Higher correspondences, simplicial maps and inner fibrations Redi Haderi [email protected] May 2020 Abstract In this essay we propose a realization of Lurie’s claim that inner fi- brations p : X → C are classified by C-indexed diangrams in a ”higher category” whose objects are ∞-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all sim- plicial maps. Correspondences between ∞-categories, and simplicial sets in general, are a generalization of the concept of profunctor (or bimodule) for cat- egories. While categories, functors and profunctors are organized in a double category, we will exibit simplicial sets, simplicial maps, and corre- spondences as part of a simpliclal category. This allows us to make precise statements and proofs. Our main tool is the theory of double colimits. arXiv:2005.11597v1 [math.AT] 23 May 2020 1 Notation and terminology We will mostly observe standard notation and terminology. Generic categories are denoted by calligraphic capital letters C, D etc., while particular categories are denoted by the name of their objects in bold characters. For example Set, Cat, sSet are the categories of sets, categories and simplicial sets. ∆ denotes the category of finite ordinals and order preserving maps as usual, n while ∆ denotes the standard n-simplex. For a simplicial set X, Xn refers to its set of n-simplices. Given the nature of our discussion the term ”simplicial category” means simplicial object in the category of categories. We refer to simplicially enriched categories as sSet-categories. Also, given our references, the term ∞-category means quasi-category, that is a simplicial set with the inner horn filling property. 1 Introduction and summary 1.1 Correspondences Besides functors, another interesting notion of morphism between categories is that of a profunctor. A profunctor u between two categories C and D, or (C, D)-profunctor, is a functor u : Cop × D → Set where Set is the category of sets and functions. A profunctor records a right action of C and a left action of D simultaniously. If C and D are groups profunctors are also known as bisets. If we enrich the above definition over abelian groups and let C and D be rings then profunctors are simply bimodules. If we enrich over the category 2 = {0 → 1}, in which case C and D are posets, when C and D are discrete sets profunctors are simply relations. This is why they are reffered to as relators sometimes. The perspective of interest from our point of view is one of a more combi- natorial flavour. Given a profunctor u we may record all its information in a category called the collage of u. This category is simply constructed by first starting with a copy of C and D and then considering the elements of the sets u(c, d) for c ∈ C, d ∈ D as actual arrows c → d. Composition in the collage is given by the functoriality of u. C D x c d f g c′ d′ (f, g)(x) It is easy to see that 2 defining a (C, D)-profunctor is the same as constructing a new cat- egory U from C and D by adding new morphisms from objects of C to those of D (but not in the reverse direction) Hence, profunctors are simply collages. To make things even more interesting, observe that a collage U is naturally equipped with a map p : U → ∆1 ,where ∆1 = {0 → 1} is the usual categorical 1-simplex, with p−1(0) =∼ C and p−1(0) =∼ D. It is easy to see that U is the collage of a profunctor if and only if it comes equipped with a map p to ∆1. So we see that profunctors are simply maps to ∆1 We would like to reserve the term correspondence when we have the latter perspective in mind. Nonetheless all the above terms are synonimous (at least for categories) and these labels are purely the author’s own preference. The reader can work out a lot of examples by picking one of the above perspectives. For example given a functor F : C → D there is a profunctor F ∗ given by F ∗(c, d)= D(Fc,d) We may interpret F ∗ the induced bimodule, but also as the correspondence representing the mapping cylinder of F . In a formal double categorical setting F ∗ is obtained as a certain Kan extension and it is also called the companion of F . The author still marvels at the fact that all of the above are equivalent and considers the true value of category theory to be precisely the unveiling of such patterns in mathematics. This allows us not only to reinterpret known constructions more conceptually but to prove new theorems as well. For example in this essay we will consider correspondences between simplicial sets and prove a classification result about simplicial maps (and inner fibrations). We want to think of profunctors as morphisms between categories so we need to specify a composition. Given that profunctors are bimodules, their composition will be a tensor product. Let u be a (C, D)-profuntor and v be a (D, E)-profunctor. We define their composition to be ”the” (C, E)-profunctor v ⊗D u whose evaluation at a pair of objects c ∈C, e ∈E is given by the coend formula (v ⊗D u)(c,e)= v(d, e) × u(c, d) Zd∈D While this coend formula has the virtue of applying in any enrichment it might not be very illuminating. If we view profunctors as collages we may interpret their composition as follows. Consider col(u) and col(v) and juxtapose them along D. What results is not a category because we cannot compose x ∈ u(c, d) with y ∈ v(d, e). Resolve this issue by generating a free category out of the data by declaring a new morphism y ⊗ x : c → e serving as a composite (subject to the obvious relations). Finally we remove the objects of D to obtain the collage of v ⊗D u. 3 D d x y C E c e y ⊗ x If we think of profunctors as maps to ∆1 we obtain yet another description of composition. Let p : U → ∆1 and q : V → ∆1 be composable correspondences, i.e. p−1(1) =∼ D =∼ q−1(0) for some D. By taking the pushout along D we 2 ∼ 1 1 obtain a map p D q : U D V → ∆ = ∆ ∆0 ∆ (this is precisely the above picture). Then obtain the tensor product by taking the pullback ` ` ` V ⊗D U U D V ` d1 ∆1 ∆2 It is easy to see that the above three descriptions are equivalent. This composition operation does not produce a category because it is unital and associative only up to canonical isomorphism (we defined it using universal properties after all). Given C, the identity profunctor is defined to be op homC : C × C → Set which assingns to a pair of objects c,c′ the set of morphisms in C between them. With this data we obtain a weak 2-category Prof with categories as objects, profunctors as 1-morphisms and their natural transformations as 2-morphisms. 1.2 The meaning of inner fibrations Let p : X → C be an inner fibration between simplicial sets. For an n-simplex −1 σ ∈ Cn consider the fiber p (σ) obtained by the pullback square p−1(σ) X pσ p σ ∆n C Fibrations are stable under pullback, so pσ is an inner fibration as well. Moreover, if the target of an inner fibration is a category then the source is an ∞-category. In our case we conclude that fibers p−1(σ) over each simplex σ 4 have to be ∞-categories. It is not difficult to see that the converse is also true: if a map p : X → C of simplicial sets is such that fibers over each simplex are ∞-categories then it is an inner fibration. Referring to the above observations we quote the authors of [BS18] ”So in a strong sense, we’ll understand the ”meaning” of inner fi- brations once we understand the ”meaning” of functors from ∞- categories to ∆n” Given that for n = 1 these maps are understood as correspondences between ∞-categories, we will refer to functors to ∆n as higher correspondences. A higher correspondence p : X → ∆n may be thought of as consisting of −1 n +1 ∞-categories Xi = p (i), i =0, 1 ...n, and a big collage between them. By the latter we mean that X is formed from the Xi’s by ”adding” 1-simplicies which join vertices of Xi and Xj only if i < j, and higher simplicies after that. Given an inner fibration p : X → C for each simplex σ ∈ Cn we obtain a correspondence p−1(σ). We would like to see the association σ 7→ p−1(σ) as being functorial. This means there should be a ”higher category” in which objects are ∞-categories, morphisms are correspondences and higher morphisms are higher correspondences. Then the above would yield a map from C to this higher category. As Lurie points out ([Lur09]), this higher category cannot be realized as an ∞-category because higher morphisms do not have to be invertible. 1.3 Our strategy and results We will propose a (1-categorical) realization of the above. Our treatment has three characteristics which distinguish it from the rest of the literature (to the best of our knowledge): • We will treat the relationship between inner fibrations and higher corre- spondences as part of a larger pattern.