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Notes on Locally Cartesian Closed Categories

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Notes on Locally Cartesian Closed Categories Notes on Locally Cartesian Closed Categories Sina Hazratpour 13/05/2019 Contents 1 Locally cartesian closed categories1 1.1 The enrichment aspect................................ 14 1.2 Locally cartesian closed structures on the presheaf categories.......... 14 1 Locally cartesian closed categories We begin by having some insights into slice and comma categories. Construction 1.1. Suppose C is category. For an object I of C the slice category C=I is defined as comma category (IdC X): we denote the objects of C=I by tuples (X; p: X I). Proposition 1.2. 1. The monomorphisms is the slice category C=I are exactly the monomor- # ! phisms in C. 2. A morphism e:(Y; q) (X; p) is an epimorphisms is the slice category C=I if it is an epimor- phism in the category C. However, the converse is not true. Therefore, the forgetful functor dom C=I C does not! preserve epimorphisms, and hence colimits. Proof. 1. Suppose m:(A; f) (B; g) is a monomorphism in the slice category C=I. Assume ! a; a0 : X A such that ma = ma0. Therefore, fa = gma = gma0 = fa0. Hence, a; a0 :(X; fa) (A; f) is a! morphism in C=I with ma = ma0 and by our supposition we have a =!a0. The converse is obvious. ! 2. The implication is obvious. A counter-example for the converse statement is constructed as follows: Consider the category C generated by the following directed graph γ f A B C g β I subject to the equation f ◦ γ = g ◦ γ. Obviously γ is an epimorphism in C=I =∼ f0 1g but not in C. ! 1 The situation for cartesian categories is better due to the following observation: Proposition 1.3. Suppose I is an object of category C in which all binary products I × A exists for all objects A of C. The forgetful functor dom: E =I E preserves and creates colimits, and therefore it preserves and creates epimorphisms. ! Proof. right adjoint. Recall that Definition 1.4. A category C is • cartesian if C has all finite products (including the empty product which is the terminal object). • locally cartesian if C has a terminal object 1 and moreover, for any object A of C, the slice category C=A is cartesian. • cartesian closed if C is cartesian and moreover, C has all exponential, i.e. for every object A of C, the functor (-) × A: C C has a right adjoint. • locally cartesian closed if C has! a terminal object 1 and moreover, for any object A of C, the slice category C=A is cartesian closed. Proposition 1.5. 1. Any locally cartesian category is cartesian. 2. Any locally cartesian category has all finite limits. 3. Any locally cartesian closed category is cartesian closed and therefore cartesian as well. 4. Any locally cartesian closed category is locally cartesian, and therefore has all finite limits. 5. The slices of a locally cartesian category are locally cartesian. 6. The slices of a locally cartesian closed category are locally cartesian closed. Proof. For (1) and (3) use the isomorphism C=1 =∼ C of categories to transport the involved structure. For (2), notice that products in slices are pullbacks in the underlying category. (4) follows from the fact that every cartesian closed category, by definition, is cartesian. For (5) and (6), notice that for any objectd I of C, the category C=I has a terminal object, namely (I; id: I I), and furthermore, for every object (X; p: X I) of C=I, we have the isomorphism of categories in below. ! ! (C=I)=(X; p) =∼ C=X ((X0; p0); u:(X0; p0) (X; p)) 7- (X0; u: X0 X) (1) h: ((X00; p00); u0) ((X0; p0); u) 7- h:(X00; u0) (X0; u) ! ! ! ! ! ! 2 From the theorem above we have the following embeddings of 2-categories: Catlcc Catcc Catlc Catc Here Catlcc is the 2-category of locally cartesian closed categories, Catcc is the 2-category of cartesian closed categories, Catlc is the 2-category of locally cartesian categories, and Catc is the 2-category of cartesian categories. In each case the morphisms are the functors which preserve the appropriate structures. Suppose C is a locally cartesian category and consider the self-indexing pseudo-functor C: Cop Cat associated to C: it takes an object I of C to the slice category C=I and takes a morphism f: J I of C to the functor ∆f : C=I C=J where ∆f is given by the pullback along ! op f (aka reindexing along f). Notice that C: C Cat factors through Catc Cat since each slice is cartesian! and moreover, the reindexing! preserves finite limits. More is true: C factors through Catlc because of part (5) of theorem (1.5!). Remark 1.6. Given a locally cartesian closes category C, the self-indexing pseudo-category op C: C Cat factors through Catlcc Cat. This is because the reindexing functors preserve the exponentials in the slices. To see this, first note that[ ♠1:♠] ! Example 1.7. The category of small categories is an example of a cartesian closed category which is not locally cartesian closed. To see why, for a natural number m, let [m] be the standard m-simplex considered as a category, i.e. [m] has m objects 0; 1; : : : ; m and has exactly one arrow between any two objects (i.e. it is a linear order). ∗ Consider the inclusion [1] [2] fixing the endpoints. Now, observe that p : Cat =[2] Cat =[1] can not have a right adjoint since it does not preserve colimits. To see the latter, notice that the following pushout square in Cat =[2] ! ([0]; f1g) 1 ([1]; [0; 1]) 0 y [0≤1] ([1]; [1; 2]) ([2]; id) [1≤2] once pulled back to Cat =[1] yields the diagram ([0]; f1g) 1 ([1]; [0; 1]) 0 y [0≤1] ([1]; [1; 2]) ([2]; id) [1≤2] which clearly is not a pushout. Construction 1.8. Here we show that how to construct 1. the re-indexing from the binary cartesian products in slices, 3 2. the dependent products from the exponentials in slices. For (1), define (∆f(X); f × p) , (X; p) × (J; f) in C=I, and let the C-morphisms ∆f(p): f × p f and "p : f × p p be the product projection. Now, the universal property of (∆f(X); f × p) says exactly that the diagram below is a pullback square in C. ! ! "f(p) ∆f(X) X p ∆f(p) p (2) J I f Let’s take a moment to see "f(p) in terms of maps of families indexed by I: indeed, "f(p) ( X j i : I) ≡ ( X j i : I) ≡ (J × X j i : I) corresponds to the map of families j:Ji f(j) j:Ji i i i (X j i : I) (j; x) x i which projects a pair toP its second coordinateP . According to the table (1), we think of the dependent product Πf(Y) as the family! ( Y j i : I) j:Ji j . Therefore, we can identify an element of this dependent product by an assignment Ji Yj whose first coordinate assignment is identity on Ji, i.e. a map Q j:Ji f Σ (q) = f ◦ q C=I q J f inP whose composite with is the identity morphism on . Therefore, for! (2), given the left hand side diagram in C, we obtain Πf(q) in the category C=I!, as shown in below on the right hand side. if(q) (J;f) Yj j i : I (Π Y; Π (q)) (Y; Σ q) j : Ji Y f f f Yj j i : I Ji j : Ji P q p qf = Q p J I (J;f) f (I; 1I) (J; f) (1 j i : I) (JJi j i : I) idcf i idcf (3) ∼ (J;f) where idcf is the transpose of (I; 1I)×(J; f) = (J; f) through the adjunction (-)×(J; f) a (-) . Remark 1.9. It is easy to see that the construction above is functorial w.r.t. q. Alternatively, the functoriality of Πf can be seen an special case of Proposition in below. Remark 1.10. For the unique morphism !I : I 1 to the terminal object, we write ∆I for ∆f. ∼ We have the natural isomorphism ΣI ◦ ∆I = (-) × I of functors. We note that the functor ∆I is not full, since! even between two constant families of spaces there are usually many non-constant I-indexed families of maps: In fact, ∼ ∼ X C=I(∆I(X); ∆I(Y)) = C(X × I; Y) = C(I; Y ) The table below provides the internal familial version of the operations Σ, ∆, Π in a locally cartesian closed category. 4 Internal (familial) logic of C Category C (Xi j i : I)(X; p: X I) 2 C=I ! (fi : Xi Yi j i : I)(X; p) (Y; q) 2 C=I ! ! (xi : Xi j i : I) (section) x:(I; 1I) (X; p) 2 C=I (Y; q: Y J) 2 C=J (J; f: J I) 2 C=I ! f: J I (Yj j j : J) , ( j:J Yj j i : I) i (Y; f ◦ q: Y I) 2 C=I P ! P ! ! p∗f ∆f(X) X ! ∆ (X j i : I) (X j j : J) p (f: J I) i , f(j) ∆f(p) p ! J I f Y Πf(Y) (Y j j : J) ( Y j i : I) q f: J I j , j:Ji j Πf(q) Q ! Q J I f Parametric equations Commutativities in C ∆g(W) Σv∆gW ∗ p∗g Σu(p) f B ≡ (Ai × Ji j i : I) ∆g(p) W ΣuW ∼ Wa = Wg(b) v Σ (p) a2A B J u Xf(j) bX2B(j) p f g A u I ∆g(W) Πv∆gW ∗ p∗g Πu(p) f B ≡ (Ai × Ji j i : I) ∆g(p) W ΠuW ∼ Wa = Wg(b) v Π (p) a2A b B(j) B J u Yf(j) Y2 p f g A u I Table 1: Internal logic of locally cartesian closed categories (Y jj : J) Remark 1.11.
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