# Key Ideas - categories are the ambient context of discourse - it's all about the arrows - arrows are about externally observervable behaviour - arrows / behavior *are* the relationships / structure - things often come in pairs (due to dual constructions) - Consider objects in tandem with their appropriate structure preserving morphisms # Notational Conventions - `->` arrow e.g. `f: x -> y` - `=>` natural transformation - `...>` induced arrow (suggests it's drawn after) - `f^-1` the left and right inverse of `f` - since inverses of iso's are unique and themselves iso - `f^*` post composition by `f: x -> y` `f^*: C(c, x) -> C(c,y)` - `f_*` pre composition by `f: x -> y` `f^*: C(y, c) -> C(x, c)` - `gf` composition applicative order, shorthand for `g . f` - `f;g` composition diagram order - `1_X` the `X` idendity arrow aka `id_X` - The `1` notation alludes to the relationship between id and initial objects - `hook_arrow` inclusion morphism - `~=` isomorphism - `C(x,y)` the set of arrows in `f: x -> y in C` - aka the hom-set (even though arrows aren't always homomorphs) - `C(c, -)`; `C(-,c)` blank holes => a familly of arrows (ie the hom-functor (co and contra variant) - `C(c, -) : C -> Set` `C(-, c) : C^op -> Set` - `F: C^op -> D` - will refer to arrows `f: c -> c' in C` (rather than C^op) - `Ff: Fc' -> Fc` in D running in the opposite direction to reflect the contravariance. # Catalog of category adjectives / constructions - Groupoid: - cat with all arrows iso - Group: - a groupoid with 1 opbject - a cat with 1 object and with all arrows iso - Subcategory: - D is a subcat of C if obj(D) is a subcollection of obj(C), arr(D) a subcollection of arr(C) and D is still a category (includes enough) - Maximal groupoid - Any category has a maximal groupoid subcategory. All objects and only isos - e.g. `Fin_iso` is the maximal groupoid subcat of `Fin` with all finite sets bijections (useful as a categorification of the natural numbers and algebraic view of the laws of arithmetic) - `c/C` co-slice (C under c) - the objects are arr(C) whose sources are c `C(c, -)`; `f: c -> X` - the arrows are arrows in C between targets of c that commute - `g: c -> Y` and `h: X -> Y` st `g = hf` c / \ X -> Y - dually `C/c` slice (C over c) X -> Y \ / c - Both are specific cases of comma-categories ## Catalog of morphism behaviours - Iso (isomorphism): an invertable arrow `f`. The inverse `g` must be both he left and right inverse. - f : X -> Y s.t. g: Y -> X, `gf = 1_X` and `fg = 1_Y` - `g` is unique so often called `f^-1` - idendity arrows are degenerate iso (id is its own inverse, id composed with itself = id) - Isos are self-dual (`f: x -> y in C` is iso iff `f^op: y -> x in C^op`) - Endo (endomorphism): same domain / codomain `f: x -> x` - Auto (automorphism): iso + endo --- - Links: [[Math]] [[Type Theory]] [[Logic]] [[Philosophy]]