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CATEGORICAL ASPECTS of TORIC TOPOLOGY
Nigel Ray nige@ma.man.ac.uk Scho ol of Mathematics University of Manchester Manchester M13 9PL

Includes joint work with Victor Buchstab er, Dietrich Notb ohm, Taras Panov, Rainer Vogt

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OVERVIEW
Aims: (i) to describ e categorical asp ects of toric objects, and (ii) to give examples of useful calculations in this framework.

1. THE CATEGORICAL VIEWPOINT

2. TORIC OBJECTS

3. HOMOTOPY THEORY

4. FORMALITY

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This is why we are all here ..
Z
K T m-n

2 2 M1 n , . . . , M k n

T

n

Pn Lower quotients are strict; P n = Cone(K ).

Z

K T

m-n

2 2 M1 n , . . . , M k n

T

n

DJ (K ) Lower quotients are homotopy quotients.
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1. THE CATEGORICAL VIEWPOINT Some p eople love categories, and others hate them; but they are here to stay! A category c has objects X , and a set of morphisms c(X, Y ) b etween every pair of objects. Some categories are large, such as top, the category of top ological spaces and continuous maps. Others are small (finite, even!), such as the category cat(K ) of faces of a simplicial complex K and their inclusions. Functors are morphisms b etween categories, such as the singular co chain algebra functor C (-; R) : top - dgaR over a nice ring R.
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Toric Top ology exists within two categorical frameworks, which may seem indep endent . . . but they are deeply intertwined!

(i) Lo cal: many toric spaces admit natural decomp ositions into simpler subspaces; and these are often indexed by small categories such as cat(K ).

(ii) Global: as problems vary, our spaces may lie in the category of smo oth manifolds and diffeomorphisms; or CW-complexes and homotopy classes of maps; or . . . .

The lo cal viewp oint considers toric spaces as diagrams, whereas the global viewp oint interprets their invariants as functors from geometric to algebraic categories.
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As well as cat(K ), we like the small category , with objects (n) = {0, 1, . . . , n} for n 0, and morphisms the non-decreasing maps. We denote their opp osites by catop and op. We like geometric categories such as top+ : p ointed top ological spaces tmon : top ological monoids. We also like algebraic categories such as dgaR : differential graded algebras cdgaQ : commutative dgas dgcQ : differential graded colagebras, usually with (co)augmentations. Differentials go down in dga and dgc, and up in cdga.

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Given a small indexing category a, we may view diagrams in c as functors D : a c; then the collection of all such diagrams also forms a category [a, c]. If a is , then [, c] and [op, c] are the categories of cosimplicial and simplicial objects in c, often denoted by cc and sc resp ectively. For example: (i) the cosimplicial simplex · : - top maps (n) to the standard n-simplex n ; (ii) the singular chain complex C·(X ) : op - sset maps (n) to the set of continuous functions f : n X for any space X .
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In nice categories, pushouts and pullbacks are universal objects arising from diagrams on {1} - - {2} and {1} - - {2};

these are cat(· ·) and catop(· ·) resp ectively! In tmon, the pushout of the diagram T1 - {1} - T2 of circles is the free pro duct T1 T2 T1 â T2; in top+, the pushout of the diagram B T1 - - B T2 of classifying spaces is B T1 B T2 B T â B T . Copro ducts (or sums) are sp ecial cases of pushouts, which are themselves examples of colimits of arbitrary diagrams in a category. Similarly, pro ducts are sp ecial cases of pullbacks, which are examples of limits.
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2. TORIC OBJECTS We start with a simplicial complex K on vertices V = {v1, . . . , vm}, and construct two asso ciated top ological spaces:

· the Davis-Januszkiewicz space DJ (K ) · the moment-angle complex ZK .

The top ologists amongst us are interested in their prop erties up to homotopy equivalence, so we have some freedom in making the constructions.

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The vertices determine · an m-torus T V · its classifying space B T V For any face V of K , there is · a co ordinate subtorus T T V · its classifying space B T B T V · the space D = (D2) â T V \ . So there are diagrams · T K : cat(K ) - tmon · B T K : cat(K ) - top+ · DK : cat(K ) - top+ which map an inclusion of faces to · the monomorphism T T · the inclusion B T B T · the inclusion D D resp ectively.
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(C P )V .


To construct our first toric spaces, we take colimits of diagrams. We obtain colimtmon T K = Cir (K (1)) as top ological groups; and colimtop+ B T K =
K

B T = DJ (K )

and colimtop+ DK =
K

D = ZK

as p ointed top ological spaces. We may also define a diagram T V \K by mapping to the projection T V \ - T V \ . In this case, colimtop+ T V \K = {1} is a single p oint.
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For algebraic purp oses, we write the vertices v1,. . . , vm as 2-dimensional variables; their desusp ensions u1, . . . , um are 1-dimensional. In either case, we denote the commutative monomials wi by w, for any multiset : V N. The symmetric algebra SR (V ) is p olynomial over R, with basis elements v. The symmetric algebra R (U ) is exterior, with basis elements u for genuine subsets U . With d = 0, b oth are objects of cdga; and so is (U ) S ( ), with dui = vi for all vi . The graded duals SR (V ) and R (U ) have dual basis elements v and u over R. In either case, their copro ducts satisfy (w ) =
1 2 =

w 1 w

2

With d = 0, b oth are objects of cdgc.
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We can define a diagram cat(K ) dga by: · K maps to the monomorphism ( ) - ( ), and diagrams catop(K ) cdga by: · SK maps to the epimorphism S ( ) - S ( ), · SK maps to the epimorphism (U ) S ( ) - (U ) S ( ). . . . and a diagram cat(K ) cdgc by: · (S K ) maps to the monomorphism S ( ) - S ( ) .
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To define algebraic toric objects, we consider colimdgaK = T (u1, . . . , where I = u2, [ui, h also limcdga SK = R[K ], the Stanley-Reisner algebra of K ; and limcdga SK = ( R[K ], d), where dui = vi for 1 i m; and colimcdgc (S K ) = R K , the Stanley-Reisner coalgebra of K . We can also define a diagram U \K by mapping to the monomorphism (U \ ) - (U \ ). In this case, limcdga U \K = R is simply the ground ring in dimension 0.
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um ) / I , uj ] : h, {i, j } K ;


3. HOMOTOPY THEORY Classical homotopy theory do es not interact well with limits and colimits! Taking colimits in top+, we have that colim DK = ZK and colim T V \K = {1}

However, the projections D = (D2) â T V \ - T V \ , induce a morphism DK T V \K , which is a homotopy equivalence for each face of K . The simplest example of this case is P 1 = 1, so K = · ·. Then DK is the pushout diagram
2 2 T1 â D2 - T1 â T2 - D1 â T2, and colim DK = S 3.

But T V \K is the pushout T1 - T1 â T2 - T2, and colim T V \K = {1}.
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Algebraically, we take limits in cdga and find: lim SK = R[K ] and lim U \K = R However, the monomorphisms (U \ ) - (U ) S ( ) induce a morphism U \K SK , which is a quasi-isomorphism for each face of K . Both have cohomology (U \ ). Again, the simplest example is K = · ·, for which SK is the pullback diagram (u1, u2) S (v2) - (u1, u2) - (u1, u2) S (v1), and lim SK = (u1v2). But U \K is the pullback (u1) - (u1, u2) - (u2), and lim U \K = R.
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In b oth geometric and algebraic contexts, we learn that objectwise weak equivalences do not preserve colimits or limits. In order to understand this situation prop erly, we follow Quillen's inspired ideas for axiomatising categories in which we can "do homotopy theory". This is the world of mo del category theory. In any such category, three classes of sp ecial morphism are defined; the fibrations, the cofibrations, and the weak equivalences. They ob ey axioms that are suggested by the prop erties of top, and allow us to pass to a homotopy category, where the weak equivalences are invertible. The b eauty of the axioms is that many algebraic categories also admit natural mo del structures, as well as more obvious geometric examples such as top+ and sset.
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Given a mo del category mc and a nice indexing category a, the category [a, mc] admits a canonical mo del structure, and a weak equivalence of diagrams is an objectwise weak equivalence. Recent results show that lim and colim may always b e replaced by more subtle functors holim and ho colim : [a, mc] mc. To construct ho colimmc D for any diagram D : a mc we: (i) replace the objects D(a) by nicer D (a) (ii) replace the diagram D by nicer D (iii) form colimmc D . Then ho colimmc D is preserved (up to weak equivalence in mc) by weak equivalences of diagrams; and by certain functors which do not preserve colim.
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Some diagrams are given in the form D , so lim and colim are weakly equivalent to holim and ho colim. Examples are B T K and SK , for which there is also an isomorphism H (colimtop+ B T K ; R) = limcdga SK . This is b etter known as H (DJ (K ); R) = R[K ] ! The source of our weak equivalence DK T V \K is of the form D , but the target is not. So we have a zig-zag

Z

K

colim DK - · · · - ho colim T V \K

of weak equivalences in top+. The target of our weak equivalence U \K SK is also of the form D , but the source is not. So we have a zig-zag holim U \K - · · · - lim SK of weak equivalences in cdgaQ. C (ZK ; Q)

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Now consider the problem of describing the lo op space DJ (K ) as a colimit. There are homorphisms T colimtmon T K , which combine to give a homotopy homomorphism colimtop+ B T K - colimtmon T K . When K is flag, this is a weak equivalence DJ (K ) - Cir (K (1)); but not in general. There is also a homotopy homomorphism DJ (K ) - ho colimtmon T K , which is a weak equivalence for all K . So lo oping preserves homotopy colimits.

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Almost all our algebraic categories admit mo del structures, in which weak equivalences are the quasi-isomorphisms. So we hop e we can describ e structures like the Pontrjagin ring H( DJ (K ); R) as the homology of an appropriate ho colim in dga. When K is flag, there is a zig-zag of weak equivalences C( DJ (K ); Q) - · · · - colimdgaQ K ; in general, there is a zig-zag C( DJ (K ); Q) - · · · - ho colimdgaQ K . The pro of uses Adams's cobar construction : dgcQ - dgaQ, which is the algebraic analogue of taking lo ops.
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4. FORMALITY Sullivan's PL-forms functor A : top dgaQ provides a very go o d representation of the rational homotopy category as an algebraic mo del category. Understanding A(X ) in dgaQ is as go o d as understanding XQ in topQ. Certain nice spaces X (such as Eilenb erg-Mac Lane spaces, or classifying spaces of Lie groups) are formal, b ecause there exists a zig-zag of quasi-isomorphisms (A(X ), d) - · · · - (H (X ; Q), 0) in cdgaQ. In fact DJ (K ) is formal for every K . Some X are also integrally formal, b ecause a similar zig-zag of quasi-isomorphisms (C (X ; Z), d) - · · · - (H (X ; Z), 0) exists in dga, for the singular co chain algebra C (X ; Z). This is also true for every DJ (K ).
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Now we can discuss toric manifolds M 2n! We let K b e the b oundary of a simplicial convex n-p olytop e, and search for a linear system of parameters t1, . . . , tn in Z[K ]. For any such system t, an M is defined (up to weak equivalence) as the pullback of
p t DJ (K ) -- B T n -- E T n,

in top. So T n acts freely on M . Moreover, t induces t : subtorus t(T ) < T n is T ( ). So there is a cat which maps each T V T n, and each an isotropy subgroup (K )-diagram T n/K , to the projection

T n/T ( ) - T n/T ( ). Then M is ho colimtop T n/K , and the orbit space M /T n is the nerve of cat(K ), or P n.
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We may verify that A(M ) is weakly equivalent to the homotopy colimit of A(DJ (K )) - A(B T n) - A(E T n). Similarly, H (M ; Q) is weakly equivalent to the homotopy colimit of H (DJ (K ); Q) - H (B T n; Q) - Q, b ecause t is a linear system of parameters. As DJ (K ) is formal and E T n is contractible, the two diagrams are related by a zig-zag of weak equivalences. So their homotopy colimits are related by a zig-zag of weak equivalences in cdgaQ, and M is formal.

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Quillen studies XQ by means of the mo del category dglQ of differential graded Lie algebras. The homology of (L(XQ), d) gives ( X ) Q, equipp ed with the Whitehead pro duct bracket. Then X is coformal if there is a zig-zag of quasi-isomorphisms (L(XQ), d) - · · · - (( X ) Q, 0) in dglQ. By restricting to primitive elements, we find that such a zig-zag exists for DJ (K ) when and only when there is a zig-zag Q K - · · · - H( DJ (K ); Q). So DJ (K ) is coformal if and only if K is flag.
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