By Nicolas Bourbaki
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Extra info for Elements of Mathematics: General Topology, Pt.2
After all, when is the last time anyone ever actually cared about the e-tree defining an element of a smooth manifold? For the insistently set-theory minded, we will dispense with the problems usually raised concerning sizes of categories by using a sufficiently strong large-cardinal axiom—Grothenieck's Axiom of Universe. Those size problems which do not collapse in the face of this axiom, and there are some, do not arise in this study. 16 For a locally small category C, the set of all arrows with source(/) = X and t a r g e t ( / ) = Y is denoted Homc(X,Y) or simply C(X, Y), and called the hom-set from X to Y.
19 Every formal diagram in the theory of lax (resp. oplax, strong) semigroupal functors commutes. Consequently, any diagram which is a functorial image of such a formal diagram under a (strict monoidal) functor commutes. These coherence theorems are the basis for a very useful notion and notational convention: throughout our discussion of categorical deformation theory in P a r t II, we will use padded composition operators \ ]. These operators are an embodiment of the coherence theorems of Mac Lane  and Epstein  .
D are the categories of K-vectorspaces and K-linear maps, and of finite dimensional K-vectorspaces and K-linear maps, respectively, and where
K is the tensor product over K (universal target for bilinear maps), au,v,w ? s given on the usual spanning set by (u®v)®w H-> U (g) (v