aa r X i v : . [ m a t h . C T ] A ug ON (CO)ENDS IN ∞ -CATEGORIES RUNE HAUGSENG
Abstract.
In this short note we prove that two definitions of (co)ends in ∞ -categories, via twisted arrow ∞ -categories and via ∞ -categories of simplices,are equivalent. We also show that weighted (co)limits, which can be definedas certain (co)ends, can alternatively be described as (co)limits over left andright fibrations, respectively. Contents
1. Introduction 12. (Co)ends via the ∞ -Category of Simplices 33. Coends via the Twisted Arrow ∞ -Category 84. Weighted (Co)limits 11References 121. Introduction
Ends and coends were first introduced by Yoneda [Yon60] and play an importantrole in the theory of both ordinary and enriched categories. Indeed, the calculusof (co)ends can be viewed as a central organizing principle in category theory; werefer the reader to the book [Lor19a] for further discussion and many applicationsof (co)ends.Ends and coends can also be defined in the context of ∞ -categories, though itis not immediately clear how to do this if we use what seems to be the most com-mon way of defining (co)ends, in terms of so-called “extranatural transformations”.Luckily, it is not actually necessary to introduce this notion, as there are two easyways to define the end of a functor F : C op × C → D as an ordinary limit:(A) Let Tw ℓ ( C ) denote the (left) twisted arrow category of C . This has morphismsin C as objects, with a morphism from f : x → y to f ′ : x ′ → y ′ given by acommutative diagram x x ′ y y ′ . f f ′ Taking the source and target of morphisms gives a functor p : Tw ℓ ( C ) → C op × C , and the end of F is just the limit of the composite functor F ◦ p : Tw ℓ ( C ) → D . Date : August 11, 2020. (B) If D has products, the end of F can be expressed as the (reflexive) equalizerof the two morphisms(1) Y x ∈ C F ( x, x ) ⇒ Y f : x → y F ( x, y ) , given on the factor indexed by f by projecting to F ( y, y ) and F ( x, x ) and com-posing with f in the first and second variable, respectively. We can reinterpretthis (and get rid of the assumption on D ) using the category of simplices ∆ / C of C . An object here is a functor [ n ] → C with [ n ] in ∆ (i.e. a sequence of n composable morphisms in C for n ≥ n ] [ m ] C φF G for some morphism φ : [ n ] → [ m ] in ∆ . We then have a functor q : ∆ / C → C op × C that takes F : [ n ] → C to ( F (0) , F ( n )) and a morphism as above to( G (0) → G ( φ (0)) = F (0) , F ( n ) = G ( φ ( n )) → G ( m )) . The end of F can then be defined as the limit of the composite F ◦ q : ∆ / C → D . We can compute this in two stages by first taking the right Kan extensionalong the projection ∆ / C → ∆ and then taking the limit of the resultingcosimplicial diagram; for ordinary categories the inclusion of the subcategoryof ∆ containing just the two coface maps [0] ⇒ [1] is coinitial, and this recoversthe equalizer (1) we started with. Both of these definitions have natural extensions to ∞ -categories, and our maingoal in this paper is to show that these definitions of ∞ -categorical ends are in factequivalent: Theorem 1.1.
For a functor of ∞ -categories F : C op × C → D , the limits of thecomposites Tw ℓ ( C ) → C op × C → D , ∆ / C → C op × C → D are (naturally) equivalent if either exists. We introduce the ∞ -category of simplices ∆ / C and define the functor to C op × C in §
2. The definition of (co)ends in terms of twisted arrow ∞ -categories was previouslydiscussed in [Gla16, §
2] and [GHN17]; we review the definition in § ∞ -categories W : C → S and φ : C → D , we can define the limitlim W C φ of φ weighted by W as the end of the functor φ W : C op × C → D , where φ ( c ) W ( c ′ ) denotes the limit over W ( c ′ ) of the constant diagram with value φ ( c ), provided D admits such limits. In § Pulling back this coinitial map along the right fibration ∆ / C → ∆ , we see that it is enoughto consider the subcategory of ∆ / C consisting of functors [ n ] → C with n = 0 , § IX.5].
N (CO)ENDS IN ∞ -CATEGORIES 3 Theorem 1.2.
Let p : W → C be the left fibration corresponding to the functor W .Then there is an equivalence lim W C φ ≃ lim W φ ◦ p, provided either limit exists in D . As a consequence of this result, the definition of weighted limits in terms of endsagrees with that introduced by Rovelli [Rov19].
Acknowledgments.
I thank Clark Barwick for introducing me to twisted arrowcategories and Grigory Kondyrev for decreasing my ignorance about weighted col-imits. I also thank Espen Auseth Nielsen for helpful discussions about coends backin 2017 and Moritz Groth for telling me the derivation of the Bousfield–Kan formulafor homotopy colimits given in Corollary 2.18 at some even earlier date.2. (Co)ends via the ∞ -Category of Simplices In this section we define the ∞ -category of simplices ∆ / C of an ∞ -category C ,and prove that this has a canonical functor to C op × C , which allows us to give ourfirst definition of (co)ends. Definition 2.1. If C is an ∞ -category, its ∞ -category of simplices ∆ / C is definedby the pullback square ∆ / C Cat ∞ / C ∆ Cat ∞ , where the lower horizontal map is the usual embedding of ∆ in Cat ∞ (taking theordered set [ n ] to the corresponding category). Since Cat ∞ / C → Cat ∞ is a rightfibration, so is the projection ∆ / C → ∆ . Remark 2.2.
The functor ∆ op → S corresponding to the right fibration ∆ / C → ∆ is given by [ n ] Map
Cat ∞ ([ n ] , C ) . Thus this simplicial space is the ∞ -category C viewed as a (complete) Segal space. Warning 2.3. If C is an ordinary category, then ∆ / C as we have defined it here isnot quite the category of simplices we discussed in the introduction, but a variantwhere the morphisms are triangles that commute up to a specified natural isomor-phism. In other words, for us ∆ / C → ∆ is the fibration corresponding to thefunctor that takes [ n ] to the groupoid of functors [ n ] → C , rather than the set ofsuch functors. Proposition 2.4.
Suppose I is a small ∞ -category and φ : I op → S is the presheafcorresponding to a right fibration p : E → I . Then φ is the colimit of the compositefunctor E p −→ I y −→ Fun( I op , S ) , where y is the Yoneda embedding.Proof. This is essentially [Lur09, Lemma 5.1.5.3], but we include a proof. Sup-pose ψ is another presheaf on I , corresponding to a right fibration F → I . Thenunstraightening gives a natural equivalenceMap Fun( I op , S ) ( φ, ψ ) ≃ Map / I ( E , F ) ≃ Map / E ( E , E × I F ) . RUNE HAUGSENG
Here E × I F → E is the right fibration for the composite functor E op p op −−→ I op ψ −→ S ,and we can identify its ∞ -groupoid of sections with the limit of this functor by[Lur09, Corollary 3.3.3.4]. Thus we have a natural equivalenceMap Fun( I op , S ) ( φ, ψ ) ≃ lim I op ψ ◦ p op ≃ lim I op Map
Fun( I op , S ) ( y ◦ p, ψ ) , where the second equivalence follows from the Yoneda lemma. This shows that φ has the universal property of the colimit colim I y ◦ p , as required. (cid:3) Corollary 2.5.
The ∞ -category C is the colimit of the composite functor ∆ / C → ∆ → Cat ∞ . Proof.
By thinking of ∞ -categories as complete Segal spaces, we can view Cat ∞ as afull subcategory of Fun( ∆ op , S ), and the composite y : ∆ → Cat ∞ ֒ → Fun( ∆ op , S )is the Yoneda embedding. Since colimits in Cat ∞ can be computed by takingcolimits in Fun( ∆ op , S ) and then localizing, it is enough to show that C is thecolimit of the composite ∆ / C → ∆ y −→ Fun( ∆ op , S ) , which follows from Proposition 2.4. (cid:3) Definition 2.6.
Let ∆ ∗ denote the category with objects pairs ([ n ] , i ) with [ n ] ∈ ∆ and i ∈ [ n ], and a morphism ([ n ] , i ) → ([ m ] , j ) given by a morphism φ : [ n ] → [ m ]in ∆ such that φ ( i ) ≤ j . Let π : ∆ ∗ → ∆ be the obvious projection.The functor π is the cocartesian fibration for the inclusion ∆ ֒ → Cat ∞ ; thecocartesian morphisms are the morphisms over φ : [ n ] → [ m ] of the form ([ n ] , i ) → ([ m ] , φ ( i )). Thus the cocartesian fibration for the composite ∆ / C → ∆ → Cat ∞ is π C : ∆ / C , ∗ := ∆ / C × ∆ ∆ ∗ → ∆ / C . Corollary 2.7.
There is a natural equivalence of ∞ -categories ∆ / C , ∗ [cocart − ] ∼ −→ C . Proof.
This follows from the description of colimits in Cat ∞ in [Lur09, § F : I → Cat ∞ is given by inverting the cocartesian morphismsin the corresponding cocartesian fibration. (cid:3) Definition 2.8.
Let l : ∆ → ∆ ∗ be the section of π defined on objects by l ([ n ]) =([ n ] , n ); since any map φ : [ n ] → [ m ] satisfies φ ( n ) ≤ m this makes sense. Note that πl = id. Moreover, we have a natural isomorphismHom ∆ ∗ (([ n ] , i ) , l ([ m ])) ∼ = Hom ∆ ([ n ] , [ m ]) , so that l is right adjoint to π . Next, we define λ : ∆ ∗ → ∆ by λ ([ n ] , i ) = { , . . . , i } ;if φ : [ n ] → [ m ] satisfies φ ( i ) ≤ j then φ restricts to a map { , . . . , i } → { , . . . , j } ,which we define to be λ ( φ ). A map ([ n ] , n ) → ([ m ] , i ) in ∆ ∗ is determined by amap [ n ] → λ ([ m ] , i ) in ∆ , i.e. we have a natural isomorphismHom ∆ ∗ ( l ([ n ]) , ([ m ] , i )) ∼ = Hom ∆ ([ n ] , λ ([ m ] , i )) , and so λ is right adjoint to l . Note also that λl = id and there is a naturaltransformation α : λ → π given at the object ([ n ] , i ) by the inclusion { , . . . , i } ֒ → [ n ]. (This can also be defined as λ applied to the unit transformation id → lπ .) Lemma 2.9.
Let LV denote the set of last-vertex morphisms in ∆ , i.e. the maps φ : [ n ] → [ m ] such that φ ( n ) = m . Then(i) λ takes the π -cocartesian morphisms to morphisms in LV ,(ii) l takes morphisms in LV to π -cocartesian morphisms,(iii) the unit map [ n ] → λl ([ n ]) is in LV (being in fact the identity of [ n ] ), N (CO)ENDS IN ∞ -CATEGORIES 5 (iv) the counit map lλ ([ n ] , i ) = ([ i ] , i ) → ([ n ] , i ) is π -cocartesian.Moreover, the adjunction l ⊣ λ induces an adjoint equivalence ∆ [LV − ] ∼ ⇄ ∼ ∆ ∗ [cocart − ] . Proof.
Properties (i)–(iv) are immediate from the definition of the π -cocartesianmorphisms, and imply that the adjunction l ⊣ λ descends to the localized ∞ -categories where the unit and counit transformations become natural equivalences. (cid:3) We now want to lift this equivalence to ∆ / C ; we first lift the adjoint triple: Proposition 2.10.
The adjoint triple π ⊣ l ⊣ λ induces for all ∞ -categories C anadjoint triple of functors π C ⊣ l C ⊣ λ C between ∆ / C , ∗ and ∆ / C .Proof. Since πl = id, pulling back l gives a commutative diagram ∆ / C ∆ / C , ∗ ∆ / C ∆ ∆ ∗ ∆ , l C π C l π where both squares are cartesian. The unit transformation id → lπ pulls backsimilarly, and the adjunction identities hold since they lie over equivalences in ∆ and ∆ ∗ and right fibrations are conservative.We define λ C : ∆ / C , ∗ → ∆ / C by taking the cartesian pullback of α : λ → π ,which gives a filler in the diagram ∆ / C , ∗ × { } ∆ / C ∆ / C , ∗ × ∆ ∆ ∗ × ∆ ∆ π C α C α where the value of α C at an object X ∈ ∆ / C , ∗ over ([ n ] , i ) in ∆ ∗ is the cartesianmorphism in ∆ / C over α ([ n ] ,i ) with target π C X . Since α restricts to the identitytransformation along l , it follows that λ C ◦ l C ≃ id.The natural transformation lα : lλ → lπ factors as the composite lλ → id → lπ of the counit transformation for l ⊣ λ and the unit transformation for π ⊣ l . Hence l C α C factors as l C λ C → id → l C π C where the second morphism is the unit for π C ⊣ l C , since this is the unique transformation over id → lπ with target l C π C , as ∆ / C , ∗ → ∆ ∗ is a right fibration. We claim that the transformation l C λ C → id is acounit. To see this consider the commutative diagramsMap ∆ / C ( X, λ C Y ) Map ∆ / C , ∗ ( l C X, l C λ C Y ) Map ∆ / C , ∗ ( l C X, Y )Map ∆ ([ m ] , λ ([ n ] , i )) Map ∆ ∗ ( l [ m ] , lλ ([ n ] , i )) Map ∆ ∗ ( l [ m ] , ([ n ] , i )) . for objects X and Y lying over [ m ] and ([ n ] , i ), respectively. Here the left square iscartesian since l C is a pullback of l , and the right square is cartesian since ∆ / C , ∗ → ∆ ∗ is a right fibration (and hence the morphism l C λ C Y → Y is cartesian). The RUNE HAUGSENG composite in the bottom row is an equivalence, as we know that l is left adjoint to λ ,so this implies that the composite in the top row is an equivalence, as required. (cid:3) Corollary 2.11.
Let LV C denote the morphisms in ∆ / C that lie over last-vertexmorphisms in ∆ . Then(1) λ C takes π C -cocartesian morphisms to morphisms in LV C ,(2) l C takes morphisms in LV C to π C -cocartesian morphisms,(3) the unit map X → λ C l C X is in LV C (since it is an equivalence),(4) the counit map l C λ C X → X is π C -cocartesian.Moreover, the adjunction l C ⊣ λ C induces an adjoint equivalence ∆ / C [LV − ] ∼ ⇄ ∼ ∆ / C , ∗ [cocart − ] . Proof.
The π C -cocartesian morphisms are precisely the morphisms in ∆ / C , ∗ thatlie over π -cocartesian morphisms in ∆ ∗ , so this follows from Lemma 2.9 and thefact that the unit and counit for l C ⊣ λ C lie over the unit and counit for l ⊣ λ . (cid:3) Combining this with the equivalence of Corollary 2.7, we have proved:
Proposition 2.12.
There is a natural equivalence of ∞ -categories ∆ / C [LV − ] ≃ C , and hence a natural transformation L C : ∆ / C → C . (cid:3) To obtain a natural map ∆ / C → C op , we combine this with the order-reversingautomorphism of ∆ : Definition 2.13.
Let rev : ∆ → ∆ be the order-reversing automorphism of ∆ ,i.e. rev([ n ]) = [ n ] but for φ : [ n ] → [ m ] we have rev( φ )( i ) = m − φ ( n − i ). Lemma 2.14.
We have a natural pullback square ∆ / C op ∆ / C ∆ ∆ . rev C rev Since rev is an equivalence, so is rev C .Proof. The pullback of ∆ / C → ∆ along rev is the right fibration for the composite ∆ op rev op −−−→ ∆ op C −→ S and this composite is precisely the complete Segal space corresponding to C op . (cid:3) Under the equivalence rev, the last-vertex morphisms in LV correspond to the initial-vertex morphisms
IV, i.e. the maps φ : [ n ] → [ m ] such that φ (0) = 0. Wethus get: Corollary 2.15.
There is a natural equivalence of ∞ -categories ∆ / C [IV − C ] ≃ C op , where IV C are the morphisms in ∆ / C that lie over IV . Hence there is a naturaltransformation I C : ∆ / C → C op . Proposition 2.16.
Suppose L : C → C [ W − ] is the localization of an ∞ -category C at a collection of morphisms W . Then the functor L is coinitial and cofinal. N (CO)ENDS IN ∞ -CATEGORIES 7 Proof.
Without loss of generality the morphisms in W are closed under composi-tion and contain all equivalences; we can then let W denote the subcategory of C containing the morphisms in W . By definition of C [ W − ] we then have a pushoutsquare W k W k C C [ W − ] , where k W k denotes the ∞ -groupoid obtained by inverting all morphisms in W . By[Lur09, Corollary 4.1.2.6] the map W → k W k is cofinal, so by [Lur09, Corollary4.1.2.7] the pushout C → C [ W − ] is also cofinal. To see that it is also coinitial, weapply the same argument on opposite ∞ -categories. (cid:3) Corollary 2.17.
For any ∞ -category C , the functors L C : ∆ / C → C , I C : ∆ / C → C op are both coinitial and cofinal. (cid:3) We can use this to obtain an ∞ -categorical version of the Bousfield–Kan formulafor homotopy colimits: Corollary 2.18 (Bousfield–Kan formula) . Let D be a cocomplete ∞ -category. Thecolimit of a functor F : C → D is equivalent to the colimit of a simplicial object ∆ op → D given by [ n ] colim α ∈ Map([ n ] , C ) F ( α (0)) . Proof.
We can compute the colimit of F after composing with the cofinal map I op C : ∆ op / C → C , which takes α : [ n ] → C to α (0). This colimit we can in turncompute in two stages, by first taking the left Kan extension along the projection ∆ op / C → ∆ op , which produces a simplicial object of the given form, and then takingthe colimit of this simplicial object. (cid:3) We are now in a position to define ends and coends:
Definition 2.19.
Given a functor F : C × C op → D , its coend R C F is the colimitof the composite functor ∆ op / C ( I op C , L op C ) −−−−−−→ C × C op F −→ D . Dually, the end R ∗ C F of F is the limit of the composite functor ∆ / C ( L C , I C ) −−−−−→ C × C op → D . Lemma 2.20. If D is a cocomplete ∞ -category, then the coend of a functor F : C × C op → D can be computed as the colimit of a simplicial object ∆ op → D given by [ n ] colim α ∈ Map([ n ] , C ) F ( α (0) , α ( n )) . Proof.
The colimit over ∆ op / C can be computed in two steps by first taking the leftKan extension along the projection ∆ op / C → ∆ op , which gives the desired simplicialobject, and then taking the colimit of this simplicial object. (cid:3) Remark 2.21.
This simplicial colimit formula for coends is an ∞ -categoricalversion of the definition of homotopy-coherent coends studied by Cordier andPorter [CP97] in the context of simplicial categories. We use the original notational convention of [Yon60] rather than the “Australian” convention,where the coend is denoted R C F and the end is denoted R C F — after all, it is the co end of F that is somewhat analogous to an integral, not the end. RUNE HAUGSENG
A key property of ends is the “Fubini theorem” for iterated ends. This wasproved for ∞ -categories by Loregian [Lor19b], using the definition of (co)ends viatwisted arrows. We include a proof, as it is very easy to see using ∞ -categories ofsimplices: Proposition 2.22 (“Fubini’s Theorem”) . Given a functor F : ( C × D ) op × ( C × D ) → E , there are natural equivalences of ends Z ∗ C Z ∗ D F ≃ Z ∗ C × D F ≃ Z ∗ D Z ∗ C F. Proof.
Since unstraightening preserves limits, we have a natural equivalence ∆ / C × D ≃ ∆ / C × ∆ ∆ / D . This means we have a pullback square ∆ / C × D ∆ / C × ∆ / D ∆ ∆ × ∆ , where the bottom horizontal arrow is coinitial, since ∆ op is sifted. Since the rightvertical arrow is a right fibration, this implies that the top horizontal arrow is alsocoinitial. Moreover, the composite of this functor with the projection to ∆ / C is thefunctor induced by the composition with the projection C × D → C , and similarlyfor ∆ / D . It follows that we also have a commutative triangle ∆ / C × D ∆ / C × ∆ / D C op × D op × C × D . ( I C × D , L C × D ) ( I C , I D , L C , L D ) Together with the description of limits over a product as iterated limits this impliesthe result. (cid:3) Coends via the Twisted Arrow ∞ -Category In this section we recall the definition of twisted arrow ∞ -categories, and provethat we can equivalently define (co)ends as (co)limits using these ∞ -categories. Definition 3.1.
Let ǫ : ∆ → ∆ be the endomorphism given by[ n ] [ n ] op ⋆ [ n ] , and write ι : id → ǫ , ρ : rev → ǫ for the natural transformations corresponding tothe inclusions of the factors [ n ] and [ n ] op . For C an ∞ -category, we define Tw ℓ ( C )as the simplicial space [ n ] Map([ n ] op ⋆ [ n ] , C ) , i.e. ǫ ∗ C if we view C as a complete Segal space. Restricting along ι and ρ we get aprojection η C : Tw ℓ ( C ) → C op × C . We refer to Tw ℓ ( C ) as the (left) twisted arrow ∞ -category of C , as is justified bythe following result: Proposition 3.2 ([HMS19, Proposition A.2.3], [Lur17, Proposition 5.2.1.3]) . If C is an ∞ -category then Tw ℓ ( C ) is a complete Segal space, and the projection η C : Tw ℓ ( C ) → C op × C is a left fibration. (cid:3) N (CO)ENDS IN ∞ -CATEGORIES 9 Variant 3.3.
If we instead consider the endofunctor of ∆ given by [ n ] [ n ] ⋆ [ n ] op ,we get the right twisted arrow ∞ -category Tw r ( C ) := Tw ℓ ( C ) op , whose projectionTw r ( C ) → C × C op is a right fibration. Lemma 3.4.
There is a natural pullback square ∆ / Tw ℓ ( C ) ∆ / C ∆ ∆ ǫ C ǫ Proof.
The pullback of ∆ / C → ∆ along ǫ is the right fibration for the composite ∆ ǫ −→ ∆ C −→ S , which is by definition ∆ / Tw ℓ ( C ) . (cid:3) Proposition 3.5.
There is a natural commutative square (2) ∆ / Tw ℓ ( C ) ∆ / C Tw ℓ ( C ) C op × C . ǫ C L Tw ℓ ( C ) ( I C , L C ) η C Proof.
Observe that the definition of ǫ implies that we have ǫ (LV) ⊆ IV ∩ LV. Thecomposite ∆ op / Tw ℓ ( C ) ǫ C −→ ∆ op / C ( I C , L C ) −−−−−→ C op × C hence takes the morphisms in LV Tw ℓ ( C ) to equivalences, and so this compositefactors uniquely through the localization L Tw ℓ ( C ) : ∆ op / Tw ℓ ( C ) → ∆ op / Tw ℓ ( C ) [LV − ] ≃ Tw ℓ ( C ) , giving a natural commutative square ∆ / Tw ℓ ( C ) ∆ / C Tw ℓ ( C ) C op × C . ǫ C L Tw ℓ ( C ) ( I C , L C ) υ C It remains to show that the induced functor υ C is naturally equivalent to η C .Viewing the natural transformation ι as a functor ∆ × ∆ → ∆ , the projection p : Tw ℓ ( C ) → C is described as a morphism of complete Segal spaces by ι ∗ C : ∆ op × (∆ ) op → S , corresponding to the right fibration obtained as a pullback(3) ι ∗ ∆ / C ∆ / C ∆ × ∆ ∆ ι C ι The composite functor ι ∗ ∆ / C → ∆ is a cartesian fibration, and corresponds tothe functor q : ∆ / Tw ℓ ( C ) → ∆ / C given by composition with p , which fits in acommutative square ∆ / Tw ℓ ( C ) ∆ / C Tw ℓ ( C ) C q L Tw ℓ ( C ) L C p It therefore suffices to show that q is equivalent to ǫ C ; to see this we observe thatthe pullback square (3) induces a commutative triangle ι ∗ ∆ / C ∆ / C × ∆ ∆ , where the diagonal functors are both cartesian fibrations and the horizontal functorpreserves cartesian morphisms. We can straighten this to a commutative square of ∞ -categories ∆ / Tw ℓ ( C ) ∆ / C ∆ / C ∆ / C , ǫ C q which implies that q ≃ ǫ C . A similar argument works for the projection Tw ℓ ( C ) → C op , which completes the proof. (cid:3) The following is a special case of [Bar17, Proposition 2.1].
Proposition 3.6. ǫ : ∆ → ∆ is coinitial.Proof. By [Lur09, Theorem 4.1.3.1] it suffices to show that the pullback ∆ × ∆ ∆ / [ n ] along ǫ is weakly contractible for all objects [ n ] in ∆ . This pullback we can identifywith ∆ / Tw ℓ [ n ] by Lemma 3.4, and we have a cofinal functor L Tw ℓ [ n ] : ∆ / Tw ℓ [ n ] → Tw ℓ [ n ] from Corollary 2.17. Since cofinal functors are in particular weak homotopyequivalences, it suffices to show that the category Tw ℓ [ n ] is weakly contractible.This category can be described as the partially ordered set of pairs ( i, j ) with0 ≤ i ≤ j ≤ n , with partial ordering given by( i, j ) ≤ ( i ′ , j ′ ) ⇐⇒ i ′ ≤ i ≤ j ≤ j ′ . Here (0 , n ) is a terminal object, and so this category is indeed weakly contractible. (cid:3)
Corollary 3.7.
The functor ǫ C : ∆ / Tw ℓ ( C ) → ∆ / C is coinitial.Proof. From Proposition 3.6 and Lemma 3.4 we know that this functor is the pull-back of the coinitial functor ǫ along the cartesian fibration ∆ / C → ∆ . It is thereforecoinitial by (the dual of) [Lur09, Proposition 4.1.2.15]. (cid:3) Corollary 3.8.
Given a functor F : C op × C → D , its end is given by the limit ofthe composite Tw ℓ ( C ) η C −−→ C op × C F −→ D . Proof.
In the commutative square (2) from Proposition 3.5, the functor ǫ C is coini-tial by Corollary 3.7 while the functor L Tw ℓ ( C ) is coinitial by Corollary 2.17. Wetherefore have natural equivalenceslim Tw ℓ ( C ) F ◦ η C ≃ lim ∆ / Tw ℓ ( C ) F ◦ η C ◦ L Tw ℓ ( C ) ≃ lim ∆ / Tw ℓ ( C ) F ◦ ( I C , L C ) ◦ ǫ C ≃ lim ∆ / C F ◦ ( I C , L C ) , where the latter is the end of F as we defined it above. (cid:3) N (CO)ENDS IN ∞ -CATEGORIES 11 Remark 3.9.
Dually, for a functor F : C × C op → D , its coend can be computedeither as the colimit of the composite ∆ op / C ( I op C , L op C ) −−−−−−→ C × C op F −→ D , or as the colimit of Tw r ( C ) ≃ Tw ℓ ( C ) op η op C −−→ C × C op F −→ D . Weighted (Co)limits
Weighted (co)limits can be defined as certain (co)ends. Our goal in this sectionis to show that they can also be expressed as (co)limits over left and right fibrations,respectively. The latter description agrees with the definition of weighted (co)limitsstudied by Rovelli [Rov19] in terms of an explicit construction in quasicategories.Given a presheaf W : I op → S and a functor φ : I → C , the colimit of φ weighted by W , denoted colim W I φ , can be defined as the coend of the functor W × φ : I op × I → C ,at least if C admits colimits indexed by ∞ -groupoids. Similarly, for ψ : I op → C the limit of ψ weighted by W , denoted lim W I op φ , can be defined as the end of thefunctor ψ W : I × I op → C , provided C admits limits indexed by ∞ -groupoids. Onecan also characterize weighted limits and colimits in terms of universal properties,as we have Map C (colim W I φ, c ) ≃ lim W I op Map C ( φ, c ) , Map C ( c, lim W I op ψ ) ≃ lim W I op Map C ( c, ψ ) . Since all weighted limits exist in S , this also gives a definition of weighted (co)limitswithout any assumptions on C .The key property of weighted limits is that they describe mapping spaces infunctor categories. We state this in the case of presheaves: Theorem 4.1 (Glasman) . For presheaves φ, ψ ∈ P ( I ) we have a natural equivalence Map P ( I ) ( φ, ψ ) ≃ lim φ I op ψ. Proof.
This is a special case of [Gla16, Proposition 2.3] or [GHN17, Proposition5.1]. (cid:3)
Remark 4.2.
As a consequence, the Yoneda lemma implies that we express anypresheaf as a colimit weighted by itself:(4) φ ≃ colim φ I y I for φ ∈ P ( I ), where y I : I → P ( I ) is the Yoneda embedding. This follows from theequivalencesMap P ( I ) (colim φ I y I , ψ ) ≃ lim φ I op Map P ( I ) ( y I , ψ ) ≃ lim φ I op ψ ≃ Map P ( I ) ( φ, ψ ) . Proposition 4.3.
Suppose q : V → J is the left fibration corresponding to a functor V : J → S . Then for a functor ψ : J → C there is an equivalence (5) lim V J ψ ≃ lim V ψ ◦ q, provided either side exists.Proof. By the universal mapping properties of the two sides it suffices to show thereis an equivalence lim V J Map C ( c, ψ ) ≃ lim V Map C ( c, ψ ◦ q ) , natural in c ∈ C . In other words, it suffices to show there is a natural equivalence(5) for functors Ψ : J → S . Using Theorem 4.1 and the straightening equivalence, we can rewrite the left-hand side as lim V J Ψ ≃ Map
Fun( J , S ) ( V, Ψ) ≃ Map / J ( V , E ) , where E → J is the left fibration for Ψ. We now have an obvious equivalenceMap / J ( V , E ) ≃ Map / V ( V , q ∗ E ) , so our weighted limit is naturally equivalent to the space of sections of the left fibra-tion q ∗ E . Since pullback of left fibrations corresponds to composition of functors to S , this is the left fibration for Ψ ◦ q . Moreover, the space of sections of a left fibra-tion is equivalent to the limit of the corresponding functor to S by [Lur09, Corollary3.3.3.4], so that we have Map / V ( V , q ∗ E ) ≃ lim V Ψ ◦ q, as required. (cid:3) Corollary 4.4.
Suppose p : W → I is the right fibration corresponding to a presheaf W : I op → S . Then for a functor φ : I → C there is an equivalence (6) colim W I φ ≃ colim W φ ◦ p, provided either side exists.Proof. By Proposition 4.3 we have an equivalencelim W I op Map C ( φ, c ) ≃ lim W op Map C ( φp, c ) , natural in c ∈ C . This implies (6) by the universal mapping properties of the twoobjects. (cid:3) Corollary 4.5.
The definition of weighted limit from [Rov19] agrees with the def-inition as an end.Proof.
Combine Proposition 4.3 with [Rov19, Theorem D]. (cid:3)
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