One of the methods for extracting information from a data set is clustering the data set according to some rule. In this paper, datasets are represented as finite metric spaces. A finite metric space is a pair (X, d X ), where X is a finite set and d X : X × X → R + is a distance function. We denote by M the collection of all finite metric spaces.We start by providing a definition of a clustering method with some examples. For any n ∈ N, we denote the set {1, 2, . . . , n} by [1 : n]. Given (X, d X ) ∈ M, we denote by P (X), the collection of all partitions of X. Precisely, every P ∈ P (X) is a family of setsas a block of P . We denote by P, the collection of all pairs (X, P X ), where X ∈ M and P X ∈ P (X). Formally,Example 2.2 An example of a clustering method is the discrete clustering that partitions every metric space into singletons. Precisely, we have C disc : M → P with C disc ((X, d X )) = (X, S X ), where S X ∈ P (X) is the partition of X into singletons.Example 2.3 Another example of a clustering method is the full clustering that partitions every metric space into a single block. Precisely, we have C full : M → P with C full ((X, d X )) = (X, {X}).There are various other examples of clustering methods such as partitioning into clusters whose diameter is bounded above by a constant, or partitioning into clusters whose diameter is bounded below by a constant, and so on [JS72]. Since we are working with finite metric spaces, the metric structure is the only information we have for determining a partition. Thus, 3 it seems natural that for (X, d X ), (Y, d Y ) ∈ M and a structure preserving map f : X → Y , a partition of Y induced by a clustering method C can be determined, at least partially, using the map f and a partition of X induced by the same clustering method C. Precisely, we want a clustering method C to be a functor, see [CM13].In order to view a clustering method C as a functor, we need to view M and P as categories. We refer the readers to [Jac12, Spi14] for an account on category theory. We define the categorical structure on M and P as follows:Definition 2.4 (Category of Finite Metric Spaces). Let M, by abuse of notation, denote the category of finite metric spaces. The objects of M are finite metric spaces (X, d X

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We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension δ ∈ (1, d), for any ε > 0 and c ≥ 1, any c-spanner must have treewidth at least, matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:For any δ ∈ (1, d) and any ε > 0, d-dimensional Euclidean TSP on n points with fractal dimension at most δ cannot be solved in time 2 O(n 1−1/(δ−ε) ) . The best-known upper bound is 2 O(n 1−1/δ log n) . For any δ ∈ (1, d) and any ε > 0, the problem of finding k-pairwise non-intersecting ddimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most δ cannot be solved in time f (k)n O(k 1−1/(δ−ε) ) for any computable function f . The best-known upper bound is n O(k 1−1/δ log n) . The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014].