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This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties.

On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 which has an indecomposable object with precisely ℓ mutations.

The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A .

{S}} . (3) The extension closure of 𝖲 {\mathsf{S}} is functorially finite. It is interesting to note that ( 2 ) {(2)} and ( 3 ) {(3)} are also a feature of ( w - 1 ) {(w-1)} -cluster tilting subcategories in the positive CY case, cf. [ 12 , Lemma 3.4]. It was not known thus far whether the notions of maximal systems of stable orthogonal bricks and simple-minded systems coincided, cf. [ 9 ]. As a byproduct of our combinatorial classification, we can conclude that these notions do not coincide. We shall then use the classification of w -Hom configurations, w

dimensions, both little and big, of L can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory P<yðL-modÞ consisting of the finitely generated left L-modules of finite pro- jective dimension is completely determined by a finite number of, possibly infinite dimen- sional, string modules—one for each simple L-module—which are algorithmically con- structible from quiver and relations of L. Namely, P<yðL-modÞ is contravariantly finite in L-mod precisely

{\mathrm{Hom}}_{\Lambda}(\widetilde{\sigma},N)} is surjective, where σ ~ = [ σ   0 ] {\widetilde{\sigma}=[\sigma\quad 0]} in the projective presentation P 1 ⊕ P → σ ~ P 0 → M . P_{1}\oplus P\xrightarrow{\widetilde{\sigma}}P_{0}\xrightarrow{\phantom{% \widetilde{\sigma}}}M. A result of [ 16 ] relates τ-tilting finiteness with functorial finiteness of torsion classes; we refer the reader to, for example, [ 34 ] for the definition of functorial finiteness. Theorem 2.9 ([ 16 , Theorem 3.8]). A finite-dimensional algebra Λ is τ-tilting finite if and only if every torsion class (equivalently, every torsionfree class

{Hom}}_{\mathcal{A}}(-,X)|_{\mathcal{C}}} ) is a finitely generated 𝒞 {\mathcal{C}} -module for any X in 𝒜 . {\mathcal{A}.} We call 𝒞 {\mathcal{C}} functorially finite if it is both contravariantly and covariantly finite. It is known that if 𝒞 {\mathcal{C}} is a contravariantly finite subcategory of abelian category 𝒜 {\mathcal{A}} , then mod ⁢ - ⁢ 𝒞 {{\mathrm{{mod\mbox{-}}}}\mathcal{C}} is an abelian category. Let 𝒜 {\mathcal{A}} be an abelian category with enough projectivs and 𝒞 {\mathcal{C}} contains all projective objects of 𝒜 . {\mathcal{A}.} We consider the stable category

group of automorphisms of the triangle functor Wn; ð1Þn1Wn1 acts simply transi- tively on the class of pre-n-angulations of ðF;SÞ. Proof. If Y : ðS; sÞ ! Wn; ð1Þn1Wn1 is an isomorphism of triangle functors, we have by Lemma 3.3, dð YÞ ¼ Y. If dð 0Þ ¼ Y, we have by construction 0H Y, but then 0 ¼ Y by Proposition 2.5 (c). r 108 Geiss, Keller and Oppermann, n-angulated categories 4. Standard construction 4.1. Definition. Let T be a triangulated category with suspension S3. A full sub- category SLT is called d-cluster tilting if it is functorially finite [2], p. 82

subcategories XB of A-Mod (respectively, Mod-A), i.e., strict full subcategories of A-Mod (respectively, Mod-A) closed under products, co- products, kernels and cokernels. If A is a finite dimensional K-algebra, this bijection can be restricted between: (1) ring epimorphisms A! B up to equivalence, where B is a finite dimensional K-algebra, (2) bireflective subcategories XB of A-mod (respectively, mod-A), i.e., strict full functorially finite subcategories of A-mod (respectively, mod-A) closed under kernels and cokernels. From ring epimorphisms to universal localisations 1143

.1 essentially tells us how to put the Taylor approximations in the multivariable setting together to obtain the Taylor approximations in the single variable setting. The latter theorem uses the classification of homogeneous multivariable functors (Theorem 5.19). In Section 7, we give a non-functorial finite model for the Taylor approximations in the case where F is the embedding or link maps functor andM D ` i I is a dis- joint union of intervals (so F is the space of string links or homotopy string links respectively). These are precisely the functors which will be studied

triangulated category, and we suppose that ℳ ⊆ 𝒵 \mathcal{M}\subseteq\mathcal{Z} are full (not necessarily triangulated) subcategories of 𝒞 \mathcal{C} . Theorem 1.1. With the setup as above, assume further that M \mathcal{M} is functorially finite in Z \mathcal{Z} , and that Z \mathcal{Z} is closed under cones of M \mathcal{M} -monomorphisms and cocones of M \mathcal{M} -epimorphisms (see Section 2.1 for more details). Then Z / [ M ] \mathcal{Z}/[\mathcal{M}] has the structure of a triangulated category. We then show that 𝒵 / [ ℳ ] \mathcal