], (cos(2nt), sin(2nt)) < (cos(2nt'), sin(2nt')) so t £~t ——► (cos(2nt), sin(2 nt)) which is anticlockwise would also be directed. Marco Grandis approach consists of equiping a topological space X with a set of distinguished paths denoted dX and submited to some conditions. The elements of dX are naturaly called the directed paths. Then it suffices to equip S1 with the set of all clockwise paths to obtain a model of the directed circle. It is also possible to have a model of directed circle by covering S1 with open subsets, each of which being suitably equiped with an order relation < that locally makes S1 a pospace.

28obviously we could have chosen the anticlokwise ones

Besides, the fact that a pospace does not have loops makes its fundamental category loop-free, in particular it has no endomorphisms. As a direct consequence, trying to define the "fundamental monoid" of a pospace X as —1(—X)[x,x] is sound but pointless because f—1(—X)[x,x] = {idx}. Introducing loops in our models, the "fundamental monoid" becomes relevant.

Ideas related to the definition of local pospaces are borrowed from the ones of differential geometry and smooth manifold theory, for a deeper analogy see [21] and [23].

Definition 7.1 [Local Pospaces] A local pospace is a triple (X,tx, <x) such that (X, tx) is a topological space, <x a relation on X and Vx £ X3U an open neighbourhood of x such that (U,tu, <u) is a pospace. tu and <u are respectively the restriction of tx and <x to U. An atlas of (X,tx , <x) is an open covering (Ui)iei of (X,tx) such that Vi £ I, (Ui,Tui, <ui) is a pospace. A local dimap f : (X,tx, <x) x (Y,ty, <y) is a continuous map between underlying topological spaces such that 3(Uj)jej atlas of (X,tx, <x) 3(Vj)jeJ atlas of (Y,ty, <y) satisfying Vj £ J, fuj—v3 : x £ Uj -—► f (x) £ Vj is a dimap (i.e. a morphism of POSPC). Local pospaces and local dimaps organize themselves in a category denoted LPOSPC.

As [0,1] is the standard example of pospace, the directed circle ff1 is the standard example of local pospace, its relation is described by means of maps 90 : x £ ]0, 2n[—> (cos(x), sin(x)) £ S1 and 91 : x £] — —> (cos(x), sin(x)) £ S1 setting Vx, y £]0,2n[, 90(x) < 90(y) if x < y and Vx, y £] — n, n[, 91(x) < 01(y) if x < y. The next definition is due to Marco Grandis in [14] 29

Definition 7.2 [d-spaces] A directed space or d-space is a triple (X,tx,dX) where (X, tx) is a topological space and dX C {paths of (X, tx)} with the following conditions

(i) {constant paths} C dX

(ii) for all 9 : [0,1] x [0,1] continuous and increasing, for all Y £ dX, y ° 0 £ dX (dX is stable under di-reparametrization)

(iii) for all Y1,Y2 £ dX, y2 ° Y1 £ dX (dX is stable under concatenation)

A d-map from (X,tx,dX) to (Y,ty,dY) is a continuous map f from (X,tx) to (Y, ty) such that Vy £ dX f ° y £ dY d-spaces and d-maps organize themselves in a category denoted dSPC.

Remark that we have the "obvious" inclusion functors

POSPCc-> LPOSPC-^dSPC

Now let us focus on two examples:

29 [14] also contains a definition of local pospace which differs from the presently given one.

Denoting n1 (S ) the fundamental category of S , we have yx £ -S, n1(-S)[x,x] isomorphic to N. Compare N to the fundamental group of the circle. Precisely, can be described the following way, for each x,y £ S1 there is a distinguished arrow axy and the family of distinguished arrows is submitted to the following axiom, yx,y,z £ S1, ay,z o ax,y = axz, where y £ (x,z). Here, (x, z) is the clockwise open arc from x to z. Intuitively, the distinguished arrow from x £ -S1 to y £ -S1 is the clockwise path from x to y on the directed circle, see the left side figure. Then £ n1(-S)[x,y] Bin £ N such that Y = (ay,y)n o axy and y^ £ n1(—1')[x,xj Bin £ N such that y = (ax , x)n. Hence we could define the fundamental monoid of -S1 as (N, +). The idea of the fundamental monoid is attractive but does not work because, in general, it depends on the base point x:

The left side picture can easily be described as a local pospace or a directed space denoted 1 in both cases. Adapting the description of the fundamental category of -S, it is easy to describe the one of XX. Then we observe that 7r1(XX)[x,x] = (N, +) in MON - the category of monoids - while 7f1(XX)[y, y] = {•}. The base point dependence makes impossible to define the fundamental monoid of XX as the straight>— forward generalization of the fundamental group.

In addition, the component category of S1 is its fundamental one. Indeed, none of the morphisms axy of n1 (S*1) is Yoneda inversible. By definition, if axy were Yoneda inversible then, since n1(S^)[y,x] = 0, we would have a morphism g from y to x such that g o ax, y = idx, which is impossible. Hence, as any arrow of ni(-1) can be written as a composition of a's, none of them is Yoneda inversible. In particular the component category, as it has been defined previously, does not efficiently reduce the size of the fundamental category of a local pospace or a directed space which "contains" —1. Still, the next result may provide a way to solve this problem:

Proposition 7.3 Let C be a small category. Suppose that a : x —► y is a morphism

of C :

(i) If Vô G C [x,x\ 3!y g C [y, y] such that a o ô = 7 o a then the map : ô G C[x, x\ 1—► 7 G C[y, y\ is a morphism of monoids

(ii) If Vy G C[y,y\ 3!ô G C[x,x\ such that y ◦ a = a o 5 then the map : 7 G C[y, y\ 1—► ô G C[x,x\ is a morphism of monoids

(iii) If Vô GC[x,x] 3!y g C[y, y\ such that a o 5 = 7 o a and Vy G C[y, y\ 3!ô G C[x, x\ such that y o a = a o 5 then o = Idc[x,x] and o = Idcyy

Proof. aoidx = idyoa, thus (idx) = idy. Moreover, ao(ô2oôi) = (ô2oôi)^ oa

and a o (Ô2 o ôi) = (a o Ô2) o ôi = (Ô2) o a^ o ôi = (Ô2) o (a o ôi) = (Ô2)) o

^ct(ôi)^ oa. By uniqueness, (ô2 oôi) = (ô2) o(ôi), hence is a morphism of monoids. The same holds for dualizing everything. Suppose we have the hypothesis of the third point, then a o ô = (ô) o a = a o (ô)), hence, by

uniqueness, (ô)) = ô. The same way, (7)) = 7. □

Proposition 7.4 Let C be a small category. Suppose that a : x —► y is a morphism of C such that fa : ô G C[x, x\ 1—► aoô G C[x, y\ and ga : 7 G C[x, x\ 1—> 7oa G C[x, y\ are bijective. Then the hypothesis of the third point of proposition 7.3 are satisfied.

Proof. Given ô G C[x,x\, by definition of the bijections f and g, 7 := g-i(a o ô) is the only element of C[y, y\ such that a o ô = 7 o a. Of course, given 7 G C[y, y\, by defintion of the bijections f and g, ô := f -i(y o a) is the only element of C[x, x\ such that y o a = a o ô. In particular, = g-i o fCT and ^ = f-i o gCT. □

Corollary 7.5 Any Yoneda inversible morphism satisfies the hypothesis of proposition 7.4

For example, we remark that Vx, y G Si, ax,y satisfies the hypothesis of the third point of proposition 7.3 and of proposition 7.4, nevertheless, as we have already seen, they are not Yoneda inversible. This remark leads to settle some definitions, the former Yoneda inversible morphisms are, from now, called strong Yoneda inversible, the morphisms satisfying the hypothesis of proposition 7.4 are called Yoneda inversible and the ones enjoying the hypothesis of the third point of proposition 7.3 are called weak Yoneda inversible. The next step consists of adapting the definition of inessential system, putting Yoneda or weak Yoneda instead of strong Yoneda. In the new context, there is no doubt that the definition of the component category will require some changes too.

7.1 Component category of the directed torus with a hole

Take the directed square with a hole (see figure 1) then identify [0,1\ x {0} « [0,1\ x {1} and {0} x [0,1\ « {1} x [0,1\. We obtain a local pospace whose underlying topological space is a torus with a hole and where the local order is clockwise on the "small" and "large" generators, denote T this local pospace. Figure 7 represents T with the identifications described above. No morphism is

Fig. 7. Directed torus T2 -unfold representation-

Yoneda inversible, for the same reason as in the directed circle case. Still, all the morphisms of the fundamental category they induce are both monic and epic making the correponding set theoretic maps are one-to-one but not onto (see definition 4.5). For example, the dipaths a and 3 on figure 7 are not dihomotopic, the right hand part shows the "only" dihomotopy one could image. In fact, a and 3 are not even homotopic, it is a classical algebraic topology problem. The consequence is that V5, morphism of ^(T2), OB o (C ^ A) = (C ^ A) o 5. Where (C ^ A) is an arrow whose beginning is in the interior of C and the end in the interior of A, and O is a loop starting from A going through B and coming back to its initial point. In particular, any dipath crossing the fronteer of A, B or C, cannot be weakly Yoneda inversible. Conversely, it seems that any dipath staying in the same "component" is weakly Yoneda inversible. Then, ideologically, we "should" have three components30 and the component category "should" be the

free category generated by B A C •

These last statements are just prospective thoughts and conjectures.

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