nLab
pointed derivator

Context

Homotopy theory

(,1)(\infty,1)-Category theory

Pointed derivators

Idea

A derivator is pointed if it has a zero object in the sense appropriate to a derivator.

Definition

A derivator D:Dia opCatD\colon Dia^{op} \to Cat is pointed, or has a zero object, if each category D(X)D(X) has a zero object.

Note that these zero objects are automatically preserved by all the functors u !,u *,u *u_!,u^*,u_*, since they are all adjoints on one side or the other.

Properties

This simple definition hides a lot of structure.

Relative diagram categories

If DD is a pointed derivator and i:ABi\colon A\hookrightarrow B is a full subcategory, we write D(B,A)D(B,A) for the full subcategory of DBD B on those diagrams XDBX\in D B such that i *Xi^*X is null.

Lemma

If ii is the inclusion of a sieve, then i *:D(A)D(B)i_*\colon D(A) \to D(B) is fully faithful and its essential image is D(B,BA)D(B,B\setminus A). Similarly, if ii is a cosieve, then i !i_! identifies D(A)D(A) with D(B,BA)D(B,B\setminus A).

Proof

Suppose ii is a sieve. Since ii is in particular fully faithful, so is i *i_* by general arguments (see homotopy exact square). Moreover, since ii is a sieve, the square

A i BA j B\array{ \emptyset & \to & A\\ \downarrow & & \downarrow^i\\ B\setminus A & \underset{j}{\to} & B}

is homotopy exact (where jj is the inclusion). Therefore j *i *Xj^* i_* X is terminal for any XD(A)X\in D(A), so that i *i_* lands inside D(B,BA)D(B,B\setminus A). Conversely, if XD(B,BA)X\in D(B,B\setminus A), then the map Xi *i *XX \to i_* i^* X is an isomorphism on AA (since i *i_* is fully faithful) and on BAB\setminus A (since both are terminal there), hence is itself an isomorphism; so XX is in the essential image of i *i_*.

The following theorem is stated as Proposition 7.4 of Heller’s paper “Stable homotopy theories and stabilization”.

Theorem

D(B,A)D(B,A) is a reflective and coreflective subcategory of D(B)D(B).

Let MiM i denote the “codirected mapping cylinder” of i:ABi\colon A \hookrightarrow B, by which we mean the full subcategory of B×IB\times I on the objects (x,0)(x,0) for xBx\in B and (i(x),1)(i(x),1) for xAx\in A (here II is the interval category (01)(0\to 1)). This is also the same as the cocomma object? (Bi)(B \uparrow i). Write BuMipBB \overset{u}{\to} M i \overset{p}{\to} B for the evident inclusion and projection.

Consider the adjunction

D(Mi)p *p !D(B). D(M i) \underoverset{p^*}{p_!}{\rightleftarrows} D(B).

We claim that this restricts to an adjunction

(1)D(Mi,A)p *p !D(B,A) D(M i,A) \underoverset{p^*}{p_!}{\rightleftarrows} D(B,A)

where AA denotes the subcategory of MiM i on the objects (i(x),1)(i(x),1). Certainly p *p^* takes D(B,A)D(B,A) into D(Mi,A)D(M i,A), since i:ABi\colon A \hookrightarrow B is the composite AMipBA \hookrightarrow M i \overset{p}{\to} B. On the other hand, the square

A Mi p A i B\array{A & \hookrightarrow & M i \\ \downarrow & & \downarrow^p\\ A & \underset{i}{\to} & B }

is homotopy exact, so p !p_! takes D(Mi,A)D(M i,A) into D(B,A)D(B,A).

Now since uu is the inclusion of a sieve, by the previous lemma, u *u_* identifies D(B)D(B) with D(Mi,A)D(M i,A). Therefore, the inclusion D(B,A)D(B)D(B,A)\hookrightarrow D(B) can be identified with the restricted right adjoint p *:D(B,A)D(Mi,A)p^*\colon D(B,A) \to D(M i,A), which therefore has a left adjoint p !p_!. Explicitly, the left adjoint is p !u *p_! u_*.

The proof of coreflectivity is of course dual, using the directed mapping cylinder instead of the codirected one.

Note that if i:ABi'\colon A'\hookrightarrow B' is another full inclusion and

A f A i i B g B\array{A & \overset{f}{\to} & A'\\ ^i\downarrow && \downarrow^{i'}\\ B& \underset{g}{\to} & B'}

commutes, then we have a restriction (g,f) *:D(B,A)D(B,A)(g,f)^*\colon D(B',A') \to D(B,A), which has a left adjoint (g,f) !(g,f)_! given by the composite

D(B,A)D(B)g !D(B)L B,AD(B,A) D(B,A) \hookrightarrow D(B) \overset{g_!}{\to} D(B') \overset{L_{B',A'}}{\to} D(B',A')

where L B,A:D(B)D(B,A)L_{B,A}\colon D(B) \to D(B,A) denotes the reflection. It also has a right adjoint defined dually.

Pointed exactness

Define a category with zeros to be a category BB equipped with a full subcategory B 0B_0. By the preceeding remarks, any pointed derivator gives rise to a contravariant pseudofunctor defined on the 2-category of categories with zeros (and functors preserving the specified subcategories), all of whose transition functors have both adjoints. It is natural to look for exactness conditions, analogous to (Der4) and the characterization of homotopy exact squares, which apply to these adjoints.

Since the Beck-Chevalley transformation relating composites of these adjoints is the composite of the corresponding transformation for the unpointed diagram and a Beck-Chevalley transformation relating the restriction functors to the reflections D(B)D(B,B 0)D(B) \to D(B,B_0), we clearly need to study when restriction commutes with these reflections. This is the purpose of the following definition and lemma.

Definition

A functor f:ABf\colon A \to B between categories with zeros is locally null-final if for every aAa\in A, every b 0B 0b_0 \in B_0, and every morphism ϕ:b 0f(a)\phi\colon b_0 \to f(a), the category of triples (a 0A 0,a 0αa,b 0βf(a 0))(a_0\in A_0, a_0 \xrightarrow{\alpha} a, b_0 \xrightarrow{\beta} f(a_0)) such that ϕ=f(α).β\phi = f(\alpha).\beta has a contractible nerve.

Lemma

If f:ABf\colon A \to B is locally null-final and DD is a pointed derivator, then the functors f *:D(B)D(A)f^*\colon D(B) \to D(A) and f *:D(B,B 0)D(A,A 0)f^*\colon D(B,B_0) \to D(A,A_0) commute with the reflections of D(B)D(B) and D(A)D(A) into D(B,B 0)D(B,B_0) and D(A,A 0)D(A,A_0), respectively. In other words, the canonical natural transformation

D(A) f * D(B) L A L B D(A,A 0) f * D(B,B 0) \array{ D(A) & \xleftarrow{f^*} & D(B) \\ ^{L_A}\downarrow & \seArrow & \downarrow^{L_B} \\ D(A,A_0) & \xleftarrow{f^*} & D(B,B_0) }

is an isomorphism.

Proof

Recall that L AL_A is computed as the composite D(A)u *D(M A,A 0)p !D(A,A 0)D(A) \xrightarrow{u_*} D(M_A, A_0) \xrightarrow{p_!} D(A,A_0), where M AM_A is the codirected mapping cylinder of A 0AA_0\to A. We can therefore factor the above square as

D(A) f * D(B) u * u * D(M A,A 0) f * D(M B,B 0) p ! p ! D(A,A 0) f * D(B,B 0)\array{ D(A) & \xleftarrow{f^*} & D(B) \\ ^{u_*}\downarrow & & \downarrow^{u_*} \\ D(M_A,A_0) & \xleftarrow{f^*} & D(M_B,B_0)\\ ^{p_!}\downarrow & & \downarrow^{p_!} \\ D(A,A_0) & \xleftarrow{f^*} & D(B,B_0)}

It suffices, therefore, to show that the squares

A u M A f f B u M BandM A f M B p p A f B\array{A & \xrightarrow{u} & M_A \\ ^f \downarrow & & \downarrow^f \\ B & \xrightarrow{u} & M_B} \qquad\text{and}\qquad \array{M_A & \xrightarrow{f} & M_B\\ ^p\downarrow & & \downarrow^p\\ A & \xrightarrow{f} & B}

are homotopy exact.

For the first square, consider first a bBb\in B and aAM Aa\in A\subset M_A and a φ:f(a)b\varphi\colon f(a)\to b in BM BB\subset M_B. The category of triples (a,aa,f(a)b)(a', a\to a', f(a')\to b) which compose to φ\varphi is contractible, since (a,id a,φ)(a, id_a, \varphi) is an initial object. Second, we should consider a bBb\in B and an a 0A 0M Aa_0\in A_0 \subset M_A, and a φ:f(a 0)b\varphi\colon f(a_0)\to b in B 0M BB_0 \subset M_B — but by definition of M BM_B, no such morphism φ\varphi can exist. Thus, the first square is always homotopy exact.

In fact, however, the preceeding argument is unnecessary, because we have seen that with their targets restricted as above, the functors u *u_* are equivalences, and the mate of an isomorphism with respect to any pair of adjoint equivalences is again an isomorphism. Note that the fact that u *u_* is an equivalence was already used in making the assertion that p !u *p_! u_* is a left adjoint to D(A,A 0)D(A)D(A,A_0) \hookrightarrow D(A), and therefore the inverse of the top square already factors into the definition of the transformation appearing in the statement of the lemma. (This may reassure a reader who was worried about the fact that the canonical transformation in the first square goes the “wrong direction”.)

For the second square, consider first an aAa\in A and a bBM Bb\in B\subset M_B, and a φ:bf(a)\varphi\colon b \to f(a). The category we must investigate containts two types of objects:

  • triples (aA,bf(a),aa)(a'\in A, b\to f(a'), a' \to a) which compose to φ\varphi, and
  • triples (a 0A 0,bf(a 0),a 0a)(a_0 \in A_0, b\to f(a_0), a_0 \to a) which compose to φ\varphi.

By definition of M AM_A, the morphisms between these objects are the obvious ones, except that there are no morphisms from the second type to the first. Now the full subcategory on the first type of object is coreflective, since A 0A_0 is coreflective in M AM_A. And that full subcategory is contractible, since (a,φ,id a)(a, \varphi, id_a) is a terminal object. Thus, since adjunctions induce homotopy equivalences of nerves, the category in question is also contractible.

Finally, consider the case of an aAa\in A and a b 0B 0M Bb_0\in B_0 \subset M_B, and a φ:b 0f(a)\varphi\colon b_0 \to f(a). The category in question consists of triples (a 0A 0,b 0f(a 0),a 0a)(a_0\in A_0, b_0 \to f(a_0), a_0\to a) which compose to φ\varphi (since there are no morphisms in M BM_B from b 0B 0b_0\in B_0 to anything in the image of AM AA\subset M_A). But this is precisely the category asserted to be contractible in the assumption that ff is locally null-final.

Theorem

If a square

I f J h k K g L\array{ I & \xrightarrow{f} & J \\ ^h\downarrow & \swArrow & \downarrow^k\\ K & \xrightarrow{g} & L}

of categories with zeros has the properties that

  1. it is homotopy exact as a square of categories, when the full subcategories are ignored, and

  2. gg is locally null-final,

then for any pointed derivator, the induced transformation

D(I,I 0) f * D(J,J 0) h ! k ! D(K,K 0) g * D(L,L 0)\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\ ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\ D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}

is an isomorphism.

Proof

By definition, the displayed functors h !h_! and k !k_! are obtained by applying the h !h_! and k !k_! of a derivator followed by the reflection into the relative diagram categories. Thus the given square factors into

D(I,I 0) f * D(J,J 0) h ! k ! D(K) g * D(L) L K L L D(K,K 0) g * D(L,L 0)\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\ ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\ D(K) & \xleftarrow{g^*} & D(L)\\ ^{L_K}\downarrow & \seArrow & \downarrow^{L_L}\\ D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}

in which the first square is an isomorphism by the first assumption, and the second square by the second assumption and Lemma 2.

For example, we can conclude:

Corollary

If f:ABf\colon A\to B is a fully faithful and locally null-final functor between categories with zeros, then for any pointed derivator DD, the functor f !:D(A,A 0)D(B,B 0)f_!\colon D(A,A_0) \to D(B,B_0) is fully faithful.

Proof

The square

A id A id f A f B\array{ A & \xrightarrow{id} & A \\ ^{id}\downarrow & \swArrow & \downarrow^f\\ A & \xrightarrow{f} & B}

satisfies the hypotheses of the previous theorem.

Theorem 2 is not best possible, however. Its two conditions characterize when the two squares into which the Beck-Chevalley transformation factors are separately isomorphisms. However, it might happen that the composite is an isomorphism even though one or the other of the transformations is not separately an isomorphism.

In particular, the condition of “local null-finality” on the bottom morphism gg uses no information about the categories II and JJ. If we know some things about them, then we can correspondingly weaken this condition.

Theorem

Suppose given a square

I f J h k K g L\array{ I & \xrightarrow{f} & J \\ ^h\downarrow & \swArrow & \downarrow^k\\ K & \xrightarrow{g} & L}

of categories with zeros, where LL is equipped with a full subcategory L^ 0L 0\hat{L}_0 \subseteq L_0 of its category of zeros. Write Lˇ 0=L 0L^ 0\check{L}_0 = L_0 \setminus \hat{L}_0, and suppose the following.

  1. The square is homotopy exact as a square of categories, when the categories of zeros are ignored.

  2. For any pointed derivator DD, the functor k !:D(J)D(L)k_!\colon D(J) \to D(L) maps D(J,J 0)D(J,J_0) into D(L,L^ 0)D(L,\hat{L}_0). For instance, this is the case if for each zL^ 0z\in \hat{L}_0, the category k/zk/z has a terminal object lying in J 0J_0.

  3. For each xLL 0x\in L\setminus L_0 and yL^ 0y\in \hat{L}_0 and morphism xφyx\xrightarrow{\varphi} y in LL, the following category has a contractible nerve: its objects are triples (zL 0,xαz,zβy)(z\in L_0, x\xrightarrow{\alpha} z, z\xrightarrow{\beta} y) such that βα=φ\beta\alpha=\varphi, and its morphisms are morphisms z 1z 2z_1\to z_2 in LL commuting with the given morphisms and such that either z 1L^ 0z_1 \in \hat{L}_0 or z 2L^ 0z_2 \notin \hat{L}_0.

  4. For every xLˇ 0x\in \check{L}_0 and yKy\in K and xφg(y)x \xrightarrow{\varphi} g(y) in LL, the category of triples (zK 0g 1(Lˇ 0),xαg(z),zβy)(z\in K_0 \cap g^{-1}(\check{L}_0), x\xrightarrow{\alpha} g(z), z \xrightarrow{\beta} y) such that g(β).α=φg(\beta).\alpha= \varphi has a contractible nerve (“relative local null-finality”).

Then for any pointed derivator, the induced transformation

D(I,I 0) f * D(J,J 0) h ! k ! D(K,K 0) g * D(L,L 0)\array{ D(I,I_0) & \xleftarrow{f^*} & D(J,J_0) \\ ^{h_!}\downarrow & \seArrow & \downarrow^{k_!}\\ D(K,K_0) & \xleftarrow{g^*} & D(L,L_0)}

is an isomorphism.

Proof

(Sketch) Because of the second condition in the theorem, instead of the reflection D(L)D(L,L 0)D(L) \to D(L,L_0) it will suffice to consider the reflection D(L,L 0^)D(L,L 0)D(L,\hat{L_0}) \to D(L,L_0). We use the third condition to give an alternate way to compute this reflection, and the fourth condition to ensure that restriction along gg commutes with this reflection.

Let M^ L\hat{M}_L be the codirected mapping cylinder of Lˇ 0L\check{L}_0 \hookrightarrow L. If LuM^ LpLL \xrightarrow{u} \hat{M}_L \xrightarrow{p} L are as before, then u *u_* identifies D(L,L^ 0)D(L,\hat{L}_0) with the subcategory of D(M^ L)D(\hat{M}_L) consisting of diagrams which are zero on the copy of Lˇ l\check{L}_l that is the target end of the mapping cylinder and on the copy of L^ 0\hat{L}_0 inside the copy of LL that is the source end; call this subcategory D(M^ L,L^ 0Lˇ 0)D(\hat{M}_L, \hat{L}_0 \cup \check{L}_0).

Now if the adjunction p !:D(M^ L)D(L):p *p_! \colon D(\hat{M}_L) \rightleftarrows D(L) \;: p^* restricts to an adjunction D(M^ L,L^ 0Lˇ 0)D(L,L 0)D(\hat{M}_L, \hat{L}_0 \cup \check{L}_0)\rightleftarrows D(L,L_0), we will be able to compute the reflection D(L,L^ 0)D(L,L 0)D(L,\hat{L}_0) \to D(L,L_0) as p !u *p_! u_*, as before. This will follow if the following square is exact:

L^ 0Lˇ 0 M^ L L 0 L\array{ \hat{L}_0 \cup \check{L}_0 & \to & \hat{M}_L \\ \downarrow & & \downarrow \\ L_0 & \to & L }

where L^ 0Lˇ 0\hat{L}_0 \cup \check{L}_0 denotes the same full subcategory of M^ L\hat{M}_L as above. Exactness of this square can be verified to be equivalent to the third condition in the theorem.

Finally, modifying the proof of Lemma 2, we see that the fourth condition in the theorem implies commutativity of g *g^* with this reflection p !u *p_! u_*. Therefore, the theorem follows as before.

If L^ 0=\hat{L}_0 = \emptyset, then the second and third conditions of Theorem 3 are vacuous and it reduces to Theorem 2. At the other extreme, if L^ 0=L 0\hat{L}_0 = L_0, then the second and fourth conditions of Theorem 3 are vacuous and the third becomes “k !k_! maps D(J,J 0)D(J,J_0) into D(L,L 0)D(L,L_0).”

Extraordinary inverse images

Theorem

A derivator DD is pointed if and only if whenever u:ABu\colon A\to B is a sieve in DiaDia, the functor u *:D(A)D(B)u_* \colon D(A) \to D(B) has a right adjoint u !u^!, and dually whenever uu is a cosieve, the functor u !u_! has a left adjoint u ?u^?.

The “if” direction of this is easy, since u:Bu\colon \emptyset \to B is always a sieve, while u *:*U(B)u_*\colon * \to U(B) assigns the terminal object of D(B)D(B). Thus, if u *u_* has a further right adjoint, it must preserve colimits, and in particular its value on the unique object of D()D(\emptyset) must be an initial object (as well as a terminal one).

Conversely, if DD is pointed and uu is a sieve, then by Lemma 1, u *u_* identifies D(A)D(A) with D(B,BA)D(B,B\setminus A). But by Theorem 1, the inclusion D(B,BA)D(B)D(B,B\setminus A)\hookrightarrow D(B) has a right adjoint. The other case is dual.

The functors u !u^! and u ?u^? are sometimes referred to as extraordinary and co-extraordinary inverse image functors. Our proof shows that u ?=u *p !v *u^? = u^* p_! v_*, where vv and pp are respectively the inclusion of BB in the codirected mapping cylinder of BABB\setminus A \hookrightarrow B, and its projection to BB.

In the literature, the existence of these functors is often taken as the definition of when a derivator is pointed. The equivalence of the two definitions is a “super-difficult” excercise in Maltsionitis’ notes.

Loop and suspension

In a pointed derivator, we have a suspension functor Σ:D1D1\Sigma\colon D 1 \to D 1 defined as the composite

D1a *DΓabc !Dd *D1 D 1 \overset{a_*}{\to} D \Gamma \overset{abc_!}{\to} D \square \overset{d^*}{\to} D 1

where \square denotes the category

a b c d \array{ a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d }

and Γ\Gamma its full subcategory on {a,b,c}\{a,b,c\}.

Equivalently, the suspension can be defined as the composite

D1=D(1,)(a,) !D(,bc)(d,) *D(1,)=D1. D 1 = D(1,\emptyset) \overset{(a,\emptyset)_!}{\to} D(\square,bc) \overset{(d,\emptyset)^*}{\to} D(1,\emptyset) = D 1.

This follows because we have D1D(Γ,bc)D 1 \simeq D(\Gamma,bc), and thus (a,) !(a,\emptyset)_! is isomorphic to (abc,bc) !:D(Γ,bc)D(,bc)(abc,bc)_!\colon D(\Gamma,bc) \to D(\square,bc), which by definition is L ,bcabc !L_{\square,bc} \circ abc_!. But abc !abc_! is fully faithful since abc:Γabc\colon \Gamma \hookrightarrow \square is, and so it already takes D(Γ,bc)D(\Gamma,bc) into D(,bc)D(\square,bc). Thus L ,bcL_{\square,bc} is doing nothing, so (a,) !(a,\emptyset)_! is essentially just abc !abc_! which appeared in our previous definition of Σ\Sigma. (The other functor a *a_* essentially implements the equivalence D1D(Γ,bc)D 1 \simeq D(\Gamma,bc).)

Similarly, we have a loop space object functor Ω:D1D1\Omega\colon D 1 \to D 1 defined as the composite

D1d !DΓ opbcd *Da *D1. D 1 \overset{d_!}{\to} D \Gamma^{op} \overset{bcd_*}{\to} D \square \overset{a^*}{\to} D 1.

where Γ op\Gamma^{op} is the full subcategory of \square on {b,c,d}\{b,c,d\}, and it can equivalently be given as the composite

D1=D(1,)(d,) *D(,bc)(a,) *D(1,)=D1. D 1 = D(1,\emptyset) \overset{(d,\emptyset)_*}{\to} D(\square,bc) \overset{(a,\emptyset)^*}{\to} D(1,\emptyset) = D 1.

The description in terms of relative diagram categories makes it clear that ΣΩ\Sigma \dashv \Omega.

We can also describe Σ\Sigma and Ω\Omega in terms of the extraordinary inverse image functors. Since the square

Γ Γ r abc * d \array{ \Gamma & \to & \Gamma \\ ^r\downarrow & & \downarrow^{abc}\\ * & \underset{d}{\to} & \square}

is homotopy exact, the composite d *abc !d^* abc_! in our first definition is the same as r !r_!. The other functor a *a_* can in turn be decomposed as a composite ab *s *ab_* s_*, where s:*Is\colon * \to I is the inclusion of ss into the interval category (st)(s\to t), and ab:IΓab\colon I\to \Gamma is the inclusion into {a,b}\{a,b\}.

Therefore we have Σr !ab *s *\Sigma \cong r_! ab_* s_*. But Γ\Gamma can be identified with MsM s and abab with the inclusion u:IMsu\colon I \hookrightarrow M s, while rr factors as MspIq*M s \overset{p}{\to} I \overset{q}{\to} * so that r !q !p !r_! \cong q_! p_!. But finally, the square

* t I q * *\array{* & \overset{t}{\to} & I\\ \downarrow & & \downarrow^q\\ *& \to & *}

is homotopy exact, so q !t *q_! \cong t^* and thus

Σt *p !u *s *t ?s *.\Sigma \cong t^* p_! u_* s_* \cong t^? s_*.

Dually, we can identify Ωs !t !\Omega \cong s^! t_!, from which it again follows directly that ΣΩ\Sigma \dashv \Omega.

Enrichment over pointed sets

Every pointed derivator can be canonically “enriched” over pointed sets, in the sense that we can extend it to a functor

D:Set *CatSet *CATD': \; Set_* Cat \; \to\; Set_* CAT

from small Set *Set_*-enriched categories to large ones, satisfying analogues of the derivator axioms. Specifically, if CC is a Set *Set_*-category, define C¯\bar{C} to be CC with a zero object adjoined. (Note that zero objects are absolute colimits in Set *Set_*-categories.) Then set D(C)=D(C¯,0)D'(C) = D(\bar{C},0), the “relative diagram category” as defined above, i.e. the full subcategory of D(C¯)D(\bar{C}) on those diagrams which send the zero object to zero.

Any Set *Set_*-functor automatically preserves zero objects (since an object xx of a Set *Set_*-category is zero iff its identity id xid_x is the basepoint of the pointed homset hom(x,x)hom(x,x)). In particular, any Set *Set_*-functor u:ABu:A\to B induces a relative functor (u,0):(A¯,0)(B¯,0)(u,0):(\bar{A},0) \to (\bar{B},0), and hence a pullback u *=(u,0) *:D(B)D(A)u^* = (u,0)^*:D'(B) \to D'(A). According to the general theory of relative diagram categories, this functor has left and right adjoints u !u_! and u *u_*, obtained from left and right extension along uu followed by reflection or coreflection from D(B¯)D(\bar{B}) into D(B¯,0)D(\bar{B},0). In fact, however, in this case the reflection/coreflection is unnecessary: since the comma category (u,0)/0(u,0)/0 has a terminal object 00, the functor u !u_! already maps D(A¯,0)D(\bar{A},0) into D(B¯,0)D(\bar{B},0).

Thus we have a Set *Set_*-analogue of the derivator axiom (Der3), existence of adjoints. For the enriched axiom (Der2), conservativity, observe that if II is the unit Set *Set_*-category having one object xx with hom(x,x)=S 0={1 x,0}hom(x,x) = S^0 = \{1_x, 0\}, then D(I)D(*)D'(I) \simeq D(*). So the family of Set *Set_* functors IAI \to A for any Set *Set_*-category AA gives rise a family *A¯*\to \bar{A} picking out the nonzero objects, but since any two zero objects are isomorphic, (Der2) for DD implies that the resulting family of functors D(A)D(I)D'(A) \to D'(I) are jointly conservative.

We can also conclude a version of the axiom (Der4) about exact squares from the above theorems about exactness for categories with zeros. Namely, if

A B C D\array{ A & \to & B \\ \downarrow & \swArrow & \downarrow \\ C & \to & D }

is a comma square of Set *Set_*-categories, then

A¯ B¯ C¯ D¯\array{ \bar{A} & \to & \bar{B} \\ \downarrow & \swArrow & \downarrow \\ \bar{C} & \to & \bar{D} }

is homotopy exact when considered as a square of ordinary categories. Moreover, the functor C¯D¯\bar{C} \to \bar{D} is always locally null-final. Thus, by Theorem 2, our “Set *Set_*-enriched derivator” satisfies the Beck-Chevalley condition for any comma square of Set *Set_*-categories.

It remains to consider the axiom (Der1) regarding coproducts. The coproduct “ABA\vee B” of two Set *Set_*-categories AA and BB, as Set *Set_*-categories, is not quite a “disjoint” union, but rather includes zero morphisms in both directions from each category to the other. The inclusions i,ji,j of AA and BB are fully faithful, however, and the induced functors i¯:A¯AB¯\bar{i}\colon \bar{A}\to \overline{A\vee B} and j¯:B¯AB¯\bar{j}\colon \bar{B}\to \overline{A\vee B} are locally null-final; hence the left extensions i !:D(A)D(AB)i_! \colon D'(A) \to D'(A\vee B) and j !:D(B)D(AB)j_!\colon D'(B) \to D'(A\vee B) are also fully faithful.

By exactness with zeros, we can show that j *i !j^* i_! and i *j !i^* j_! send everything to zero. Therefore, if we define a functor (i,j) !:D(A)×D(B)D(AB)(i,j)_!\colon D'(A) \times D'(B) \to D'(A\vee B) by (i,j) !(X,Y)=(i !X+j !Y)(i,j)_!(X,Y) = (i_! X + j_! Y), then we have (i *,j *)(i,j) !Id(i^*, j^*)(i,j)_! \cong Id. On the other hand, by fully-faithfulness of i !i_! and j !j_!, for any ZD(AB)Z\in D'(A\vee B) the map i !i *ZZi_! i^* Z \to Z is an isomorphism on objects of AA, and similarly for jj and BB; hence (i,j) !(i *,j *)Id(i,j)_! (i^*,j^*) \cong Id as well. In particular, (i *,j *)(i^*,j^*) is an equivalence of categories, providing the Set *Set_*-enriched version of (Der1).

What about (Der5)?

Conversely, given a functor DD' as above satisfying the axioms as given above, we can define D(A)=D(A)D(A) = D'(A'), where AA' denotes the free Set *Set_*-category on an ordinary category AA (i.e. adjoin a new zero morphism between each pair of objects). This DD will automatically satisfy axioms (Der3) (homotopy Kan extensions) and (Der2) (conservativity on objects) since DD' does, and it satisfies (Der1) on coproducts since DD' does and since (A+B)A+B(A+B)' \cong A' + B'.

Does (Der4) on Beck-Chevalley conditions carry back over? Does this require a “pointed version of Cisinski’s theorem”?

Finally, of course each category D(A)=D(A)D(A) = D'(A') has a zero object, so DD is a pointed derivator. Thus, it seems that Set *Set_*-enrichment is an essentially equivalent way to express the notion of pointed derivator. (This approach was used by Franke (see below).)

However, what is not immediately clear is that passing from DD to DD' and back to DD again gives the same derivator. At first sight this seems to require a stronger theorem about homotopy exactness with zeros than is stated above, one in which the square need not be homotopy exact without zeros.

The pointed reflection

Every derivator can be made pointed in a universal way; given DD we define D *(A)D_*(A) to be the full subcategory of D(A×I)D(A\times I) which is terminal when restricted along 0:AA×I0\colon A\to A\times I. It requires a little work to show that this is a derivator. The main observation being that the inclusion D *(A)D(A×I)D_*(A) \hookrightarrow D(A\times I) has a left adjoint (the “mapping cone”), which can be constructed just as in the proof of Theorem 1. See III.5 of Heller’s memoir.

References

See derivator for general references. The pointed reflection is discussed in III.5 of

The definition of relative diagram categories is taken from:

The Set *Set_*-enriched approach is taken as basic in

  • Jens Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, K-theory archive

See also

  • Moritz Groth, Derivators, pointed derivators, and stable derivators (pdf)

Revised on September 28, 2012 17:41:51 by Tim Porter (95.147.236.176)