Descent for internal multicategory functors

We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal multicategories as an equalizer of categories of diagrams. The second approach extends the techniques developed by Ivan Le Creurer in his study of descent for internal essentially algebraic structures.


Introduction
Let B be a category and p : x → y a morphism in B such that pullbacks along p exist.We say that p is an effective descent (descent) morphism whenever the change-of-base functor p * : B ↓ y → B ↓ x is monadic (premonadic).The main subject of this note, the study of effective descent morphisms, is at the core of Grothendieck Descent Theory (see e.g.[12,14]) and its applications (see, for instance, [2]).
Except for the case of locally cartesian closed categories, the full characterization of effective descent morphisms is far from trivial in general.The topological descent case is the main example of such a challenging problem (see the characterization in [18] and the reformulation in [5]).
The notion of (T, V)-categories, introduced in [10], generalizes both enriched categories and various notions of spaces.By studying effective descent morphisms in categories of (T, V)-categories, Clementino and Hofmann were able to give further descent results and understanding in various contexts, including, for instance, the reinterpretation of the topological results mentioned above and many other interesting connections (see, for instance, [6,9,7,8]).
On one hand, since they were mainly concerned with topological results, their study focused on the case where V is a quantale, and there is no obvious way to generalize their approach to more general monoidal categories V. On the other hand, their work, together with the characterization of effective descent morphisms for the category of internal categories (see [14,Section 6] and [15]), have raised interest in further studying effective descent morphisms in categories of generalized categorical structures.
With this in mind, [17,Lemma 9.10] showed that we can embed the category of Venriched categories (with V lextensive) in the category of internal categories in V. From this embedding, [17,Theorem 1.6] provides sufficient conditions for effective descent morphisms in V-categories.However, the literature still lacks results for (T, V)-categories for a non-trivial T and an extensive V.
The present note is part of a project which aims to study descent and Janelidze-Galois theory within the realm of generalized multicategories and other categorical structures.The first aim of this project consists of studying effective descent morphisms in categories of generalized multicategories.
While the definition of (T, V)-categories generalizes that of enriched categories, the definitions of internal T -multicategories in B, for T a (cartesian) monad and B with pullbacks, introduced in [3, p. 8] and [11,Definition 4.2], generalize the notion of internal categories.Following this viewpoint and the approach of [17,Theorem 1.6], in order to study effective descent morphisms between more general (T, V)-categories, the first step is to study effective descent morphisms of categories of internal T -multicategories, which is the aim of the present paper.
The main contributions of our present work consist of two approaches to the problem of finding effective descent morphisms between internal multicategories.We explain, below, the key ideas of our first approach, which is the main subject of Section 4.
As a special case of [17,Theorem 9.2] (see Proposition 4.1), given a pseudo-equalizer (iso-inserter ) of categories with pullbacks and pullback preserving functors, p is of effective descent whenever F Ip is of descent and Ip is of effective descent.Therefore, whenever (effective) descent morphisms in C and D are well-understood, we find tractable, sufficient conditions for effective descent in PsEq(F, G).
We establish the category Cat(T, B) of internal T -multicategories in B as an equalizer consisting of a category of models of a finite limit sketch and categories of diagrams (Lemma 3.1), which is fully embedded in the corresponding pseudo-equalizer (Theorem 3.3).Since descent in categories of models of a finite limit sketch were studied in [15, Section 3.2], and categories of diagrams are well-understood, we obtain sufficient conditions for effective descent in the pseudo-equalizer by the result mentioned above (Lemma 4.3).
Finally, we find that the embedding of Cat(T, B) into the pseudo-equalizer reflects effective descent morphisms (Lemma 4.4), getting, then, our first result.Namely, a functor p of internal T -multicategories is effective for descent whenever where p i is the component of p between the objects of i-tuples of composable morphisms (Theorem 4.5).
Our second approach to the problem is presented in Section 5, which extends the work of [15] on effective descent morphisms between internal structures.We observe that the same techniques employed in Le Creurer's work can be applied to the "sketch" of internal T -multicategories.With these techniques, we were able to refine our result on effective descent morphisms.We prove that functors p such that p 1 is an effecive descent morphism in B, p 2 is a descent morphism in B, p 3 is an almost descent morphism in B, are effective descent morphisms in Cat(T, B).
The techniques exploited in Section 5 proved to be more suitable to our context of internal structures.However, the approach given there cannot be trivially applied to other generalized (enriched) categorical structures.Thus, Section 4 has expository value and its techniques are especially relevant to our future work in descent theory of generalized (enriched) categorical structures.
After fixing some notation on Section 1, we recall some basic aspects on effective descent morphisms in Section 2.Then, we study the equalizer that gives the category of internal T -multicategories and its corresponding pseudo-equalizer in Section 3. Afterwards, we discuss each approach to our main problem in the two subsequent sections.We end the paper with a discussion of examples of cartesian monads and internal multicategories.

Preliminaries
Let J : B → C be a diagram with a limit (lim J, λ).For any cone γ b : x → Jb, there exists a unique morphism f : x → lim J such that γ b = λ b • f for all b in B. We denote f as (γ b ) b∈ob B .As an example, let B be a category with pullbacks, and C an internal category.The object of pairs of composable morphisms is given by the pullback: Thus, if we have morphisms g : X → C 1 and f :

Effective descent morphisms
We recall some known facts about effective descent morphisms.In a category B with chosen pullbacks along p, the category Desc(p) of descent data for a morphism p : x → y in B is defined as the category of algebras for the monad p * p ! .Explicitly, objects are pairs of morphisms (a : , so that we may apply p * ), γ • (a, id) = id, the unit law, where (a, id) is the unique morphism such that Here, U p is the forgetful functor, and K p is commonly denoted the comparison functor.We say a morphism p is -an almost descent morphism if K p is faithful, -a descent morphism if K p is fully faithful, -an effective descent morphism if K p is an equivalence.By the Bénabou-Roubaud theorem (originally proven in [1], see, for instance, [12, p. 258] or [17,Theorem 7.4 and Theorem 8.5] for generalizations), this is equivalent to the classical formulation of the descent category w.r.t. the basic (bi)fibration.
As a consequece of Beck's monadicity theorem, we may characterize (almost) descent morphisms (also check [15,Corollary 0.3.4]and [14,Theorem 3.4]): Proposition 2.1.In a category B with finite limits, pullback-stable epimorphisms are exactly the almost descent morphisms, and pullback-stable regular epimorphisms in are exactly the descent morphisms.
Proof.Let p : x → y be a morphism in B. K p is (fully) faithful if and only if ǫ is a pointwise (regular) epimorphism (in B ↓ y), which happens if and only if p is a universal (regular) epimorphism in B ↓ y, as ǫ is given pointwise by pullback of p.
Since B has a terminal object, the forgetful functor B/y → B has a right adjoint, hence it preserves colimits.
Thus, once we have a pullback-stable regular epimorphism p, it is natural to take an interest in studying the image of K p .To do so, we make the following elementary observation.Since we have defined descent data as algebras, we restrict our attention to this context.It should be noted, however, that the result holds in much more general contexts, and hence its applicability in descent arguments does not depend on the Bénabou-Roubaud theorem.
where w is an object such that a = U w.
Proof.The algebra K T w satisfies (1) by naturality.Conversely, if an algebra (U w, γ) satisfies (1), then As a corollary, we get a fairly commonly used result in proofs about effective descent morphisms.It has been, sometimes, implicitly assumed in the literature.The instance of Le Creurer's argument in Proposition 3.2.4,where he implicitly uses this result, is of particular interest for our work.
Corollary 2.3.K p is essentially surjective if and only if, for all descent data (a, γ), there is We finish this section recalling the following classical descent result (see [12, 2.7], [14, 3.9]): Proposition 2.4.Let U : C → D be a fully faithful, pullback-preserving functor, and let p be a morphism in C such that U p is effective for descent.Then p is effective for descent if and only if for all pullback diagrams of the form there exists an isomorphism U y ∼ = z for y an object of C.
The following consequence is of particular interest: Corollary 2.5.Let U : C → D be a fully faithful, pullback-preserving functor.If there exists z ∼ = U y whenever there is an effective descent morphism g : U x → z, then U reflects effective descent morphisms.
Proof.Suppose ( 2) is a pullback square.If U p is an effective descent morphism, then so is f * (U p) : U x → z by pullback-stability.By hypothesis, we have z ∼ = U y, whence we conclude that p is effective for descent by Proposition 2.4.

Multicategories and pseudo-equalizers
Recall that a monad T = (T, m, e) is cartesian if T preserves pullbacks and the naturality squares of m and e are pullbacks.
As defined in [11], for T a cartesian monad on a category B with pullbacks, a Tmulticategory internal to B is a monad in the bicategory Span T (B), and a functor between two such T -multicategories is a monad morphism considering the usual proarrow equipment B → Span T (B); these define the category Cat(T, B).Explicitly, a T -multicategory is given by an object x 0 of B, together with a span and two morphisms, given by dashed arrows below which make the triangles commute, where Moreover, this data is required to satisfy certain identity and associativity conditions, which we will proceed to specify.Following the terminology of Section 1, we say that a pair g : a → x 1 , f : a → T x 1 is composable if d 1 g = (T d 0 )f , we write (g, f ) : a → x 2 for the uniquely defined morphism, and we let g The identity properties of the monad guarantee that , and the associativity property guarantees that h • (g where we are implicitly given the following pullback diagram for h : a → x 1 , g : a → T x 1 and f : a → T T x 1 such that h, g are composable and g, f are T -composable.Moreover, a functor p : x → y between internal T -multicategories is given by a pair of morphisms p 0 : x 0 → y 0 and p 1 : Going back to an internal description, we may denote so the above data can be organized in the following diagram which is similar to [3, Figure 1].In fact, one may define T -multicategory as a diagram satisfying certain relations, a description particularly suitable for our techniques in Section 4. First, we let S be the (finite limit) sketch given by the following graph (3) x ′ 0 , and limit cones (4) with i = 0, 1 and j ≤ i. Abusing notation, we also denote by S the category generated by the graph (3) and the given relations.Writing Mod(S, B) for the category of B-models of S, we have: For a cartesian monad (T, m, e) on a category B with pullbacks, Cat(T, B) is given as the equalizer of the following composite of pullback-preserving functors: (5)  4) to be a pullback, as the equalizer condition will force x ′ i = T x i and x ′′ i = T T x i , and since T preserves pullbacks, the pullback condition for the aforementioned diagram is already guaranteed.Moreover, omitting this apparently redundant diagram, an analogous version of Lemma 3.1 would describe Burroni's notion of T -multicategories (check [3], where this extra pullback condition is not required), even when T is not cartesian, or even pullback-preserving.
In spite of the above reasons, this requirement is justified by the sharper results we obtain about effective descent in Mod(S, B) (see Proposition 4.2), and consequently, in Cat(T, B) as well (see Theorem 4.5).
Note that the inclusion Mod(S, B) → [S, B] is an iso-inserter of categories of diagrams, thus it creates limits.
The categories S I , S T , S m i , and S e i for i = 0, 1 are subcategories of S, respectively given by where T * , m and ê are the functors induced by the monad T .Note that these preserve pullbacks exactly when T is cartesian.Note that, in general, the equalizer is a full subcategory of the pseudo-equalizer: for functors F, G : C → D, the category PsEq(F, G) is the category whose objects are pairs (c, φ) where c is an object of C and ι : F y → Gy is an isomorphism, and morphisms (c, φ) → (d, ψ) are morphisms f : c → d such that Gf • φ = ψ • F f .Thus, the full embedding may be given on objects by x → (x, id).
Henceforth, we denote Given an object (y, ι) of P, ι can be explicitly described as a family of isomorphisms making the appropriate squares commute: Lemma 3.4.An object (y, ι) of P is isomorphic to a T -multicategory if and only if the following coherence conditions hold: ) ι e i 0 = id for i = 0, 1, Such an object (y, ι) satisfying these conditions is said to be coherent.
Proof.Given a coherent (y, ι), we define a T -multicategory ŷ such that ŷ0 = y 0 , ŷ1 = y 1 , and we consider the span and we let d1 = d 1 : x 2 → x 1 and ŝ0 = s 0 : x 0 → x 1 .Consider the diagram for i = 0, 1: The right square is a pullback because ι T j is an isomorphism for j = 0, 1, 2, and the left square is a pullback by definition, therefore the outer rectangle is a pullback as well.
by definition, the left triangle of (7) commutes.The left triangle of (8) also commutes, for we have e = ι e 0 1 • e 0 , ι e 0 1 = ι T 0 and e 0 = d 1 • s 0 .We claim it is possible to define and in order to verify our claim, we must show that which verifies the second.For the third and fourth, we have Recalling that the left square in ( 9) is a pullback for i = 0, 1, it follows that s 0 , s 1 : , respectively.But these are just ŝ0 , ŝ1 , d1 , d2 , respectively.The converse is implied by the result that follows.
Proof.A morphism f : (x, id) → (y, ι) is a morphism f : x → y such that Gf = ι • F f , which translates to the following equations: are epimorphisms for all i we recover the coherences; just note that T f i and T T f i are epimorphisms as well, and that T f

Descent via bilimits
Recall the pseudo-equalizer P defined in (6) from the previous section.We understand the effective descent morphisms of P via the effective descent morphisms of Mod(S, B) by the following instance of [17,Theorem 9.2].Furthermore, by the work of [15], we are able to provide sufficient conditions for effective descent in Mod(S, B) for B with finite limits: 2 are descent morphisms in B, p 3 is an almost descent morphism in B, then p is an effective descent morphism in Mod(S, B).
Proof.We refer the reader to Section 3.2 ibid if they wish to fill in the details.The sketch S may be given as an essentially algebraic theory with sorts 1 , partially defined operations d 1 :

and equation
, among other data and equations.Then apply Proposition 3.2.4ibid.
With T cartesian, diagram ( 5) is a pseudo-equalizer, so we are under the hypothesis of Proposition 4.1.Therefore: Lemma 4.3.A morphism p in P is effective for descent whenever p is effective for descent in Mod(S, B) and S * X p is a descent morphism for each X = T, m 0 , m 1 , e 0 , e 1 .In particular, if p satisfies the conditions in Proposition 4.2, then p is effective for descent in the pseudo-equalizer.
Proof.We observe that a morphism in a product of categories is of descent if and only if each component is a descent morphism.Moreover, pointwise (effective) descent in [S, B] implies pointwise descent in [S X , B] for every X.Therefore, the result follows by Proposition 4.1.By Lemma 3.3, we may apply the previous proposition to U : Cat(T, B) → P. Consequently, we can show that: Lemma 4.4.U reflects effective descent morphisms.
Proof.Since every effective descent morphism is an epimorphism, the result follows by Theorem 3.5 and Corollary 2.5.
Combining Lemmas 4.3 and 4.4, we get our main result: Theorem 4.5.For B with finite limits, let p : x → z be a T -multicategory functor internal to B. If T p 1 is an effective descent morphism, T p 2 is a descent morphism and p 3 is an almost descent morphism in B, then p is an effective descent morphism in Cat(T, V).
Proof.By the results in Appendix A, (observe that T p 1 is a T -graph morphism), we guarantee that p is an effective descent morphism in Mod(S, B).Now apply Theorem 4.4.

Descent via sketches
In this section, we extend the techniques of [15,Chapter 3] to give refined sufficient conditions for (effective) descent morphisms in Cat(T, B) in the broader sense of Burroni; that is, without requiring T to be cartesian (though we require T to preserve kernel pairs for Theorem 5.3), while keeping the definition of T -multicategory intact.We highlight that given a functor p : x → y of internal multicategories, if p 1 is a pullback-stable (regular) epimorphism, or of effective descent, then so is p 0 by Lemma A.3.Lemma 5.1.Let p : x → y be a functor of internal T -multicategories.If p 1 is an (pullback-stable) epimorphism in B, then so is p in Cat(T, B).
Proof.Given functors q, r such that qp = rp, we have q i p i = r i p i , and therefore q i = r i for i = 0, 1, hence q = r, thus p is an epimorphism.Since pullbacks are calculated pointwise, p must be pullback-stable whenever p 1 is.Lemma 5.2.Let p be a functor of internal T -multicategories.If p 1 is a (pullback-stable) regular epimorphism in B, p 2 is an (pullback-stable) epimorphism in B, then p is a (pullback-stable) regular epimorphism in Cat(T, B).
Proof.Consider the kernel pair r, s of p, and let q : x → z be a functor such that q • r = q • s.Then there exist unique morphisms k 0 , k 1 such that k i p i = q i for i = 0, 1.We claim these morphisms define a functor y → z.We have and since p 1 , p 2 are epimorphisms, cancellation allows us to conclude that k is a functor (we note that k 2 is defined as k 2 (g, f ) = (k 1 g, k 1 f ), and hence q 2 = k 2 p 2 ).
Again, pointwise calculation of pullbacks guarantees pullback stability.
Theorem 5.3.Let p be a functor of internal T -multicategories, and assume T preserves kernel pairs.If p 1 is an effective descent morphism in B, p 2 is a descent morphism in B, p 3 is an almost descent morphism in B, then p is effective for descent in Cat(T, B).
Proof.By the previous lemma, and Proposition 2.1, the comparison functor K p is fully faithful.Hence, we aim to prove that K p is also essentially surjective under our hypotheses, thereby concluding that p is effective for descent.Suppose we are given a p * p ! -algebra (a, γ), where a : v → x is a functor and γ : u → v is the algebra structure.We have equivalences K i : B ↓ y i → Desc(p i ), for i = 0, 1, and (a, γ) then determines algebras (a i , γ i ) for i = 0, 1.Hence, there exist f i : w i → y i and is a pullback square, and moreover, we have We claim that h 0 , h 1 determine a functor h : v → w, f 0 , f 1 determine a functor f : w → y, so that the above lifts to a pullback diagram of T -multicategories.
The hypothesis that p 1 , p 2 are pullback-stable regular epimorphisms implies that h 1 , h 2 are regular epimorphisms.Taking kernel pairs and noting that T preserves them, we get therefore there exist unique morphisms making every right hand side square commute.
We note that we define h 2 (g, f ) = (h 1 g, (T h 1 )f ).Assuming that w is in fact a Tmulticategory, we may already conclude that h is a functor.The hypothesis that p 1 , p 2 , p 3 are pullback-stable epimorphisms implies that h 1 , h 2 , h 3 are epimorphisms.We have equations and by cancellation, we conclude w is a T -multicategory (proving our assumption) and, similarly, we can show that f is a functor, by following the same strategy as in the previous lemma.This confirms that p * is essentially surjective.Finally, it is immediate that h •a i for i = 0, 1 and pullbacks are calculated pointwise.The result now follows by Corollary 2.3.

Epilogue
There are sparse examples of cartesian monads, and therefore sparse examples of categories of internal multicategories over a monad.For B finitely extensive with finite limits and pullback-stable nested countable unions, as in [16,Appendix D], the free category monad on graphs internal to B is cartesian, and therefore so is the free monoid monad W on B. In fact, Leinster's construction is iterable, and most known examples fit into the above conditions.
A class of examples outside of the previous setting is given by free monoid monads on extensive categories with finite limits (thus, trading off the requirement of the aforementioned unions by infinitary extensivity).These are also cartesian; the idea is that the coproduct functor Fam(B) → B preserves finite limits, so we may construct the required limit diagrams in Fam(B), allowing us to conclude that such monads preserve pullbacks and that the required naturality squares are pullbacks.
Given a cartesian monad on a category B with pullbacks and C an internal Tmulticategory, we can construct a cartesian monad T C on B ↓ C 0 ; see Corollary 6.2.5 ibid.This yields an equivalence of categories ( 13) and since pullback-stable (regular) epimorphisms and effective descent remain unchanged on slice categories (more precisely, C/x → C creates each of the three types of morphism), we can deduce facts about effective descent of complicated internal multicategories in terms of simpler ones.
For the remainder of this section, we will discuss some simple examples of interest, compare our work with other literature, then mention some open problems.Despite being a more complicated structure than a category, (M × −)-functors of effective descent are not harder to come by compared to ordinary functors.A wellknown result (which can be deduced from ( 13)) is that Cat(M × −, Set) ∼ = Cat ↓ M , where we view M as a one object category.Hence, an (M × −)-functor is an effective descent morphism whenever it has the respective property as a functor.In fact, since [15] characterizes effective descent functors, we have also characterized effective descent (M ×−)-functors.The arguments remain unchanged when we replace Set by a lextensive category B (with regular epi-mono factorizations for the complete characterization).
Ordinary and operadic multicategories.A multicategory C consists of sets C 0 and C 1 of objects and multimorphisms, respectively, together with domain and codomain functions d 1 : C 1 → W C 0 , d 0 : C 1 → C 0 , together with composition and unit operations d 1 : C 2 → C 1 and s 0 : C 0 → C 1 satisfying associativity and identity properties.Here, C 2 is the set of multicomposable pairs given by the pullback of d 1 and T d 0 .Likewise, C n is the set of multicomposable n-tuples.
A multicategory functor F : C → D is given by a pair of functions on objects and multimorphisms which preserve domain, codomain, unit and composition.Our main result states that F is effective for descent whenever it is surjective on multimorphisms, multicomposable pairs, and multicomposable triples.
To extend this result using ( 13), suppose we have an operad O (a multicategory with one object).The induced monad W O is said to be an operadic monad, which is cartesian.These are related to strongly regular theories; we refer the reader to [16] and [4] for details.One could denote the category Cat(W O , Set) as the category of operadic multicategories and functors between them.These functors come with an underlying multicategory functor, and is effective for descent in Cat(W O , Set) if and only if it is effective for descent in Cat(W, Set).As in the previous case, the same arguments work for B lextensive.
State of the art.Our results have shown that three levels of "surjectivity" (of singles, pairs and triples of multimorphisms) are sufficient to determine effective descent in generalized multicategories.This is consistent with the findings of [14, 6.2 Proposition] for Cat, and in [15, Theorem 6.2.9] for Cat(C) where C has finite limits and a (regular epi, mono)-factorization, where these three levels are also necessary.This is also the case for V-categories, with V cartesian, as verified by [17, Theorem 9.11] (with suitable V lextensive), and [6, Theorem 5.4] (with V a complete Heyting lattice).In the latter case, since V is thin, surjectivity on triples of morphisms is no longer required.
In the enriched multicategory case, for T the ultrafilter monad and V = 2 (so that (T, V)-Cat = Top), we have the result of [5, Theorem 5.2], which requires only two levels of surjectivity as well.
Further work.We also take the opportunity to state some open problems.One might be interested in verifying whether the converses to Theorems 4.5 or 5.3 hold.As mentioned in the introduction, LeCreurer gave an affirmative answer for T = id and further requiring a (regular epi, mono)-factorization on B. One might also wonder if this extra condition is necessary.
Another interesting problem is to check whether LeCreurer's tools are also amenable to fully characterize effective descent morphisms of enriched categories internal to B.
Lemma A.3.Let f : x → y be a T -graph morphism, and let E be a class of epimorphisms, containing all retractions, closed under composition and cancellation.If f 1 is in E, then so is f 0 .
Proof.Since d 0 : x 1 → x 0 is a retraction, d 0 f 1 = f 0 d 0 is in E, therefore so is f 0 by cancellation.
We are interested in the cases when E is the class of pullback-stable epimorphisms, of descent morphisms and of effective descent morphisms.
pullback of f along p.It is clear that the change-of-base p * : B ↓ y → B ↓ x defines a functor right adjoint to p ! : B ↓ x → B ↓ y with counit ǫ.For a morphism h

Lemma 2 . 2 .
Let (L ⊣ U, ǫ, η) : A → B be an adjunction and let T be the induced monad.An algebra (a, γ) is in the image of the Eilenberg-Moore comparison K T : A → T -Alg if, and only if, a is in the image of U and (1)

Remark 3 . 2 .
Cat(T, B) has pullbacks and the canonical functor Cat(T, B) → Mod(S, B) preserves them.It might seem superfluous to require the right diagram of ( and write S * I , S * T , S * m i , S * e i , for the restriction functors.Also write x * 0 and x * 1 : [S, B] → B for the projections.With these, S * − , and Φ are the uniquely determined functors given by the following [S, B] [S T , B] [S I , B] [S T , B]
Of course, both right triangles commute by definition.Moreover, we have that the diagram(9)

Proposition 4 . 1 .
Suppose that we have a pseudo-equalizer of categories and pullbackf be a morphism in the pseudo-equalizer.Then f is effective for descent whenever If is effective for descent and F If ∼ = GIf is a pullback-stable regular epimorphism.

( 1 −
M × −)-multicategories.Given a monoid M , we can define a cartesian monad M × − on Set.An (M ×−)-multicategory C is, intuitively, a category with weighted morphisms.(M × −)-morphisms are of the form f : x m − → y for objects x, y and an element m ∈ M , and if g :y n − → z, then g • f : x n•m −−→ z.Identities are given by id : x → x, and these are to satisfy associativity and identity laws.
we write (g, f ) for the uniquely determined morphism X → C 2 .Furthermore, we denote the internal composition by g• f = d 1 • (g, f ), where d 1 : C 2 → C 1 is the composition morphism.Likewise, we can talk about tuples of composable morphisms, an idea we apply to T -multicategories.