Fiat categorification of the symmetric inverse semigroup \(\textit{IS}_n\) and the semigroup \(F^*_n\)
Abstract
Starting from the symmetric group \(S_n\), we construct two fiat 2categories. One of them can be viewed as the fiat “extension” of the natural 2category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2category can be viewed as the fiat “extension” of the 2category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.
Keywords
Categorification 2category Symmetric inverse semigroup Dual symmetric inverse semigroup1 Introduction and description of the results
Abstract higher representation theory has its origins in the papers [2, 4, 40, 41] with principal motivation coming from [15, 43]. For finitary 2categories, basics of 2representation theory were developed in [29, 30, 31, 32, 33, 34] and further investigated in [3, 11, 12, 16, 23, 36, 44, 45, 46, 47], see also [17] for applications. For different ideas on higher representation theory, see also [1, 6, 8, 14, 37] and references therein.
The major emphasis in [29, 30, 31, 32, 33, 34] is on the study of socalled fiat 2categories, which are 2categorical analogues of finite dimensional algebras with involution. Fiat 2categories appear naturally both in topology and representation theory. They have many nice properties and the series of papers mentioned above develops an essential starting part of 2representation theory for fiat categories.
Many examples of 2categories appear naturally in semigroup theory, see [9, 11, 12, 21]. The easiest example is the 2category associated to a monoid with a fixed admissible partial order, see Sect. 4.1 for details. Linear analogues of these 2categories show up naturally in representation theory, see [11, 12]. A classical example of an ordered monoid is an inverse monoid with respect to the natural partial order. There is a standard linearization procedure, which allows one to turn a 2category of a finite ordered monoid into a finitary 2category, see Sect. 3.2 for details.
One serious disadvantage with linearizations of 2categories associated to finite ordered monoids is the fact that they are almost never fiat. The main reason for that is lack of 2morphisms which start from the identity 1morphism. In the present paper we construct two natural “extensions” of the symmetric group to 2categories whose linearizations are fiat. One of them becomes a nice 2categorical analogue (categorification) for the symmetric inverse semigroup \(\textit{IS}_n\). The other one becomes a nice 2categorical analogue for the maximal factorizable subsemigroup \(F^*_n\) in the dual symmetric inverse semigroup \(I^*_n\).

sets of 2morphisms between elements of \(S_n\);

horizontal composition of 2morphisms;

vertical composition of 2morphisms.
Section 3 recalls the theory of \(\Bbbk \)linear 2categories and gives explicit constructions for a finitary \(\Bbbk \)linear 2category starting from a finite 2category. In Sect. 4 we establish that our constructions lead to fiat 2categories. We also recall, in more details, the standard constructions of finitary 2categories, starting from \(\textit{IS}_n\) and \(F^*_n\), considered as ordered monoids, and show that the 2categories obtained in this way are not fiat. In Sect. 5 we make the relation between our constructions and \(\textit{IS}_n\) and \(F^*_n\) precise. In fact, we show that the decategorification of our first construction is isomorphic to the semigroup algebra \({\mathbb {Z}}[\textit{IS}_n]\), with respect to the socalled Möbius basis in \({\mathbb {Z}}[\textit{IS}_n]\), cf. [42, Theorem 4.4]. Similarly, we show that the decategorification of our second construction is isomorphic to the semigroup algebra \({\mathbb {Z}}[F^*_n]\), with respect to a similarly defined basis. We complete the paper with two explicit examples in Sect. 6.
2 Two 2categorical “extensions” of \(S_n\)
2.1 2Categories

objects \({\texttt {i}},{\texttt {j}},\ldots \);

small morphism categories \(\mathscr {C}({\texttt {i}},{\texttt {j}})\);

bifunctorial compositions;

identity objects \({\mathbbm {1}}_{\texttt {i}}\in \mathscr {C}({\texttt {i}},{\texttt {i}})\);

objects are small categories;

1morphisms are functors;

2morphisms are natural transformations;

composition is the usual composition;

identity 1morphisms are identity functors.
2.2 First 2category extending \(S_n\)
For \(n\in {\mathbb {N}}:=\{1,2,3,\ldots \}\), consider the set \(\mathbf{n }=\{1,2,\ldots ,n\}\) and let \(S_n\) denote the symmetric group of all bijective transformations of \(\mathbf{n }\) under composition. We consider also the monoid \(\mathbf{B }_n=2^{\mathbf{n }\times \mathbf{n }}\) of all binary relations on \(\mathbf{n }\) which is identified with the monoid of \(n\times n\) matrices over the Boolean semiring \(\mathbf{B }:=\{0,1\}\) by taking a relation to its adjacency matrix. Note that \(\mathbf{B }_n\) is an ordered monoid with respect to usual inclusions of binary relations. We identify \(S_n\) with the group of invertible elements in \(\mathbf{B }_n\) in the obvious way.

\(\mathscr {A}\) has one object \({\texttt {i}}\);

1morphisms in \(\mathscr {A}\) are elements in \(S_n\);

composition \(\cdot \) of 1morphisms is induced from \(S_n\);

the identity 1morphism is the identity transformation \(\mathrm {id}_\mathbf{n }\in S_n\).

For \(\pi ,\sigma \in S_n\), we define \(\mathrm {Hom}_{\mathscr {A}}(\pi ,\sigma )\) as the set of all \(\alpha \in \mathbf{B }_n\) such that \(\alpha \subseteq \pi \cap \sigma \).

For \(\pi ,\sigma ,\tau \in S_n\), and also for \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\pi ,\sigma )\) and \(\beta \in \mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\), we define \(\beta \circ _1\alpha :=\beta \cap \alpha \).

For \(\pi \in S_n\), we define the identity element in \(\mathrm {Hom}_{\mathscr {A}}(\pi ,\pi )\) to be \(\pi \).

For \(\pi ,\sigma ,\tau ,\rho \in S_n\), and also for \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\pi ,\sigma )\) and \(\beta \in \mathrm {Hom}_{\mathscr {A}}(\tau ,\rho )\), we define \(\beta \circ _0\alpha :=\beta \alpha \), the usual composition of binary relations.
Proposition 1
The construct \(\mathscr {A}\) above is a 2category.
Proof
Composition \(\cdot \) of 1morphisms is associative as \(S_n\) is a group. The vertical composition \(\circ _1\) is clearly welldefined. It is associative as \(\cap \) is associative. If we have \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\pi ,\sigma )\) or \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\sigma ,\pi )\), then \(\alpha \subseteq \pi \) and thus \(\alpha \cap \pi =\alpha \). Therefore \(\pi \in \mathrm {Hom}_{\mathscr {A}}(\pi ,\pi )\) is the identity element.
Let us check that the horizontal composition \(\circ _0\) is welldefined. From \(\alpha \subseteq \pi \) and \(\beta \subseteq \tau \) and the fact that \(\mathbf{B }_n\) is ordered, we have \(\beta \alpha \subseteq \tau \alpha \subseteq \tau \pi \). Similarly, from \(\alpha \subseteq \sigma \) and \(\beta \subseteq \rho \) and the fact that \(\mathbf{B }_n\) is ordered, we have \(\beta \alpha \subseteq \rho \alpha \subseteq \rho \sigma \). It follows that \(\beta \alpha \in \mathrm {Hom}_{\mathscr {A}}(\tau \pi ,\rho \sigma )\) and thus \(\circ _0\) is welldefined. Its associativity follows from the fact that usual composition of binary relations is associative.
Before proving the general case, we will need the following two lemmata:
Lemma 2
 (i)
Left composition with \(\pi \) induces a bijection from \(\mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\) to \(\mathrm {Hom}_{\mathscr {A}}(\pi \sigma ,\pi \tau )\).
 (ii)For any \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\) and \(\beta \in \mathrm {Hom}_{\mathscr {A}}(\tau ,\rho )\), we have$$\begin{aligned} \pi \circ _0(\beta \circ _1\alpha )=(\pi \circ _0\beta )\circ _1(\pi \circ _0\alpha ). \end{aligned}$$
Proof
Left composition with \(\pi \) maps an element (y, x) of \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\) to \((\pi (y),x)\in \mathrm {Hom}_{\mathscr {A}}(\pi \sigma ,\pi \tau )\). As \(\pi \) is an invertible transformation of \(\mathbf{n }\), multiplying with \(\pi ^{1}\) returns \((\pi (y),x)\) to (y, x). This implies claim (i). Claim (ii) follows from claim (i) and the observation that composition with invertible maps commutes with taking intersections. \(\square \)
Lemma 3
 (i)
Right composition with \(\pi \) induces a bijection from \(\mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\) to \(\mathrm {Hom}_{\mathscr {A}}(\sigma \pi ,\tau \pi )\).
 (ii)For any \(\alpha \in \mathrm {Hom}_{\mathscr {A}}(\sigma ,\tau )\) and \(\beta \in \mathrm {Hom}_{\mathscr {A}}(\tau ,\rho )\), we have$$\begin{aligned} (\beta \circ _1\alpha )\circ _0\pi =(\beta \circ _0\pi )\circ _1(\alpha \circ _0\pi ). \end{aligned}$$
Proof
Analogous to the proof of Lemma 2. \(\square \)
Using Lemmata 2 and 3 together with associativity of \(\circ _0\), right multiplication with \(\sigma ^{1}\) and left multiplication with \(\mu ^{1}\) reduces the general case of (2.1) to the case \(\sigma =\mu ={\mathrm {id}}_\mathbf{n }\) considered above. This completes the proof. \(\square \)
2.3 Second 2category extending \(S_n\)

Consider \(\rho \) as a partition of \(\{1^{\prime },2^{\prime },\ldots ,n^{\prime },1^{\prime \prime },2^{\prime \prime },\ldots ,n^{\prime \prime }\}\) using the map \(x\mapsto x^{\prime }\) and \(x^{\prime }\mapsto x^{\prime \prime }\), for \(x\in \mathbf{n }\).
 Let \(\tau \) be the minimum, with respect to inclusions, partition ofsuch that each part of both \(\rho \) and \(\pi \) is a subset of a part of \(\tau \).$$\begin{aligned} \left\{ 1,2,\ldots ,n,1^{\prime },2^{\prime },\ldots ,n^{\prime },1^{\prime \prime },2^{\prime \prime },\ldots ,n^{\prime \prime }\right\} , \end{aligned}$$
 Let \(\sigma \) be the maximum, with respect to inclusions, partition ofsuch that each part of \(\sigma \) is a subset of a part of \(\tau \).$$\begin{aligned} \left\{ 1,2,\ldots ,n,1^{\prime \prime },2^{\prime \prime },\ldots ,n^{\prime \prime }\right\} , \end{aligned}$$

Define the product \(\rho \pi \) as the partition of \(\mathbf{n }\) induced from \(\sigma \) via the map \(x^{\prime \prime }\mapsto x^{\prime }\), for \(x\in \mathbf{n }\).
A part of \(\rho \in \mathbf{P }_n\) is called a propagating part provided that it intersects both sets \(\{1,2,\ldots ,n\}\) and \(\{1^{\prime },2^{\prime },\dots ,n^{\prime }\}\). Partitions in which all parts are propagating are called propagating partitions. The set of all propagating partitions in \(\mathbf{P }_n\) is denoted by \(\mathbf{PP }_n\), it is a submonoid of \(\mathbf{P }_n\).
The monoid \(\mathbf{P }_n\) is naturally ordered with respect to refinement: \(\rho \le \tau \) provided that each part of \(\rho \) is a subset of a part in \(\tau \). With respect to this order, the partition of \(\mathbf n \) with just one part is the maximum element, while the partition of \(\overline{\mathbf{n }}\) into singletons is the minimum element. This order restricts to \(\mathbf{PP }_n\). As elements of \(\mathbf{P }_n\) are just equivalence relations, the poset \(\mathbf{P }_n\) is a lattice and we denote by \(\wedge \) and \(\vee \) the corresponding meet and join operations, respectively. The poset \(\mathbf{PP }_n\) is a sublattice in \(\mathbf{P }_n\) with the same meet and join. As \(S_n\subset \mathbf{PP }_n\), all meets and joins in \(\mathbf{P }_n\) of elements from \(S_n\) belong to \(\mathbf{PP }_n\).

\(\mathscr {B}\) has one object \({\texttt {i}}\);

1morphisms in \(\mathscr {B}\) are elements in \(S_n\);

composition of 1morphisms is induced from \(S_n\);

the identity 1morphism is the identity transformation \(\mathrm {id}_\mathbf{n }\).

For \(\pi ,\sigma \in S_n\), we define \(\mathrm {Hom}_{\mathscr {B}}(\pi ,\sigma )\) as the set of all \(\alpha \in \mathbf{PP }_n\) such that we have both, \(\pi \le \alpha \) and \(\sigma \le \alpha \).

For \(\pi ,\sigma ,\tau \in S_n\), and also any \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\pi ,\sigma )\) and \(\beta \in \mathrm {Hom}_{\mathscr {B}}(\sigma ,\tau )\), we define \(\beta \circ _1\alpha :=\beta \vee \alpha \).

For \(\pi \in S_n\), we define the identity element in \(\mathrm {Hom}_{\mathscr {B}}(\pi ,\pi )\) to be \(\pi \).

For \(\pi ,\sigma ,\tau ,\rho \in S_n\), and also any \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\pi ,\sigma )\) and \(\beta \in \mathrm {Hom}_{\mathscr {B}}(\tau ,\rho )\), we define \(\beta \circ _0\alpha :=\beta \alpha \), the usual composition of partitions.
Proposition 4
The construct \(\mathscr {B}\) above is a 2category.
Proof
The vertical composition \(\circ _1\) is clearly welldefined. It is associative as \(\vee \) is associative. If \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\pi ,\sigma )\) or \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\sigma ,\pi )\), then \(\pi \le \alpha \) and thus \(\alpha \vee \pi =\alpha \). Therefore \(\pi \in \mathrm {Hom}_{\mathscr {B}}(\pi ,\pi )\) is the identity element.
Let us check that the horizontal composition \(\circ _0\) is welldefined. From \(\pi \le \alpha \) and \(\tau \le \beta \) and the fact that \(\mathbf{P }_n\) is ordered, we have \(\tau \pi \le \tau \alpha \le \beta \alpha \). Similarly, from \(\sigma \le \alpha \) and \(\rho \le \beta \) and the fact that \(\mathbf{P }_n\) is ordered, we have \(\rho \sigma \le \rho \alpha \le \beta \alpha \). It follows that \(\beta \alpha \in \mathrm {Hom}_{\mathscr {B}}(\tau \pi ,\rho \sigma )\) and thus \(\circ _0\) is welldefined. Its associativity follows from the fact that usual composition of partitions is associative.
It remains to check the interchange law (2.1). For this we fix any 1morphisms \(\pi ,\) \(\sigma ,\) \(\rho ,\) \(\tau ,\) \(\mu ,\) \(\nu \) and any \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\pi ,\sigma )\), \(\beta \in \mathrm {Hom}_{\mathscr {B}}(\tau ,\mu )\), \(\gamma \in \mathrm {Hom}_{\mathscr {B}}(\sigma ,\rho )\) and \(\delta \in \mathrm {Hom}_{\mathscr {B}}(\mu ,\nu )\). Assume first that \(\sigma =\mu =\mathrm {id}_\mathbf{n }\). In this case both \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are partitions containing the identity relation \(\mathrm {id}_\mathbf{n }\). Note that, given two partitions x and y containing the identity relation \(\mathrm {id}_\mathbf{n }\), their product xy as partitions equals \(x\vee y\). Hence, in this particular case, both sides of (2.1) are equal to \(\alpha \vee \beta \vee \gamma \vee \delta \).
Before proving the general case, we will need the following two lemmata:
Lemma 5
 (i)
Left composition with \(\pi \) induces a bijection between the sets \(\mathrm {Hom}_{\mathscr {B}}(\sigma ,\tau )\) and \(\mathrm {Hom}_{\mathscr {B}}(\pi \sigma ,\pi \tau )\).
 (ii)For any \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\sigma ,\tau )\) and \(\beta \in \mathrm {Hom}_{\mathscr {B}}(\tau ,\rho )\), we have$$\begin{aligned} \pi \circ _0(\beta \circ _1\alpha )=(\pi \circ _0\beta )\circ _1(\pi \circ _0\alpha ). \end{aligned}$$
Proof
Left composition with \(\pi \) simply renames elements of \(\{1^{\prime },2^{\prime },\dots ,n^{\prime }\}\) in an invertible way. This implies claim (i). Claim (ii) follows from claim (i) and the observation that composition with invertible maps commutes with taking unions. \(\square \)
Lemma 6
 (i)
Right composition with \(\pi \) induces a bijection between the sets \(\mathrm {Hom}_{\mathscr {B}}(\sigma ,\tau )\) and \(\mathrm {Hom}_{\mathscr {B}}(\sigma \pi ,\tau \pi )\).
 (ii)For any \(\alpha \in \mathrm {Hom}_{\mathscr {B}}(\sigma ,\tau )\) and \(\beta \in \mathrm {Hom}_{\mathscr {B}}(\tau ,\rho )\), we have$$\begin{aligned} (\beta \circ _1\alpha )\circ _0\pi =(\beta \circ _0\pi )\circ _1(\alpha \circ _0\pi ). \end{aligned}$$
Proof
Analogous to the proof of Lemma 5. \(\square \)
Using Lemmata 5 and 6 together with associativity of \(\circ _0\), right multiplication with \(\sigma ^{1}\) and left multiplication with \(\mu ^{1}\) reduces the general case of (2.1) to the case \(\sigma =\mu =\mathrm {id}_\mathbf{n }\) considered above. This completes the proof. \(\square \)
3 2Categories in the linear world
For more details on all the definitions in Sect. 3, we refer to [12].
3.1 Finitary 2categories
Let \(\Bbbk \) be a field. A \(\Bbbk \)linear category \(\mathcal {C}\) is called finitary provided that it is additive, idempotent split and Krull–Schmidt (cf. [38, Section 2.2]) with finitely many isomorphism classes of indecomposable objects and finite dimensional homomorphism spaces.
 (I)
\(\mathscr {C}\) has finitely many objects;
 (II)
each \(\mathscr {C}({\texttt {i}},{\texttt {j}})\) is a finitary \(\Bbbk \)linear category;
 (III)
all compositions are biadditive and \(\Bbbk \)linear whenever the notion makes sense.
 (IV)
all identity 1morphisms are indecomposable.
3.2 \(\Bbbk \)Linearization of finite categories
For any set X, let us denote by \(\Bbbk [X]\) the vector space (over \(\Bbbk \)) of all formal linear combinations of elements in X with coefficients in \(\Bbbk \). Then we can view X as the standard basis in \(\Bbbk [X]\). By convention, \(\Bbbk [X]=\{0\}\) if \(X=\varnothing \).

the objects in \(\mathcal {C}_{\Bbbk }\) and \(\mathcal {C}\) are the same;

we have \(\mathcal {C}_{\Bbbk }({\texttt {i}},{\texttt {j}}):=\Bbbk [\mathcal {C}({\texttt {i}},{\texttt {j}})]\);

composition in \(\mathcal {C}_{\Bbbk }\) is induced from that in \(\mathcal {C}\) by \(\Bbbk \)bilinearity.
3.3 \(\Bbbk \)additivization of finite categories
 objects in \(\mathcal {C}_{\Bbbk }^{\oplus }\) are elements in \(\mathbb {Z}_{\ge 0}^k\), we identify \((m_1,m_2,\ldots ,m_k)\in \mathbb {Z}_{\ge 0}^k\) with the symbol$$\begin{aligned} \underbrace{\texttt {1}\oplus \cdots \oplus \texttt {1}}_{m_1 \text { times}}\oplus \underbrace{\texttt {2}\oplus \cdots \oplus \texttt {2}}_{m_2 \text { times}}\oplus \dots \oplus \underbrace{\texttt {k}\oplus \cdots \oplus \texttt {k}}_{m_k \text { times}}; \end{aligned}$$
 the set \(\mathcal {C}_{\Bbbk }^{\oplus }(\texttt {i}_1\oplus \texttt {i}_2\oplus \cdots \oplus \texttt {i}_l,\texttt {j}_1\oplus \texttt {j}_2\oplus \cdots \oplus \texttt {j}_m)\) is given by the set of all matrices of the formwhere \(f_{st}\in \mathcal {C}_{\Bbbk }(\texttt {i}_t,\texttt {j}_s)\);$$\begin{aligned} \left( \begin{array}{cccc} f_{11}&{}\quad f_{12}&{}\quad \cdots &{}\quad f_{1l} \\ f_{21}&{}\quad f_{22}&{}\quad \cdots &{}\quad f_{2l} \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ f_{m1}&{}\quad f_{m2}&{}\quad \cdots &{}\quad f_{ml} \end{array}\right) \end{aligned}$$

composition in \(\mathcal {C}_{\Bbbk }^{\oplus }\) is given by the usual matrix multiplication;

the additive structure is given by addition in \(\mathbb {Z}_{\ge 0}^k\).
3.4 \(\Bbbk \)Linearization of finite 2categories

\(\mathscr {C}_{\Bbbk }\) and \(\mathscr {C}\) have the same objects;

we have \(\mathscr {C}_{\Bbbk }({\texttt {i}},{\texttt {j}}):=\mathscr {C}({\texttt {i}},{\texttt {j}})_{\Bbbk }^{\oplus }\);

composition in \(\mathscr {C}_{\Bbbk }\) is induced from composition in \(\mathscr {C}\) by biadditivity and \(\Bbbk \)bilinearity.
3.5 \(\Bbbk \)Finitarization of finite 2categories

\(\Bbbk \mathscr {C}\) and \(\mathscr {C}_{\Bbbk }\) have the same objects;

\(\Bbbk \mathscr {C}({\texttt {i}},{\texttt {j}})\) is defined to be the idempotent completion of \(\mathscr {C}_{\Bbbk }({\texttt {i}},{\texttt {j}})\);

composition in \(\Bbbk \mathscr {C}\) is induced from composition in \(\mathscr {C}\).
3.6 Idempotent splitting
Let \(\mathscr {C}\) be a prefinitary 2category. If \(\mathscr {C}\) does not satisfy condition (IV), then there is an object \({\texttt {i}}\in \mathscr {C}\) such that the endomorphism algebra \(\mathrm {End}_{\Bbbk \mathscr {C}}({\mathbbm {1}}_{\texttt {i}})\) is not local, that is, contains a nontrivial idempotent. In this subsection we describe a version of “idempotent splitting”, for all \(\mathrm {End}_{\Bbbk \mathscr {C}}({\mathbbm {1}}_{\texttt {i}})\), to turn \(\mathscr {C}\) into a finitary 2category which we denote by \(\overline{\mathscr {C}}\).
For \({\texttt {i}}\in \mathscr {C}\), the 2endomorphism algebra of \({\mathbbm {1}}_{\texttt {i}}\) is equipped with two unital associative operations, namely, \(\circ _0\) and \(\circ _1\). These two operations satisfy the interchange law. By the classical Eckmann–Hilton argument (see, for example, [5] or [19, Subsection 1.1]), both these operations, when restricted to the 2endomorphism algebra of \({\mathbbm {1}}_{\texttt {i}}\), must be commutative and, in fact, coincide. Therefore we can unambiguously speak about the commutative 2endomorphism algebra \(\mathrm {End}_{\mathscr {C}}({\mathbbm {1}}_{\texttt {i}})\). Let \(\varepsilon _{\texttt {i}}^{(j)}\), where \(j=1,2,\ldots ,k_{\texttt {i}}\), be a complete list of primitive idempotents in \(\mathrm {End}_{\mathscr {C}}({\mathbbm {1}}_{\texttt {i}})\). Note that the elements \(\varepsilon _{\texttt {i}}^{(j)}\) are identities in the minimal ideals of \(\mathrm {End}_{\mathscr {C}}({\mathbbm {1}}_{\texttt {i}})\) and hence are canonically determined (up to permutation).

Objects in \(\overline{\mathscr {C}}\) are \({\texttt {i}}^{(s)}\), where \({\texttt {i}}\in \mathscr {C}\) and \(s=1,2,\ldots ,k_{\texttt {i}}\).

1morphisms in \(\overline{\mathscr {C}}({\texttt {i}}^{(s)},{\texttt {j}}^{(t)})\) are the same as 1morphisms in \({\mathscr {C}}({\texttt {i}},{\texttt {j}})\).
 for 1morphisms \(\mathrm {F},\mathrm {G}\in \overline{\mathscr {C}}({\texttt {i}}^{(s)},{\texttt {j}}^{(t)})\), the set \(\mathrm {Hom}_{\overline{\mathscr {C}}}(\mathrm {F},\mathrm {G})\) equals$$\begin{aligned} \varepsilon _{\texttt {j}}^{(t)}\circ _0 \mathrm {Hom}_{{\mathscr {C}}}(\mathrm {F},\mathrm {G})\circ _0 \varepsilon _{\texttt {i}}^{(s)}. \end{aligned}$$

The identity 1morphism in \(\overline{\mathscr {C}}({\texttt {i}}^{(s)},{\texttt {i}}^{(s)})\) is \({\mathbbm {1}}_{\texttt {i}}\).

All compositions are induced from \(\mathscr {C}\).
Lemma 7
Let \(\mathscr {C}\) be a prefinitary 2category. Then the construct \(\overline{\mathscr {C}}\) is a finitary 2category.
Proof
The fact that \(\overline{\mathscr {C}}\) is a 2category follows from the fact that \(\mathscr {C}\) is a 2category, by construction. For \(\overline{\mathscr {C}}\), conditions (I), (II) and (III) from the definition of a prefinitary 2category, follow from the corresponding conditions for the original category \(\mathscr {C}\).
Starting from \(\overline{\mathscr {C}}\) and taking, for each \({\texttt {i}}\in \mathscr {C}\), a direct sum of \({\texttt {i}}^{(s)}\), where \(s=1,2,\ldots ,k_{\texttt {i}}\), one obtains a 2category biequivalent to the original 2category \(\mathscr {C}\). The 2categories \(\mathscr {C}\) and \(\overline{\mathscr {C}}\) are, clearly, Morita equivalent in the sense of [32].
Warning: Despite of the fact that \(\overline{\mathscr {C}}({\texttt {i}}^{(s)},{\texttt {j}}^{(t)})\) and \({\mathscr {C}}({\texttt {i}},{\texttt {j}})\) have the same 1morphisms, these two categories, in general, have different indecomposable 1morphisms as the sets of 2morphisms are different. In particular, indecomposable 1morphisms in \({\mathscr {C}}({\texttt {i}},{\texttt {j}})\) may become isomorphic to zero in \(\overline{\mathscr {C}}({\texttt {i}}^{(s)},{\texttt {j}}^{(t)})\).
We note that the operation of idempotent splitting is also known as taking Cauchy completion or Karoubi envelope.
4 Comparison of \(\overline{\Bbbk \mathscr {A}_n}\) and \(\overline{\Bbbk \mathscr {B}_n}\) to 2categories associated with ordered monoids \(\textit{IS}_n\) and \(F^*_n\)
4.1 2Categories and ordered monoids

\(\mathscr {S}\) has one object \({\texttt {i}}\);

1morphisms are elements in S;

for \(s,t\in S\), the set \(\mathrm {Hom}_{\mathscr {S}}(s,t)\) is empty if \(s\not \le t\) and contains one element (s, t) otherwise;

composition of 1morphisms is given by \(\cdot \);

both horizontal and vertical compositions of 2morphism are the only possible compositions (as sets of 2morphisms are either empty or singletons);

the identity 1morphism is 1.
A canonical example of the above is when S is an inverse monoid and \(\le \) is the natural partial order on S defined as follows: \(s\le t\) if and only if \(s=e t\) for some idempotent \(e\in S\).
4.2 (Co)ideals of ordered semigroups
Lemma 8
For any subsemigroup \(T\subset S\), both \(T^{\downarrow }\) and \(T^{\uparrow }\) are subsemigroups of S.
Proof
We prove the claim for \(T^{\downarrow }\), for \(T^{\uparrow }\) the arguments are similar. Let \(a,b\in T^{\downarrow }\). Then there exist \(s,t\in T\) such that \(a\le s\) and \(b\le t\). As \(\le \) is admissible, we have \(ab\le sb\le st\). Now, \(st\in T\) as T is a subsemigroup, and thus \(ab\in T^{\downarrow }\). \(\square \)
4.3 The symmetric inverse monoid
For \(n\in \mathbb {N}\), we denote by \(IS_n\) the symmetric inverse monoid on \(\mathbf n \), see [10]. It consists of all bijections between subsets of \(\mathbf n \). Alternatively, we can identify \(\textit{IS}_n\) with \(S_n^{\downarrow }\) inside the ordered monoid \(\mathbf B _n\). The monoid \(IS_n\) is an inverse monoid. The natural partial order on the inverse monoid \(\textit{IS}_n\) coincides with the inclusion order inherited from \(\mathbf B _n\). The group \(S_n\) is the group of invertible elements in \(\textit{IS}_n\).
4.4 The dual symmetric inverse monoid
For \(n\in \mathbb {N}\), we denote by \(I^*_n\) the dual symmetric inverse monoid on \(\mathbf n \), see [7]. It consists of all bijections between quotients of \(\mathbf n \). Alternatively, we can identity \(I^*_n\) with \(\mathbf{PP }_n\) in the obvious way. The monoid \(I^*_n\) is an inverse monoid. The natural partial order on the inverse monoid \(I^*_n\) coincides with the order inherited from \(\mathbf{P }_n\). The group \(S_n\) is the group of invertible elements in \(I^*_n\).
We also consider the maximal factorizable submonoid \(F^*_n\) of \(I^*_n\), that is the submonoid of all elements which can be written in the form \(\sigma \varepsilon \), where \(\sigma \in S_n\) and \(\varepsilon \) is an idempotent in \(I^*_n\). Idempotents in \(I^*_n\) are exactly the identity transformations of quotient sets of \(\mathbf n \), equivalently, idempotents in \(I^*_n\) coincide with the principal coideal in \(\mathbf{P }_n\) generated by the identity element.
Lemma 9
The monoid \(F^*_n\) coincides with the subsemigroup \(S_n^{\uparrow }\) of \(\mathbf{PP }_n\).
Proof
As \(S_n^{\uparrow }\) contains both \(S_n\) and all idempotents of \(I^*_n\), we have \(F^*_n\subset S_n^{\uparrow }\). On the other hand, let \(\rho \in S_n^{\uparrow }\). Then \(\sigma \le \rho \) for some \(\sigma \in S_n\). This means that \(\mathrm {id}_\mathbf{n }\le \rho \sigma ^{1}\). Hence \(\rho \sigma ^{1}\) is an idempotent and \(\rho =(\rho \sigma ^{1})\sigma \in F^*_n\). \(\square \)
4.5 Fiat 2categories
There are various classes of 2categories whose axiomatization covers some parts of the axiomatization of fiat 2categories, see, for example, compact categories, rigid categories, monoidal categories with duals and 2categories with adjoints.
4.6 Comparison of fiatness
Theorem 10
 (i)
Both 2categories, \(\overline{\Bbbk \mathscr {A}_n}\) and \(\overline{\Bbbk \mathscr {B}_n}\), are fiat.
 (ii)
Both 2categories, \(\Bbbk \mathscr {S}_{IS_n}\) and \(\Bbbk \mathscr {S}_{F_n^*}\), are finitary but not fiat.
Proof
The endomorphism algebra of any 1morphism in \(\Bbbk \mathscr {S}_{IS_n}\) is \(\Bbbk \), by definition. Therefore \(\Bbbk \mathscr {S}_{IS_n}\) is finitary by construction. The category \(\Bbbk \mathscr {S}_{IS_n}\) cannot be fiat as it contains noninvertible indecomposable 1morphisms but it does not contain any nonzero 2morphisms from the identity 1morphism to any noninvertible indecomposable 1morphism. Therefore adjunction 2morphisms for noninvertible indecomposable 1morphisms cannot exist. The same argument also applies to \(\Bbbk \mathscr {S}_{F_n^*}\), proving claim (ii).
By construction, the 2category \(\Bbbk \mathscr {A}_n\) satisfies conditions (I), (II) and (III) from the definition of a finitary 2category. Therefore the 2category \(\overline{\Bbbk \mathscr {A}_n}\) is a finitary 2category by Lemma 7. Let us now check existence of adjunction 2morphisms.
Recall that an adjoint to a direct sum of functors is a direct sum of adjoints to components. Therefore, as \(\overline{\Bbbk \mathscr {A}_n}\) is obtained from \((\mathscr {A}_n)_{\Bbbk }\) by splitting idempotents in 2endomorphism rings, it is enough to check that adjunction 2morphisms exist in \((\mathscr {A}_n)_{\Bbbk }\). Any 1morphism in \((\mathscr {A}_n)_{\Bbbk }\) is, by construction, a direct sum of \(\sigma \in S_n\). Therefore it is enough to check that adjunction 2morphisms exist in \(\mathscr {A}_n\). In the latter category, each 1morphism \(\sigma \in S_n\) is invertible and hence both left and right adjoint to \(\sigma ^{1}\). This implies existence of adjunction 2morphisms in \(\mathscr {A}_n\).
The above shows that the 2category \(\overline{\Bbbk \mathscr {A}_n}\) is fiat. Similarly one shows that the 2category \(\overline{\Bbbk \mathscr {B}_n}\) is fiat. This completes the proof. \(\square \)
5 Decategorification
5.1 Decategorification via Grothendieck group

\([\mathscr {C}]\) has the same objects as \(\mathscr {C}\).

For \({\texttt {i}},{\texttt {j}}\in \mathscr {C}\), the set \([\mathscr {C}]({\texttt {i}},{\texttt {j}})\) coincides with the split Grothendieck group \([\mathscr {C}({\texttt {i}},{\texttt {j}})]_{\oplus }\) of the additive category \(\mathscr {C}({\texttt {i}},{\texttt {j}})\).

The identity morphism in \([\mathscr {C}]({\texttt {i}},{\texttt {i}})\) is the class of \({\mathbbm {1}}_{{\texttt {i}}}\).

Composition in \([\mathscr {C}]\) is induced from composition of 1morphisms in \(\mathscr {C}\).
5.2 Decategorifications of \(\overline{\Bbbk \mathscr {A}_n}\) and \(\Bbbk \mathscr {S}_{\textit{IS}_n}\)
Theorem 11
We have \(A_{\overline{\Bbbk \mathscr {A}_n}}\cong A_{\Bbbk \mathscr {S}_{IS_n}}\cong \mathbb {Z}[IS_n]\).
Proof
Indecomposable 1morphisms in \(\Bbbk \mathscr {S}_{\textit{IS}_n}\) correspond exactly to elements of \(\textit{IS}_n\), by construction. This implies that \(A_{\Bbbk \mathscr {S}_{\textit{IS}_n}}\cong \mathbb {Z}[\textit{IS}_n]\) where an indecomposable 1morphism \(\sigma \) on the left hand side is mapped to itself on the right hand side. So, we only need to prove that \(A_{\overline{\Bbbk \mathscr {A}_n}}\cong \mathbb {Z}[\textit{IS}_n]\).
Assume that \(\sigma (X)\ne Y\). Then the inclusionexclusion formula implies that any subrelation of \(\sigma \) appears in the linear combination above with coefficient zero. This means that the 1morphism \(\sigma \in \overline{\Bbbk \mathscr {A}_n}({\texttt {i}}^{(Y)},{\texttt {i}}^{(X)})\) is zero if and only if \(\sigma (X)\ne Y\).
Consequently, isomorphism classes of indecomposable 1morphisms in the category \(\overline{\Bbbk \mathscr {A}_n}({\texttt {i}}^{(Y)},{\texttt {i}}^{(X)})\) correspond precisely to elements in \(\textit{IS}_n\) with domain X and image Y. Composition of these indecomposable 1morphisms is inherited from \(S_n\). By comparing formulae (5.1) and (5.2), we see that composition of 1morphisms in \(\overline{\Bbbk \mathscr {A}_n}\) corresponds to multiplication of the Möbius basis elements in \(\mathbb {Z}[IS_n]\). This completes the proof of the theorem. \(\square \)
Theorem 11 allows us to consider \(\overline{\Bbbk \mathscr {A}_n}\) and \(\mathscr {S}_{\textit{IS}_n}\) as two different categorifications of \(IS_n\). The advantage of \(\overline{\Bbbk \mathscr {A}_n}\) is that this 2category is fiat.
The construction we use in our proof of Theorem 11 resembles the partialization construction from [20].
5.3 Decategorifications of \(\overline{\Bbbk \mathscr {B}_n}\) and \(\Bbbk \mathscr {S}_{F_n^*}\)
Theorem 12
We have \(A_{\overline{\Bbbk \mathscr {B}_n}}\cong A_{\Bbbk \mathscr {S}_{F_n^*}}\cong \mathbb {Z}[F_n^*]\).
Proof
Using the Möbius function for the poset of all quotients of \(\mathbf n \) with respect to \(\le \) (see, for example, [39, Example 1]), Theorem 12 is proved mutatis mutandis Theorem 11. \(\square \)
Theorem 12 allows us to consider \(\overline{\Bbbk \mathscr {B}_n}\) and \(\mathscr {S}_{F_n^*}\) as two different categorifications of \(F_n^*\). The advantage of \(\overline{\Bbbk \mathscr {B}_n}\) is that this 2category is fiat.
The immediately following examples are in low rank, but show that these constructions can be worked with at the concrete as well as the abstract level. In particular, they illustrate the difference between the two constructions.
6 Examples for \(n=2\)
6.1 Example of \(F^*_2\)
\(y{\setminus }x\)  \(\epsilon \)  \(\sigma \)  \(\tau \) 

\(\epsilon \)  \(\epsilon ,\tau \)  \(\tau \)  \(\tau \) 
\(\sigma \)  \(\tau \)  \(\sigma ,\tau \)  \( \tau \) 
\(\tau \)  \(\tau \)  \(\tau \)  \(\tau \) 
The 2endomorphism algebra of both \(\epsilon \) and \(\sigma \) in \((\mathscr {B}_2)_{\Bbbk }\) is isomorphic to \(\Bbbk \oplus \Bbbk \) where the primitive idempotents are \(\tau \) and \(\epsilon \tau \), in the case of \(\epsilon \), and \(\tau \) and \(\sigma \tau \), in the case of \(\sigma \).
The 2category \(\Bbbk \mathscr {B}_2\) has three isomorphism classes of indecomposable 1morphisms, namely \(\tau \), \(\epsilon \tau \) and \(\sigma \tau \).
\(y{\setminus } x \)  \( \texttt {i}_{\epsilon \tau } \)  \(\texttt {i}_{\tau }\) 

\(\texttt {i}_{\epsilon \tau }\)  \( \epsilon \tau ,\sigma \tau \)  \( \varnothing \) 
\(\texttt {i}_{\tau }\)  \(\varnothing \)  \( \tau \) 
6.2 Example of \(\textit{IS}_2\)
\(y{\setminus }x \)  \( \epsilon \)  \(\sigma \)  \( \tau \)  \( \alpha \)  \(\beta \)  \(\gamma \)  \( \delta \) 

\(\epsilon \)  \( \epsilon ,\alpha ,\delta ,\tau \)  \( \tau \)  \(\tau \)  \(\alpha ,\tau \)  \(\tau \)  \(\tau \)  \(\delta ,\tau \) 
\(\sigma \)  \(\tau \)  \( \sigma ,\beta ,\gamma ,\tau \)  \(\tau \)  \(\tau \)  \(\beta ,\tau \)  \(\gamma ,\tau \)  \(\tau \) 
\(\tau \)  \(\tau \)  \( \tau \)  \(\tau \)  \(\tau \)  \(\tau \)  \(\tau \)  \(\tau \) 
\(\alpha \)  \(\alpha ,\tau \)  \( \tau \)  \(\tau \)  \(\alpha ,\tau \)  \(\tau \)  \(\tau \)  \(\tau \) 
\(\beta \)  \(\tau \)  \( \beta ,\tau \)  \(\tau \)  \(\tau \)  \(\beta ,\tau \)  \(\tau \)  \(\tau \) 
\(\gamma \)  \(\tau \)  \( \beta ,\tau \)  \(\tau \)  \(\tau \)  \(\tau \)  \(\gamma ,\tau \)  \(\tau \) 
\(\delta \)  \(\alpha ,\tau \)  \( \tau \)  \(\tau \)  \(\tau \)  \(\tau \)  \(\tau \)  \(\delta ,\tau \) 
The 2endomorphism algebra of \(\epsilon \) in \((\mathscr {A}_2)_{\Bbbk }\) is isomorphic to \(\Bbbk \oplus \Bbbk \oplus \Bbbk \oplus \Bbbk \) where the primitive idempotents are \(\tau \), \(\alpha \tau \), \(\delta \tau \) and \(\epsilon \alpha \delta +\tau \). Similarly one can describe the 2endomorphism algebra of \(\sigma \) in \((\mathscr {A}_2)_{\Bbbk }\). The 2endomorphism algebra of \(\alpha \) in \((\mathscr {A}_2)_{\Bbbk }\) is isomorphic to \(\Bbbk \oplus \Bbbk \) where the primitive idempotents are \(\tau \) and \(\alpha \tau \). Similarly one can describe the 2endomorphism algebras of \(\beta \), \(\gamma \) and \(\delta \).
\(y{\setminus } x \)  \( \texttt {i}_{\epsilon \alpha \delta +\tau } \)  \(\texttt {i}_{\alpha \tau }\)  \(\texttt {i}_{\delta \tau }\)  \(\texttt {i}_{\tau }\) 

\(\texttt {i}_{\epsilon \alpha \delta +\tau }\)  \(\epsilon \alpha \delta +\tau ,\sigma \beta \gamma +\tau \)  \( \varnothing \)  \( \varnothing \)  \( \varnothing \) 
\(\texttt {i}_{\alpha \tau }\)  \(\varnothing \)  \( \alpha \tau \)  \( \gamma \tau \)  \( \varnothing \) 
\(\texttt {i}_{\delta \tau }\)  \(\varnothing \)  \( \beta \tau \)  \( \delta \tau \)  \( \varnothing \) 
\(\texttt {i}_{\tau }\)  \(\varnothing \)  \( \varnothing \)  \( \varnothing \)  \( \tau \) 
This table can be compared with [35, Figure 1].
Notes
Acknowledgements
The main part of this research was done during the visit of the VM to University of Leeds in October 2014. Financial support of EPSRC and hospitality of University of Leeds are gratefully acknowledged. PM is partially supported by EPSRC under Grant EP/I038683/1. VM is partially supported by the Swedish Research Council and Göran Gustafsson Foundation. We thank Stuart Margolis for stimulating discussions.
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