## Pseudo algebras and pseudo double categories

Venue: | J. Homotopy Relat. Struct |

Citations: | 17 - 2 self |

### BibTeX

@ARTICLE{Fiore_pseudoalgebras,

author = {Thomas M. Fiore},

title = {Pseudo algebras and pseudo double categories},

journal = {J. Homotopy Relat. Struct},

year = {},

pages = {2369164}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.

### Citations

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Citation Context ...egory consists of two sets with composition defined in terms of pullback. For algebraic structures on several sets, one can use limit theories, sketches, and multi-sorted theories as in [1], [11], or =-=[12]-=-, or schemes of operators as in [47]. In this paper we consider categories as algebras over a 2-theory. This adds a new ingredient to categorification that we do not see in the one-object case of Lawv... |

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Citation Context ...sms of D, and we call the objects and morphisms of D1 the horizontal morphisms and squares of D.�� �� � �� �� � �� PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES 11 We can expand this definition as in =-=[51]-=-. A double category D consists of a set of objects, a set of horizontal morphisms, a set of vertical morphisms, and a set of squares equipped with various sources, targets, and associative and unital ... |

124 | The algebra of cubes
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Citation Context ...symmetric double categories, a connection pair (with trivial holonomy) is the same as a thin structure as shown in [20], and in higher dimensions in [48]. Edge-symmetric foldings were used already in =-=[15]-=- to prove that the category of crossed complexes is equivalent to the category of cubical ω-groupoids, and were generalized to all dimensions in [2]. More recently, foldings found important applicatio... |

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Citation Context ...iod, a full theory of double categories is beginning to emerge. Classics in the subject include [8], [21], [22], [32]-[37], and [57]. For recent work on double categories and related topics, see [3], =-=[13]-=--[20], [25]-[31], [44]-[46], [56], and [62]-[65]. We recall double categories and foldings, as well as their morphisms and transformations. Foldings allow us to compare double categories with I-catego... |

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Citation Context ...n period, a full theory of double categories is beginning to emerge. Classics in the subject include [8], [21], [22], [32]-[37], and [57]. For recent work on double categories and related topics, see =-=[3]-=-, [13]-[20], [25]-[31], [44]-[46], [56], and [62]-[65]. We recall double categories and foldings, as well as their morphisms and transformations. Foldings allow us to compare double categories with I-... |

45 | Tensor products and homotopies for ω-groupoids and crossed complexes
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Citation Context ...the 2-category of 2-groups, functors, and vertical natural transformations. Since we are interested in double groups, we work with the latter 2-category of 2-groups instead of categorical groups. See =-=[16]-=- for the analogue of Theorem 5.11 in arbitrary dimensions. Theorem 5.11 (Brown-Spencer in [22]). The 2-category 2-Gp of 2groups, functors, and vertical natural transformations is 2-equivalent to the 2... |

44 |
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Citation Context ...way that one-object categories are monoids. A familiar example of a bicategory consists of rings, bimodules, and bimodule morphisms. Bicategories were introduced in the 1960’s in [9], [10], [35], and =-=[37]-=-. Since then, they (and their variants) have appeared in diverse areas, such as homotopy theory and high energy physics. However, the question arises: what exactly does one mean by “coherence isos sat... |

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Citation Context ...n by simply relabelling the outbound boundary components of x via k and relabelling the inbound boundary components of y via j: the holomorphic diffeomorphism stays the same. This example, along with =-=[52]-=- and [66], suggests that double categories play a role in the mathematics relating to field theories and high energy physics.�� � �� � �� � � �� �� �� PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES 51 ... |

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Citation Context ...nt years have seen widespread applications of categorification. The term categorification refers to a process of turning algebraic notions on sets into algebraic notions on categories as explained in =-=[6]-=-. Generally speaking, one takes a set-based algebraic notion, then replaces sets by categories, functions by functors, and equations by natural isomorphisms which satisfy certain coherence diagrams. F... |

38 | Algebraic models of 3-types and automorphism structures for crossed modules - Brown, Gilbert - 1989 |

37 | Conformal field theory and elliptic cohomology
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Citation Context ...bras over the 2-theory of categories have an object category I instead of an object set, as we shall see. This version of categorification in terms of pseudo algebras over 2theories was introduced in =-=[49]-=-, and further developed in [38] and [50], to give a completely rigorous approach to conformal field theory with n-dimensional modular functor. Pseudo algebras over the 2-theory of� 4 THOMAS M. FIORE ... |

34 |
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Citation Context ...In analogy to 2-groups, we shall call a one-object double groupoid a double group 5 . Brown and Spencer proved that edge-symmetric double groups with connection 6 are equivalent to crossed modules in =-=[21]-=-. We extend this to a 2-equivalence between general double groups with folding and crossed modules under groups in Theorem 5.15. Brown and Higgins showed in [15] that so-called crossed modules over gr... |

31 |
The combinatorics of n-categorical pasting
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Citation Context ...f2], �� �� which implies that [ i h j1 i h j2 ] = i h [ j 1 j2 ] since the units of X are strict. The naturality of the square αf3,f2,f1 and the interchange law follow from bicategorical pasting, see =-=[37]-=-, [55], [56], and [60] for more on pasting. Hence we conclude J (X) is a pseudo double category.� � 44 THOMAS M. FIORE The 2-functor J is defined similarly on morphisms. If G : X ��Y is a morphism in... |

29 | Double categories, 2-categories, thin structures and connections’, Theory and Applications of Categories 5
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(Show Context)
Citation Context ...ified strict case by reviewing strict categories and double categories in Section 2 and Section 3, and prove the strict versions of our desired result in Theorems 4.6, 4.8, and 4.9. Foldings, used in =-=[20]-=-, are introduced to facilitate the 2-equivalence of strict 2-algebras over the 2-theory of categories with underlying category I (I-categories for short) and certain double categories. It turns out th... |

27 | Determination of a double Lie groupoid by its core diagram
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Citation Context ... groups with connection pair and crossed modules. In Theorem 5.15 we prove that double groups (not necessarily edge symmetric) with folding are 2-equivalent to crossed modules under groups. The paper =-=[19]-=- contains a substantial generalization of [21] by giving an equivalence of ‘core diagrams’ to double groupoids with certain filling conditions. Double groupoids have recently found application in the ... |

26 | Closed and open conformal field theories and their anomalies
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Citation Context ... pseudo algebra over the appropriate theory. This version of categorification arose in the completely rigorous approach to conformal field theory with n-dimensional modular functor of [27], [35], and =-=[36]-=-. Pseudo algebras over the 2-theory of commutative monoids with cancellation were introduced in [35] to rigorize the symmetric approach to conformal field theory outlined in [57]. The notion of 2-theo... |

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Citation Context ...ies in the same way that one-object categories are monoids. A familiar example of a bicategory consists of rings, bimodules, and bimodule morphisms. Bicategories were introduced in the 1960’s in [9], =-=[10]-=-, [35], and [37]. Since then, they (and their variants) have appeared in diverse areas, such as homotopy theory and high energy physics. However, the question arises: what exactly does one mean by “co... |

23 |
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Citation Context ...nce a category consists of two sets with composition defined in terms of pullback. For algebraic structures on several sets, one can use limit theories, sketches, and multi-sorted theories as in [1], =-=[11]-=-, or [12], or schemes of operators as in [47]. In this paper we consider categories as algebras over a 2-theory. This adds a new ingredient to categorification that we do not see in the one-object cas... |

23 |
Higher dimensional categories: an illustrated guide book, 2004. Available via http://www.math.uchicago.edu/∼eugenia/guidebook
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Citation Context ... � � � PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES 3 given concept, such as category. Indeed, there already are a dozen or so different definitions of weak n-category, many of which are described in =-=[23]-=- and [55]. Lawvere theories and 2-theories provide one answer to this question. Lawvere theories, first introduced in [53], abstractly encode algebraic structure. For most familiar algebraic structure... |

20 |
Higher-dimensional algebra
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(Show Context)
Citation Context ...ted homotopy 2-types in [61]. The survey [69] contains an account of the use of crossed modules and their higher-dimensional analogues to model homotopy types. Recently, 2-groups have been studied in =-=[7]-=-. Our comparison of one-object I-categories with everything invertible will build on this result of Brown and Spencer. In fact, Brown and Spencer obtained a 2-equivalence, and we will in Theorem 5.13 ... |

20 | Pseudo limits, biadjoints, and pseudo algebras: Categorical foundations of conformal field theory
- Fiore
(Show Context)
Citation Context ...ories have an object category I instead of an object set, as we shall see. This version of categorification in terms of pseudo algebras over 2theories was introduced in [49], and further developed in =-=[38]-=- and [50], to give a completely rigorous approach to conformal field theory with n-dimensional modular functor. Pseudo algebras over the 2-theory of� 4 THOMAS M. FIORE commutative monoids with cancel... |

20 |
Algebras with a scheme of operators
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(Show Context)
Citation Context ...osition defined in terms of pullback. For algebraic structures on several sets, one can use limit theories, sketches, and multi-sorted theories as in [1], [11], or [12], or schemes of operators as in =-=[47]-=-. In this paper we consider categories as algebras over a 2-theory. This adds a new ingredient to categorification that we do not see in the one-object case of Lawvere theories. For example, pseudo al... |

18 |
General associativity and general composition for double categories
- Dawson, Paré
- 1993
(Show Context)
Citation Context ...� �� ] [ β δ Remark 3.2. A few comments about composition in a double category are in order. A compatible arrangement is intuitively a pasting diagram of squares in a double category. It was shown in =-=[30]-=- that if a compatible arrangement has a composite, then this composite does not depend on the order of composition, although there may be compatible arrangements in a given double category that do not... |

17 |
Adjoint for double categories
- Grandis
(Show Context)
Citation Context ...ore diagrams’ to double groupoids with certain filling conditions. Double groupoids have recently found application in the theory of weak Hopf algebras in [4] and [5]. The pseudo double categories of =-=[45]-=- are reviewed in Section 6. We finally prove in Theorem 7.10 and Theorem 7.11, under the assumption of strict units, the 2-equivalence of pseudo algebras over the 2-theory of categories (pseudo I-cate... |

14 | Thin elements and commutative shells in cubical ω-categories
- Higgins
(Show Context)
Citation Context ...as 3.24-3.27 and Theorem 3.28. In the case of edge-symmetric double categories, a connection pair (with trivial holonomy) is the same as a thin structure as shown in [20], and in higher dimensions in =-=[48]-=-. Edge-symmetric foldings were used already in [15] to prove that the category of crossed complexes is equivalent to the category of cubical ω-groupoids, and were generalized to all dimensions in [2].... |

9 | Group objects and internal categories
- Forrester-Barker
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(Show Context)
Citation Context ...ules are known to model pointed path-connected weak homotopy 2-types algebraically. A proof is sketched in [13]. In preparation for our theorem, we summarize Brown and Spencer’s proof as recounted in =-=[41]-=-. Brown and Spencer originally showed that categorical groups are 2-equivalent to crossed modules, crossed module morphisms, and homotopies. The 2-category of categorical groups is 2equivalent to the ... |

7 | Double categories and quantum groupoids
- Andruskiewitsch, Natale
(Show Context)
Citation Context ...zation of [21] by giving an equivalence of ‘core diagrams’ to double groupoids with certain filling conditions. Double groupoids have recently found application in the theory of weak Hopf algebras in =-=[4]-=- and [5]. The pseudo double categories of [45] are reviewed in Section 6. We finally prove in Theorem 7.10 and Theorem 7.11, under the assumption of strict units, the 2-equivalence of pseudo algebras ... |

7 | Paths in double categories - Paré |

7 | What is a free double category like
- Dawson, Paré
(Show Context)
Citation Context ...ry of double categories is beginning to emerge. Classics in the subject include [8], [21], [22], [32]-[37], and [57]. For recent work on double categories and related topics, see [3], [13]-[20], [25]-=-=[31]-=-, [44]-[46], [56], and [62]-[65]. We recall double categories and foldings, as well as their morphisms and transformations. Foldings allow us to compare double categories with I-categories in the next... |

7 | Polycategories via pseudo-distributive laws - Garner |

7 |
Adjoint for double categories. Addenda to: “Limits in double categories
- Grandis, Pare
- 1999
(Show Context)
Citation Context ...sition of 2-cells. Remark 7.8. The requirement that units be strict in a pseudo double category is not as rigid as it first seems, since this can be arranged in most examples. The authors of [45] and =-=[46]-=- also assume that units are strict, and arrange it in most of their examples. Theorem 7.9. Let I be a category. The 2-category Y of pseudo double categories D with strict units equipped with a folding... |

6 | and ˙ Ilhan ˙ Içen. Towards a 2-dimensional notion of holonomy - Brown |

6 |
crossed modules and the fundamental groupoid of a topological group
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- 1976
(Show Context)
Citation Context ...h Folding Ehresmann introduced double categories in [35] and [37]. After a long gestation period, a full theory of double categories is beginning to emerge. Classics in the subject include [8], [21], =-=[22]-=-, [32]-[37], and [57]. For recent work on double categories and related topics, see [3], [13]-[20], [25]-[31], [44]-[46], [56], and [62]-[65]. We recall double categories and foldings, as well as thei... |

5 | Tensor categories attached to double groupoids
- Andruskiewitsch, Natale
- 2006
(Show Context)
Citation Context ...f [21] by giving an equivalence of ‘core diagrams’ to double groupoids with certain filling conditions. Double groupoids have recently found application in the theory of weak Hopf algebras in [4] and =-=[5]-=-. The pseudo double categories of [45] are reviewed in Section 6. We finally prove in Theorem 7.10 and Theorem 7.11, under the assumption of strict units, the 2-equivalence of pseudo algebras over the... |

5 |
Adjoining adjoints
- Dawson, Paré, et al.
(Show Context)
Citation Context ... theory of double categories is beginning to emerge. Classics in the subject include [8], [21], [22], [32]-[37], and [57]. For recent work on double categories and related topics, see [3], [13]-[20], =-=[25]-=--[31], [44]-[46], [56], and [62]-[65]. We recall double categories and foldings, as well as their morphisms and transformations. Foldings allow us to compare double categories with I-categories in the... |

5 |
On the Cobordism and Commutative Monoid with Cancellation Approaches to Conformal Field Theory
- Fiore
(Show Context)
Citation Context ...9) commutes on objects strictly, but has a coherence iso 2-cell FP(j) ∼ = P ′ (j) for each morphism j of I. Proof: Omitted. The proof relies on a construction like L(P) in the proof of Theorem 6.5 in =-=[39]-=- to remedy [[ℓ f] k] ̸= [ℓ [f k]] F([ℓ f k]) ̸= [F(ℓ) F(f) F(k)] (P(k) ◦ f) ◦ P(ℓ) ̸= P(k) ◦ (f ◦ P(ℓ)) F(P(k) ◦ f ◦ P(ℓ)) ̸= F(P(k)) ◦ F(f) ◦ F(P(ℓ)). □ This completes our comparison of strict 2-alge... |

4 |
Içen ˙ I: Homotopies and Automorphisms of Crossed Modules
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(Show Context)
Citation Context ...e on crossed modules as internal categories and their 2-cells, see [24]. Homotopies and derivations for more general crossed modules as needed for a 2-dimensional notion of holonomy are considered in =-=[17]-=-. Example 5.9. An example of a crossed module is the inclusion of a normal subgroup H into a group G where the action is conjugation by elements of G. In particular, {e} �I is a crossed module for any... |

4 |
Undecidability of the free adjoint construction
- Dawson, Pronk
(Show Context)
Citation Context ...viewed in the form below in [38]. After a long gestation period, a full theory of double categories is beginning to emerge. For recent work on double categories and related topics, see [2], [9]-[14], =-=[16]-=--[22], [29]- [33], [43], [48], and [49]. Definition 3.1. A double category D consists of a class of objects, a set of horizontal morphisms, a set of vertical morphisms, and a set of squares. Objects a... |

4 | Free extensions of double categories - Dawson, Pare, et al. |

4 |
Catégories structurées. III. Quintettes et applications covariantes. In Topol. et Géom
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(Show Context)
Citation Context ... equivalent to the data for a folding. This proof is essentially a slight generalization of an argument in Section 5 of [20]. The idea goes back to Spencer in [73] and to the quintets of Ehresmann in =-=[36]-=-. If a double category admits a folding, then that folding is unique up to isomorphism. Lemma 3.24. If (Γ, Γ ′) is a connection pair on a double category, then Λ f,k j,g (α) := [ Γ ′(j) α Γ(k) ] j �� ... |

4 |
Double clubs. Cahiers Topologie Géom
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(Show Context)
Citation Context ...double categories is beginning to emerge. Classics in the subject include [8], [21], [22], [32]-[37], and [57]. For recent work on double categories and related topics, see [3], [13]-[20], [25]-[31], =-=[44]-=--[46], [56], and [62]-[65]. We recall double categories and foldings, as well as their morphisms and transformations. Foldings allow us to compare double categories with I-categories in the next secti... |

3 | More general spans - Dawson, Pronk |

2 |
Al-Agl. Aspects of Multiple Categories
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- 1989
(Show Context)
Citation Context ...[48]. Edge-symmetric foldings were used already in [15] to prove that the category of crossed complexes is equivalent to the category of cubical ω-groupoids, and were generalized to all dimensions in =-=[2]-=-. More recently, foldings found important applications in [3] and [48]. To define foldings, we recall Brown and Spencer’s notion of holonomy in [21]: Definition 3.13. A holonomy for a double category ... |

2 |
homotopy equivalence and internal category equivalence of crossed modules in categories of groups with operations
- Whitehead
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(Show Context)
Citation Context ...p ′ 2 ,q′ 2 ) is f � �� ν2(q ′ 1 (f)) · p2(ν1(f)). Crossed modules, morphisms, and homotopies form a 2-category denoted XMod. For more on crossed modules as internal categories and their 2-cells, see =-=[24]-=-. Homotopies and derivations for more general crossed modules as needed for a 2-dimensional notion of holonomy are considered in [17]. Example 5.9. An example of a crossed module is the inclusion of a... |

2 |
Pseudo algebras with Laplaza sets
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(Show Context)
Citation Context ...tc. A pseudo monoid, pseudo semi-ring, or pseudo ring is simply a pseudo algebra over the appropriate theory. There is a systematic way to leave out some coherence diagrams to encompass more examples =-=[40]-=-. However, Lawvere theories only axiomatize algebraic structures on a single set. There is no Lawvere theory of categories, since a category consists of two sets with composition defined in terms of p... |