Theory and Applications of Categories, Vol. 11, No. 1, 2003, pp. 1–14.

CATEGORICAL MODELS AND QUASIGROUP HOMOTOPIES

GEORGE VOUTSADAKIS

ABSTRACT. As is pointed out in [Smith (1997)], in many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms. In this paper, the way homotopies may arise in the context of categorical quasigroup model theory is investigated. In this context, the algebraic structures are specified by diagram-based logics, such as sketches, and categories of models become functor categories. An idea, pioneered in [Gvaramiya & Plotkin (1992)], is used to give a construction of a model category naturally equivalent to the category of quasigroups with homotopies between them.

1. Introduction

Traditionally, when the categorical study of classes of algebras is undertaken the role of morphisms in the categories under investigation is played by the ordinary homomorphisms of universal algebra (see, e.g., [Borceux (1994)], Vol. II, Chapter 3). However, in the case of quasigroups, a different kind of morphism may be more important, depending on the application at hand, than ordinary quasigroup homomorphisms [Smith (1997)]. These are quasigroup homotopies. A homotopy from a quasigroup Q = ⟨Q, ·, /, \⟩ to a quasigroup P = ⟨P, ·, /, \⟩ is a triple (h_{1}, h_{2}, h_{3}) of set maps from Q to P, such that, for all x, y ∈ Q, h_{3}(x · y) = h_{1}(x) · h_{2}(y) (see, e.g., [Smith & Romanowska (1999)], I.4).

An alternative to equational logic for specifying algebraic structures is provided by graph-based logics, such as sketches. [Barr & Wells (1990)] and [Coppey & Lair (1984), Coppey & Lair (1988)] provide very readable introductions to sketches and a good part of the first two sections of the present paper heavily draws on their treatment. In addition, the 6th Lesson of [Coppey & Lair (1988)] contains sketches specifying many of the best known algebraic and graph-based structures. Following [Barr & Wells (1990)] and [Coppey & Lair (1984), Coppey & Lair (1988)], a sketch for quasigroups is given in the following section. Model morphisms in model categories of sketches being natural transformations, they are really tailored to capture the ordinary homomorphisms of universal algebra. Thus, as it is shown in Section 3, the category of models of the sketch for quasigroups is the category of quasigroups with homomorphisms between them.

This introduction motivates the main question we are faced with: What is the role of homotopies in categorical quasigroup model theory, or, rephrasing, how may one capture homotopies of quasigroups in the context of categorical quasigroup model theory? One feature of the sketch for quasigroups that spoils the admissibility of homotopies is the use of direct squares and of the accompanying projections. Roughly speaking, these force a natural transformation η from a model M to a model N to obey commutativity of

Received by the editors 02-12-02 and, in revised form, 02-12-11.

Transmitted by Michael Barr. Published on 03-01-10.

2000 Mathematics Subject Classification: 20N05, 18B99, 18A10.

Key words and phrases: sketches, finite product sketches, sketch models, quasigroups, homotopies.

\begin{matrix}c \\ ◯ \end{matrix} George Voutsadakis, 2003. Permission to copy for private use granted.

[figure: M(Q); N(Q); η_{Q}; M(Q^{2}); N(Q^{2}); η_{Q^{2}}; M(p_{1}); N(p_{1}); M(Q); N(Q); η_{Q}; M(Q^{2}); N(Q^{2}); η_{Q^{2}}; M(p_{2}); N(p_{2})]  (1)

whence η_{Q^{2}} = η_{Q} × η_{Q} and η_{Q} becomes necessarily a homomorphism. If this problem is to be overcome, one has to get rid of the direct squares which entails approaching the design of the sketch with a different philosophy in mind.

What comes to the rescue is an idea, first exploited by Gvaramiya and Plotkin in [Gvaramiya & Plotkin (1992)]. Its cornerstone is the introduction of different sorts for each of the arguments of the quasigroup multiplication. In the present context, this has the effect of transforming direct powers to direct products of different objects and, thus, dissolves the difficulty imposed by the previous requirement that (1) commute. Based on this idea, the notion of a ∗-automaton was defined in [Gvaramiya & Plotkin (1992)] and it was shown that every quasigroup gives rise to an invertible ∗-automaton and that every invertible ∗-automaton is isomorphic to one derived in this way by some quasigroup.

Adapting this idea to the present context, a modified sketch for quasigroups with homotopies is presented, such that its models in the category of sets are the invertible ∗-automata of [Gvaramiya & Plotkin (1992)] and the model morphisms between them, which are homomorphisms of the multi-sorted algebras, correspond to homotopies between the associated quasigroups. It is then shown in the last section that this model category and the category of quasigroups with homotopies between them are naturally isomorphic categories.

In [Smith (1997)], Smith showed, using a “semisymmetrization technique”, that the category of quasigroups with homotopies is isomorphic to a category of homomorphisms between semisymmetric quasigroups, i.e., quasigroups satisfying the semisymmetric identity (y·x)·y ≈ x. The question remains open of whether Smith’s result may be exploited, in the present context, so that a sketch be obtained having as its model category a category isomorphic to the category of quasigroups with homotopies.

2. Sketching Quasigroups

[Barr & Wells (1990)] and [Coppey & Lair (1984), Coppey & Lair (1988)] “sketch” some of the most commonly encountered algebraic structures. They are the source of the graph-theoretic and categorical definitions that are used in this and the next section in developing the standard sketch for quasigroups and showing that it corresponds to the category of quasigroups with homomorphisms between them. Definitions that pertain directly to quasigroups and their morphisms may be found in [Smith & Romanowska (1999)].

2.1. Definition. A (directed) graph G = ⟨V, E, s, t⟩ consists of a set V of nodes or vertices, a set E of edges, and two functions s, t : E → V, associating with each edge e its source vertex s(e) and its target vertex t(e), respectively. One writes e : s(e) → t(e) in this case. Let G = ⟨V, E, s, t⟩ and G′ = ⟨V′, E′, s′, t′⟩ be graphs. A graph morphism h : G → G′ is a pair h = ⟨h_{1}, h_{2}⟩, with h_{1} : V → V′ and h_{2} : E → E′ satisfying s′(h_{2}(e)) = h_{1}(s(e)) and t′(h_{2}(e)) = h_{1}(t(e)), for all e ∈ E.

As an example and for future reference we introduce the graph G_{q} with V_{q} = {Q_{1}, Q_{2}}, E_{q} = {p_{1}, p_{2}, m, l, r, ⟨p_{1}, m⟩, ⟨m, p_{2}⟩, ⟨p_{1}, l⟩, ⟨r, p_{2}⟩}, where s_{q} and t_{q} are given diagrammatically as follows:

p_{1}, p_{2}, m, l, r : Q_{2} → Q_{1},
⟨p_{1}, m⟩, ⟨m, p_{2}⟩, ⟨p_{1}, l⟩, ⟨r, p_{2}⟩ : Q_{2} → Q_{2}.

2.2. Definition. Let G be a graph. A path in G is a sequence (e_{1}, …, e_{n}) of edges in G, such that, for all i = 1, …, n − 1, t(e_{i}) = s(e_{i+1}).

Two paths p = (e_{1}, …, e_{n}) and q = (f_{1}, …, f_{m}) in G are said to be parallel if s(e_{1}) = s(f_{1}) and t(e_{n}) = t(f_{m}).

An equation in G is a pair of parallel paths p and q as above, and is usually denoted by e_{n}e_{n−1} … e_{1} ≈ f_{m}f_{m−1} … f_{1}.

The following are equations in the graph G_{q} defined previously.

\begin{matrix}p_{1}⟨p_{1}, m⟩ ≈ p_{1} & p_{2}⟨p_{1}, m⟩ ≈ m & p_{1}⟨m, p_{2}⟩ ≈ m & p_{2}⟨m, p_{2}⟩ ≈ p_{2} \\ p_{1}⟨p_{1}, l⟩ ≈ p_{1} & p_{2}⟨p_{1}, l⟩ ≈ l & p_{1}⟨r, p_{2}⟩ ≈ r & p_{2}⟨r, p_{2}⟩ ≈ p_{2} \end{matrix}

Also

\begin{matrix}l⟨p_{1}, m⟩ ≈ p_{2} & r⟨m, p_{2}⟩ ≈ p_{1} \\ m⟨p_{1}, l⟩ ≈ p_{2} & m⟨r, p_{2}⟩ ≈ p_{1} \end{matrix}

2.3. Definition. Let G be a graph. A diagram d in G is a graph morphism d : D → G, where D = ⟨U, F, σ, τ⟩ is the shape graph of d.

A cone v ◁ d in G with vertex v and base d consists of a diagram d in G, a vertex v ∈ V and a collection of edges {e_{u} : u ∈ U}, called projections, such that s(e_{u}) = v and t(e_{u}) = d(u), for all u ∈ U. The cone v ◁ d is said to be discreet or a product cone if F = ∅ and a finite product cone if, in addition, |U| < ω.

Let, for instance, D_{q} = ⟨U_{q}, F_{q}, σ_{q}, τ_{q}⟩ be the graph with U_{q} = {u_{1}, u_{2}}, F_{q} = ∅ and d_{q} : D_{q} → G_{q} the diagram in G_{q} determined by d_{q1}(u_{1}) = d_{q1}(u_{2}) = Q_{1}. Define the cone Q_{2} ◁ d_{q} in G_{q} by specifying that p_{1} : Q_{2} → Q_{1} and p_{2} : Q_{2} → Q_{1} be the two cone projections.

2.4. Definition. [Barr & Wells (1990)] A limit sketch S = ⟨G, Q, L⟩ consists of a graph G, a set Q of equations in G and a set L of cones in G. If all cones in L are product cones then S is called a product sketch and if they are all finite, then S is called a finite product sketch or an FP-sketch.

Let S_{q} = ⟨G_{q}, Q_{q}, L_{q}⟩ be the sketch with graph G_{q}, set of equations Q_{q}, containing all the equations displayed above, and set of cones L_{q} = {Q_{2} ◁ d_{q}}. S_{q} is the sketch for quasigroups.

2.5. Definition. Let S = ⟨G, Q, L⟩ be a limit sketch and C a category. A model M : S → C of S in C is a graph morphism M : G → C, where C is the underlying graph of C, such that all equations in Q become commuting diagrams in C and all cones in L become limit cones in C.

Given two models M_{1}, M_{2} of S in C, a model morphism h : M_{1} → M_{2} is a natural transformation from M_{1} to M_{2}, i.e., a family h_{v} : M_{1}(v) → M_{2}(v), v ∈ V, of morphisms in C, such that, for all e ∈ E, with s(e) = v_{1}, t(e) = v_{2}, the following rectangle commutes in C :

[figure: M_{1}(v_{2}); M_{2}(v_{2}); h_{v_{2}}; M_{1}(v_{1}); M_{2}(v_{1}); h_{v_{1}}; M_{1}(e); M_{2}(e)]

Models of S in C together with model morphisms form a category, which is denoted by Mod_{C}(S).

In case C is a category with specified limits, a model of S in C has to carry all cones in L to specified limit cones and a model morphism has to preserve all limits corresponding to limit cones on the nose.

2.6. Definition. A quasigroup Q = ⟨Q, ·, /, \⟩ is a set Q equipped with binary operations x · y or, simply, xy of multiplication, x/y of right division and x\y of left division, such that the following identities hold:

\begin{matrix}x\(x · y) ≈ y & (x · y)/y ≈ x \\ x · (x\y) ≈ y & (x/y) · y ≈ x \end{matrix}

Let Q and P be two quasigroups. A quasigroup homomorphism h : Q → P is a function h : Q → P, such that h(x · y) = h(x) · h(y), for all x, y ∈ Q.

A triple (h_{1}, h_{2}, h_{3}) : Q → P of functions from Q to P is a quasigroup homotopy if h_{3}(x · y) = h_{1}(x) · h_{2}(y), for all x, y ∈ Q.

Note that, if h : Q → P is a quasigroup homomorphism, then, for all x, y ∈ Q, h(x/y) = h(x)/h(y) and h(x\y) = h(x)\h(y). Similarly, if (h_{1}, h_{2}, h_{3}) : Q → P is a quasigroup homotopy, then h_{1}(x/y) = h_{3}(x)/h_{2}(y) and h_{2}(x\y) = h_{1}(x)\h_{3}(y), for all x, y ∈ Q (see [Smith (1997)]).

We denote by Set the category of all small sets and by Set_{→} the category of all small sets with the usual specified limits. The same notation will also be used to denote the underlying graphs of these two categories for simplicity. The following proposition is a first step in relating the notions that were introduced in this section.

2.7. Proposition. Let Q = ⟨Q, ·, /, \⟩ be a quasigroup. Define the graph morphism M_{Q} : G_{q} → Set_{→}, as follows:

M_{Q}(Q_{1}) = Q  M_{Q}(Q_{2}) = Q × Q
M_{Q}(p_{1})((x, y)) = x  M_{Q}(p_{2})((x, y)) = y
M_{Q}(m)((x, y)) = x · y  M_{Q}(l)((x, y)) = x\y
M_{Q}(r)((x, y)) = x/y
M_{Q}(⟨p_{1}, m⟩)((x, y)) = (x, x · y) M_{Q}(⟨m, p_{2}⟩)((x, y)) = (x · y, y)
M_{Q}(⟨p_{1}, l⟩)((x, y)) = (x, x\y) M_{Q}(⟨r, p_{2}⟩)((x, y)) = (x/y, y)

Then M_{Q} is a model of S_{q} in Set_{→}.

In addition, quasigroup homomorphisms give us concrete examples of model morphisms in Mod_{Set_{→}}(S_{q}):

2.8. Proposition. Let Q, P be two quasigroups, h : Q → P a quasigroup homomorphism and M_{Q}, M_{P} : S_{q} → Set_{→} the two models of S_{q} in Set_{→} defined as in Proposition 2.7. η : M_{Q} → M_{P}, defined by η_{Q_{1}} = h and η_{Q_{2}} = (h, h) is a model morphism of S_{q} in Set_{→}.

3. Homomorphisms of Quasigroups

It is now shown that the only objects in Mod_{Set_{→}}(S_{q}) are the ones given by Proposition 2.7 and, similarly, that the only model morphisms in this category are the ones provided by Proposition 2.8. What role do homotopies play in the context of the categorical model theory of quasigroups? The use of direct squares and, more generally, direct powers and the associate projections in the sketch that specifies a particular structure makes it impossible to accomodate homotopy-like morphisms in the form of natural transformations. To answer this question a new categorical specification of quasigroups will be introduced in the next section. The trick is to specify multiplication as a multisorted operation distinguishing between the values that can be substituted for each of its arguments (see [Gvaramiya & Plotkin (1992)]).

3.1. Proposition. Let M : S_{q} → Set_{→} be a model in Mod_{Set_{→}}(S_{q}). Then ⟨M(Q_{1}), M(m), M(r), M(l)⟩ is a quasigroup.

Proof. Since Q_{1} ←^{p_{1}} Q_{2} →^{p_{2}} Q_{1} is carried by M to a specified product cone, we have M(Q_{2}) = M(Q_{1})×M(Q_{1}), with M(p_{1}), M(p_{2}) the first and second coordinate projections, respectively. Now it is clear that M(m), M(r) and M(l) are binary operations on M(Q_{1}).

The pair of equations

p_{1}⟨p_{1}, m⟩ ≈ p_{1} p_{2}⟨p_{1}, m⟩ ≈ m,

interpreted in M, give, for all (x, y) ∈ M(Q_{1})^{2},

M(p_{1})(M(⟨p_{1}, m⟩)(x, y)) = M(p_{1})(x, y) and
M(p_{2})(M(⟨p_{1}, m⟩)(x, y)) = M(m)(x, y),

whence, since M(p_{i}) is the i-th projection, i = 1, 2,

M(⟨p_{1}, m⟩)(x, y) = (x, M(m)(x, y)), for all x, y ∈ M(Q_{1}).

Similarly, one obtains that

M(⟨m, p_{2}⟩)(x, y) = (M(m)(x, y), y),
M(⟨p_{1}, l⟩)(x, y) = (x, M(l)(x, y)) and
M(⟨r, p_{2}⟩)(x, y) = (M(r)(x, y), y), for all x, y ∈ M(Q_{1}).

Now from l⟨p_{1}, m⟩ ≈ p_{2} we get M(l)(M(⟨p_{1}, m⟩)(x, y)) = M(p_{2})(x, y), whence

M(l)(x, M(m)(x, y)) = y,

and, similarly,

M(r)(M(m)(x, y), y) = x, M(m)(x, M(l)(x, y)) = y, and
M(m)(M(r)(x, y), y) = x.

Thus ⟨M(Q_{1}), M(m), M(r), M(l)⟩ is a quasigroup, as claimed.

A similar result is obtained next concerning the morphisms in the model category Mod_{Set_{→}}(S_{q}).

3.2. Proposition. Let M, N : S_{q} → Set_{→} be models in Mod_{Set_{→}}(S_{q}) and η : M → N a morphism in Mod_{Set_{→}}(S_{q}). Then η_{Q_{1}} : M(Q_{1}) → N(Q_{1}) is a quasigroup homomorphism from ⟨M(Q_{1}), M(m), M(r), M(l)⟩ into ⟨N(Q_{1}), N(m), N(r), N(l)⟩.

Proof. It suffices to show that,

for all x, y ∈ M(Q_{1}), η_{Q_{1}}(M(m)(x, y)) = N(m)(η_{Q_{1}}(x), η_{Q_{1}}(y)),

i.e., that the following rectangle commutes

[figure: M(Q_{1}); N(Q_{1}); η_{Q_{1}}; M(Q_{1}) × M(Q_{1}); N(Q_{1}) × N(Q_{1}); η_{Q_{1}} × η_{Q_{1}}; M(m); N(m)]

Since M(Q_{1}) × M(Q_{1}) = M(Q_{2}) and N(Q_{1}) × N(Q_{1}) = N(Q_{2}), this would certainly be true if η_{Q_{1}} × η_{Q_{1}} = η_{Q_{2}}. But, since η : M → N is a natural transformation, we have commutativity of

[figure: M(Q_{1}); N(Q_{1}); η_{Q_{1}}; M(Q_{1}) × M(Q_{1}); N(Q_{1}) × N(Q_{1}); η_{Q_{2}}; M(p_{1}); N(p_{1}); M(Q_{1}); N(Q_{1}); η_{Q_{1}}; M(Q_{1}) × M(Q_{1}); N(Q_{1}) × N(Q_{1}); η_{Q_{2}}; M(p_{2}); N(p_{2})]

whence

η_{Q_{2}}(x, y) = (η_{Q_{1}}(x), η_{Q_{1}}(y)), for all x, y ∈ M(Q_{1}),

as required.

The results that we have obtained so far may be summarized in the following

3.3. Theorem. Mod_{Set_{→}}(S_{q}) is the category corresponding to the usual universal algebraic variety of quasigroups, i.e., the category of quasigroups with homomorphisms between them.

Next, let M : S_{q} → Set be a model of S_{q} in Set. Since Set is not assumed to have specified limits, it is not necessarily the case that M(Q_{2}) = M(Q_{1})×M(Q_{1}). One may now only conclude that M(Q_{2}) ≅ M(Q_{1})×M(Q_{1}). Denote by ϕ_{M} : M(Q_{1})×M(Q_{1}) → M(Q_{2}) the isomorphism that makes the following diagram commute

[figure: M(Q_{1}); M(Q_{2}); M(p_{1}); M(Q_{1}); M(p_{2}); M(Q_{1}) × M(Q_{1}); π_{1}; ϕ_{M}; π_{2}]

where π_{1}, π_{2} are the ordinary coordinate projections in Set.

The following propositions show that the model category obtained is essentially the same as before modulo the isomorphism ϕ_{M}.

3.4. Proposition. Let M : S_{q} → Set and ϕ_{M} : M(Q_{1}) × M(Q_{1}) → M(Q_{2}) be as above. Then ⟨M(Q_{1}), M(m)ϕ_{M}, M(r)ϕ_{M}, M(l)ϕ_{M}⟩ is a quasigroup.

Proof. It is clear that M(m)ϕ_{M}, M(r)ϕ_{M} and M(l)ϕ_{M} are binary operations on M(Q_{1}).

p_{1}⟨p_{1}, m⟩ ≈ p_{1} and p_{2}⟨p_{1}, m⟩ ≈ m

give

M(p_{1})M(⟨p_{1}, m⟩) = M(p_{1}) and M(p_{2})M(⟨p_{1}, m⟩) = M(m),

whence

M(p_{1})M(⟨p_{1}, m⟩)ϕ_{M} = π_{1} and M(p_{2})M(⟨p_{1}, m⟩)ϕ_{M} = M(m)ϕ_{M},

which together imply that

M(⟨p_{1}, m⟩)ϕ_{M} = ϕ_{M}⟨π_{1}, M(m)ϕ_{M}⟩.

Similarly, we obtain

M(⟨m, p_{2}⟩)ϕ_{M} = ϕ_{M}⟨M(m)ϕ_{M}, π_{2}⟩, M(⟨p_{1}, l⟩)ϕ_{M} = ϕ_{M}⟨π_{1}, M(l)ϕ_{M}⟩
and M(⟨r, p_{2}⟩)ϕ_{M} = ϕ_{M}⟨M(r)ϕ_{M}, π_{2}⟩.

Now, we have M(l)M(⟨p_{1}, m⟩) = M(p_{2}), whence

M(l)M(⟨p_{1}, m⟩)ϕ_{M} = M(p_{2})ϕ_{M}, i.e., M(l)ϕ_{M}⟨π_{1}, M(m)ϕ_{M}⟩ = π_{2}.

Similarly, one may obtain the remaining three identities for the quasigroup ⟨M(Q_{1}), M(m)ϕ_{M}, M(r)ϕ_{M}, M(l)ϕ_{M}⟩.

Similarly, one obtains the following proposition, whose proof is omitted.

3.5. Proposition. Let M, N : S_{q} → Set and ϕ_{M} : M(Q_{1}) × M(Q_{1}) → M(Q_{2}), ϕ_{N} : N(Q_{1}) × N(Q_{1}) → N(Q_{2}) be as above. Suppose that η : M → N is a morphism in Mod_{Set}(S_{q}). Then η_{Q_{1}} : M(Q_{1}) → N(Q_{1}) is a quasigroup homomorphism from the quasigroup ⟨M(Q_{1}), M(m)ϕ_{M}, M(r)ϕ_{M}, M(l)ϕ_{M}⟩ to the quasigroup ⟨N(Q_{1}), N(m)ϕ_{N}, N(r)ϕ_{N}, N(l)ϕ_{N}⟩.

So, what modification is needed in the quasigroup sketch, so that its model category be the category Qtp of quasigroups with quasigroup homotopies between them? In the next section a modified sketch is introduced whose model category will be shown to be naturally equivalent to Qtp. We do not know, however, whether a limit sketch exists whose category of models is isomorphic to Qtp. Smith’s semisymmetrization result in [Smith (1997)] may prove helpful in answering this question.

4. The Modified Sketch

The new graph G_{t} has vertex set

V_{t} = {Q_{1}, Q_{2}, Q_{3}, Q_{12}, Q_{13}, Q_{32}}

and its edges are given diagrammatically as follows:

p_{1}^{12} : Q_{12} → Q_{1}  p_{2}^{12} : Q_{12} → Q_{2}

and, similarly, for p_{1}^{13}, p_{3}^{13} and p_{3}^{32}, p_{2}^{32},

m : Q_{12} → Q_{3}  l : Q_{13} → Q_{2}  r : Q_{32} → Q_{1}

and, finally,

\begin{matrix}⟨m, p_{2}^{12}⟩ : Q_{12} → Q_{32} & ⟨p_{1}^{12}, m⟩ : Q_{12} → Q_{13} \\ ⟨p_{1}^{13}, l⟩ : Q_{13} → Q_{12} & ⟨r, p_{2}^{32}⟩ : Q_{32} → Q_{12}. \end{matrix}

The following is a list of equations in G_{t}. The set of these equations is denoted by Q_{t} :

\begin{matrix}p_{1}^{13}⟨p_{1}^{12}, m⟩ ≈ p_{1}^{12} & p_{3}^{13}⟨p_{1}^{12}, m⟩ ≈ m & p_{3}^{32}⟨m, p_{2}^{12}⟩ ≈ m & p_{2}^{32}⟨m, p_{2}^{12}⟩ ≈ p_{2}^{12} \\ p_{1}^{12}⟨p_{1}^{13}, l⟩ ≈ p_{3}^{13} & p_{1}^{12}⟨p_{1}^{13}, l⟩ ≈ l & p_{1}^{12}⟨r, p_{2}^{32}⟩ ≈ r & p_{2}^{12}⟨r, p_{2}^{32}⟩ ≈ p_{2}^{32} \end{matrix}

\begin{matrix}l⟨p_{1}^{12}, m⟩ ≈ p_{2}^{12} & r⟨m, p_{2}^{12}⟩ ≈ p_{1}^{12} \\ m⟨p_{1}^{13}, l⟩ ≈ p_{3}^{13} & m⟨r, p_{2}^{32}⟩ ≈ p_{3}^{32} \end{matrix}

Compare the equations in Q_{t} with those in Q_{q} displayed in Section 2.

Finally, let L_{t} be the set consisting of the following cones, given in diagrammatic form

[figure: Q_{1}  Q_{2}; Q_{12}; p_{1}^{12}; p_{2}^{12}; Q_{1}  Q_{3}; Q_{13}; p_{1}^{13}; p_{3}^{13}; Q_{3}  Q_{2}; Q_{32}; p_{3}^{32}; p_{2}^{32}]

Let S_{t} = ⟨G_{t}, Q_{t}, L_{t}⟩ be the sketch with graph G_{t}, set of equations Q_{t} and set of cones L_{t}, as constructed above. S_{t} is the sketch for quasigroup homotopies.

Now the following proposition may be easily verified.

4.1. Proposition. Let Q = ⟨Q, ·, /, \⟩ be a quasigroup. Define the graph morphism N_{Q} : G_{t} → Set_{→} as follows:

N_{Q}(Q_{1}) = N_{Q}(Q_{2}) = N_{Q}(Q_{3}) = Q,
N_{Q}(Q_{12}) = N_{Q}(Q_{13}) = N_{Q}(Q_{32}) = Q × Q,
N_{Q}(p_{1}^{12})(x, y) = N_{Q}(p_{1}^{13})(x, y) = N_{Q}(p_{3}^{32})(x, y) = x

and

N_{Q}(p_{2}^{12})(x, y) = N_{Q}(p_{3}^{13})(x, y) = N_{Q}(p_{2}^{32})(x, y) = y,

N_{Q}(m)(x, y) = x · y  N_{Q}(l)(x, y) = x\y  N_{Q}(r)(x, y) = x/y

\begin{matrix}N_{Q}(⟨m, p_{2}^{12}⟩)(x, y) = (x · y, y) & N_{Q}(⟨p_{1}^{12}, m⟩)(x, y) = (x, x · y) \\ N_{Q}(⟨p_{1}^{13}, l⟩)(x, y) = (x, x\y) & N_{Q}(⟨r, p_{3}^{13}⟩)(x, y) = (x/y, y) \end{matrix}

Then N_{Q} is a model of S_{t} in Set_{→}.

Moreover, by analogy with Proposition 2.8, we have the following:

4.2. Proposition. Let Q, P be two quasigroups, (h_{1}, h_{2}, h_{3}) : Q → P a quasigroup homotopy and N_{Q}, N_{P} : S_{t} → Set_{→} the two models of S_{t} defined as in Proposition 4.1. η : N_{Q} → N_{P}, defined by

\begin{matrix}η_{Q_{1}} = h_{1} & η_{Q_{2}} = h_{2} & η_{Q_{3}} = h_{3} \\ η_{Q_{12}} = (h_{1}, h_{2}) & η_{Q_{13}} = (h_{1}, h_{3}) & η_{Q_{32}} = (h_{3}, h_{2}), \end{matrix}

is a model morphism of S_{t} in Set_{→}.

5. Homotopies of Quasigroups

In this section, it is shown that the category of models Mod_{Set_{→}}(S_{t}) essentially contains quasigroups with homotopies between them. More precisely, that Mod_{Set_{→}}(S_{t}) is naturally equivalent to Qtp. The proof is based on the fact that the three sets in which the vertices Q_{1}, Q_{2} and Q_{3} of G_{t} are mapped in Set_{→} by any model M : S_{t} → Set_{→} are isomorphic. So up to isomorphism, i.e., a renaming of the elements corresponding to Q_{1} and Q_{2}, M will be shown to define a quasigroup with universe M(Q_{3}). Then all model morphisms in Mod_{Set_{→}}(S_{t}) between two models M and N may be appropriately translated to homotopies between the quasigroups with the universes M(Q_{3}) and N(Q_{3}).

5.1. Lemma. Let M : S_{t} → Set_{→} be a model in Mod_{Set_{→}}(S_{t}), x_{M} ∈ M(Q_{1}) and y_{M} ∈ M(Q_{2}). Then ϕ_{y_{M}} : M(Q_{1}) → M(Q_{3}) and ψ_{x_{M}} : M(Q_{2}) → M(Q_{3}), defined by

ϕ_{y_{M}} (x) = M(m)(x, y_{M}) and ψ_{x_{M}} (y) = M(m)(x_{M}, y),

for all x ∈ M(Q_{1}), y ∈ M(Q_{2}), respectively, are bijections.

Proof. We only show that ϕ_{y_{M}} : M(Q_{1}) → M(Q_{3}) is a bijection. The case of ψ_{x_{M}} may be handled similarly.

Suppose x_{1}, x_{2} ∈ M(Q_{1}), with ϕ_{y_{M}} (x_{1}) = ϕ_{y_{M}} (x_{2}). Then

M(m)(x_{1}, y_{M}) = M(m)(x_{2}, y_{M}),

whence M(r)(M(m)(x_{1}, y_{M}), y_{M}) = M(r)(M(m)(x_{2}, y_{M}), y_{M}) and, therefore, x_{1} = x_{2}. Thus ϕ_{y_{M}} is one-to-one.

Next, let z ∈ M(Q_{3}). Then, for x = M(r)(z, y_{M}) ∈ M(Q_{1}), we have

ϕ_{y_{M}} (x) = M(m)(x, y_{M}) = M(m)(M(r)(z, y_{M}), y_{M}) = z.

Thus, ϕ_{y_{M}} is also onto.

With Lemma 5.1 at hand, it may now be shown that Mod_{Set_{→}}(S_{t}) is essentially the category Qtp of quasigroups with homotopies between them. Note that, because of Lemma 5.1, to make the correspondence that is established between Mod_{Set_{→}}(S_{t}) and the category Qtp of quasigroups with homotopies between them natural we must fix a way of choosing the elements x_{M} and y_{M} in M(Q_{1}) and M(Q_{2}), respectively. Luckily enough any choice will do. To this end, an arbitrary choice function c for the class |Set_{→}| − {∅} will be fixed later in the section.

The work is divided again into two steps. In the first we deal with objects and in the second with morphisms in Mod_{Set_{→}}(S_{t}).

5.2. Proposition. Let M : S_{t} → Set_{→} be a model in Mod_{Set_{→}}(S_{t}), x_{M} ∈ M(Q_{1}) and y_{M} ∈ M(Q_{2}). Then

⟨M(Q_{3}), M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩, ϕ_{y_{M}} M(r)⟨i_{M(Q_{3})}, ψ_{x_{M}}^{−1} ⟩, ψ_{x_{M}} M(l)⟨ϕ_{y_{M}}^{−1} , i_{M(Q_{3})}⟩⟩

is a quasigroup.

Proof. Using Lemma 5.1, it is easy to verify that M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩, ϕ_{y_{M}} M(r)⟨i_{M(Q_{3})}, ψ_{x_{M}}^{−1} ⟩ and ψ_{x_{M}} M(l)⟨ϕ_{y_{M}}^{−1} , i_{M(Q_{3})}⟩ are all binary operations on M(Q_{3}). So it suffices to show that they obey the quasigroup laws. We only verify one of the four laws. The proofs of the remaining three are very similar.

ψ_{x_{M}} M(l)⟨ϕ_{y_{M}}^{−1} , i_{M(Q_{3})}⟩(x, M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩(x, y))
= ψ_{x_{M}} M(l)(ϕ_{y_{M}}^{−1} (x), M(m)(ϕ_{y_{M}}^{−1} (x), ψ_{x_{M}}^{−1} (y)))
= ψ_{x_{M}} (ψ_{x_{M}}^{−1} (y)) = y,

the second equality being valid because of the corresponding equation imposed by Q_{s}.

Given a model M : S_{t} → Set_{→}, x_{M} ∈ M(Q_{1}) and y_{M} ∈ M(Q_{2}), denote by M_{x_{M} ,y_{M}}^{∗} (Q_{3}) the quasigroup

⟨M(Q_{3}), M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩, ϕ_{y_{M}} M(r)⟨i_{M(Q_{3})}, ψ_{x_{M}}^{−1} ⟩, ψ_{x_{M}} M(l)⟨ϕ_{y_{M}}^{−1} , i_{M(Q_{3})}⟩⟩

associated with it by Proposition 5.2.

Now for the morphisms in Mod_{Set_{→}}(S_{t}) we have the following proposition.

5.3. Proposition. Let M, N : S_{t} → Set_{→} be two models in Mod_{Set_{→}}(S_{t}), x_{M} ∈ M(Q_{1}), y_{M} ∈ M(Q_{2}), x_{N} ∈ N(Q_{1}) and y_{N} ∈ N(Q_{2}). Finally, let η : M → N be a morphism in Mod_{Set_{→}}(S_{t}). Then

(ϕ_{y_{N}} η_{Q_{1}}ϕ_{y_{M}}^{−1} , ψ_{x_{N}} η_{Q_{2}}ψ_{x_{M}}^{−1} , η_{Q_{3}}) : M(Q_{3}) → N(Q_{3})

is a quasigroup homotopy from M_{x_{M} ,y_{M}}^{∗} (Q_{3}) into N_{x_{N} ,y_{N}}^{∗} (Q_{3}).

Proof. We need to show that

η_{Q_{3}}(M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩(x, y)) = N(m)⟨ϕ_{y_{N}}^{−1}, ψ_{x_{N}}^{−1}⟩(ϕ_{y_{N}} η_{Q_{1}}ϕ_{y_{M}}^{−1} (x), ψ_{x_{N}} η_{Q_{2}}ψ_{x_{M}}^{−1} (y)).

We have

N(m)⟨ϕ_{y_{N}}^{−1}, ψ_{x_{N}}^{−1}⟩(ϕ_{y_{N}} η_{Q_{1}}ϕ_{y_{M}}^{−1} (x), ψ_{x_{N}} η_{Q_{2}}ψ_{x_{M}}^{−1} (y)) = N(m)(η_{Q_{1}}ϕ_{y_{M}}^{−1} (x), η_{Q_{2}}ψ_{x_{M}}^{−1} (y))
= N(m)(⟨η_{Q_{1}}, η_{Q_{2}}⟩(ϕ_{y_{M}}^{−1} (x), ψ_{x_{M}}^{−1} (y)))
= η_{Q_{3}}(M(m)⟨ϕ_{y_{M}}^{−1} , ψ_{x_{M}}^{−1} ⟩(x, y)),

the last equality being valid because η : M → N is a natural transformation.

Now suppose that there is available a choice function c for the class |Set_{→}| − {∅} of all nonempty sets, i.e., for all X ∈ |Set_{→}|, X ≠ ∅, c(X) ∈ X. Then, it can be shown that the category Qtp of quasigroups with homotopies between them is naturally equivalent to Mod_{Set_{→}}(S_{t}).

The functor F : Qtp → Mod_{Set_{→}}(S_{t}) is defined by

F (Q) = N_{Q}, for all Q ∈ |Qtp|,

and, given (h_{1}, h_{2}, h_{3}) ∈ Qtp(Q, P), F ((h_{1}, h_{2}, h_{3})) is the model morphism in Mod_{Set_{→}}(S_{t}) defined in Proposition 4.2.

The functor G : Mod_{Set_{→}}(S_{t}) → Qtp is defined by

G(M) = M_{c(M(Q_{1})),c(M(Q_{2}))}^{∗}(Q_{3}), for all M : S_{t} → Set_{→},

and, given η ∈ Mod_{Set_{→}}(S_{t})(M, N),

G(η) = (ϕ_{c(N(Q_{2}))}η_{Q_{1}}ϕ_{c(M(Q_{2}))}^{−1}, ψ_{c(N(Q_{1}))}η_{Q_{2}}ψ_{c(M(Q_{1}))}^{−1}, η_{Q_{3}}).

It is a routine calculation to check that F and G are indeed functors. So to prove the natural equivalence it suffices to exhibit natural isomorphisms μ : I_{Qtp} → G ∘ F and ν : I_{Mod_{_{Set}_{→}}(S_{t})} → F ∘ G. Note that G(F (⟨Q, ·, /, \⟩)) is the quasigroup with universe Q and multiplication, right division and left division given, respectively, by

(x, y) ↦ (x/c(Q)) · (c(Q)\y)
(x, y) ↦ (x/(c(Q)\y)) · c(Q)
(x, y) ↦ c(Q) · ((x/c(Q))\y).

So, it is natural to define

μ_{Q1}(x) = x · c(Q), μ_{Q2}(x) = c(Q) · x, μ_{Q3}(x) = x,

i.e.,

μ_{Q1} = ϕ_{c(Q)}, μ_{Q2} = ψ_{c(Q)}, μ_{Q3} = i_{Q}.

μ_{Q} is an isotopy and, for all Q, P ∈ |Qtp|, and (h_{1}, h_{2}, h_{3}) ∈ Qtp(Q, P), commutativity of

[figure: P; G(F (P)); μ_{P}; Q; G(F (Q)); μ_{Q}; (h_{1}, h_{2}, h_{3}); G(F ((h_{1}, h_{2}, h_{3})))]

is easy to verify. For instance, for the first component, diagram chasing gives

(ϕ_{c(P)}h_{1}ϕ_{c(Q)}^{−1})ϕ_{c(Q)} = ϕ_{c(P)}h_{1}.

Next, given M : S_{t} → Set_{→}, the model F (G(M)) has

F (G(M))(Q_{1}) = F (G(M))(Q_{2}) = F (G(M))(Q_{3}) = M(Q_{3})

and, moreover,

F (G(M))(m) = M(m)⟨ϕ_{c(M(Q_{2}))}^{−1}, ψ_{c(M(Q_{1}))}^{−1}⟩
F (G(M))(r) = ϕ_{c(M(Q_{2}))}M(r)⟨i_{M(Q_{3})}, ψ_{c(M(Q_{1}))}^{−1}⟩
F (G(M))(l) = ψ_{c(M(Q_{1}))}M(l)⟨ϕ_{c(M(Q_{2}))}^{−1}, i_{M(Q_{3})}⟩.

So, now, we define

ν_{M}(Q_{1}) = ϕ_{c(M(Q_{2}))}, ν_{M}(Q_{2}) = ψ_{c(M(Q_{1}))}, ν_{M}(Q_{3}) = i_{M(Q_{3})}.

Clearly, ν_{M} is also an isotopy and, for all M, N : S_{t} → Set_{→} and η : M → N in Mod_{Set_{→}}(S_{t}), commutativity of

[figure: N; F (G(N)); ν_{N}; M; F (G(M)); ν_{M}; η; F (G(η))]

may be verified as follows:

(F (G(η)) ∘ ν_{M})_{Q_{1}} = F (G(η))_{Q_{1}}ν_{M}(Q_{1})
= ϕ_{c(N(Q_{2}))}η_{Q_{1}}ϕ_{c(M(Q_{2}))}^{−1}ϕ_{c(M(Q_{2}))}
= ϕ_{c(N(Q_{2}))}η_{Q_{1}}
= ν_{N}(Q_{1})η_{Q_{1}}
= (ν_{N} ∘ η)_{Q_{1}}

and, similarly, for Q_{2}, Q_{3}, Q_{12}, Q_{13} and Q_{32}.

We have, thus, shown the following 5.4. Theorem. The category Qtp of quasigroups with homotopies between them and the model category Mod_{Set_{→}}(S_{t}) of the product sketch S_{t} in Set_{→} are naturally isomorphic categories.

Acknowledgements

If I had not met Jonathan Smith at Iowa State I would not have known what a quasigroup is and if I had not met Charles Wells at Case Western Reserve I would not have known what a sketch is or anything about categorical model theory. Thanks to both for sharing with enthusiasm what they know best.

References

M. Barr and C. Wells (1990), Category Theory for Computing Science. Prentice-Hall International, New York.

F. Borceux (1994), Handbook of Categorical Algebra, I, II, III. Cambridge University Press.

L. Coppey and C. Lair (1984), Leçons de Théorie des Esquisses. Diagrammes, 12, i–x and 1–38.

L. Coppey and C. Lair (1988), Leçons de Théorie des Esquisses II. Diagrammes, 19, 1–68.

A.A. Gvaramiya and B.I. Plotkin (1992), The Homotopies of Quasigroups and Universal Algebras, in A. Romanowska and J.D.H. Smith (eds.), Universal Algebra and Quasigroup Theory. Heldermann, Berlin, 89–99.

J.D.H. Smith (1997), Homotopy and Semisymmetry of Quasigroups. Algebra Universalis, 38 175–184.

J.D.H. Smith and A.B. Romanowska (1999), Post-Modern Algebra. John Wiley & Sons.

School of Mathematics and Computer Science, Lake Superior State University, 650 W. Easterday Avenue, Sault Sainte Marie, MI 49783, U.S.A.

Email: gvoutsad@lssu.edu

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