S. D. V. Stephens

notes / Category Theory / chapter 1: Categories

1.5 Exercises on Categories

In progress: these notes stop partway.

Exercise 1.5.1. Find 3 examples of categories not yet mentioned.

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My first idea is the “box category”, i.e. , where in the case of finite , and in the case of infinite . Morphisms are taken componentwise: if , with the product ending in if is finite.

Composition is componentwise, , and the identity on is . Associativity holds because it holds in every component: and for every morphism , with , we have for all .

Another idea would be tensors? Since a tensor gives

A triangle with arrows from V to V⊗W, from V⊗W to W, and directly from V to W.

then maybe we can formalize this.

Category of graphs. , .

Scratch sketch of the category of graphs: a star graph mapped to another star drawn differently (same shape, different morphisms) and to a four-cycle (different shape, different morphisms), and the four-cycle redrawn with a crossing (same morphism, different shape), with the label A4.
  • object: the entire thing is an object in the category; the shape and the morphism change.
  • Two drawings of the same star graph, positioned differently, are marked . Not just ? group?
  • : 12 morphisms, .

Generally I found 4 things of interest:

  1. Category .
  2. Category (matrices ).
  3. Category monoid & category box.
  4. Übertereng category (Üb): a made-up one.

Exercise 1.5.2. Show that a morphism in a category has at most one inverse.

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Let and with and . Now let such that and , . Then Ahah! But we assumed ; thus we have a contradiction, and there cannot be more than one inverse for a map in a category.

(A cleaner approach: .)

Exercise 1.5.3. Let be categories. We defined the product category , but the definitions of composition and identity were not given. Define them.

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has composition given by , where , , and , and we compose componentwise:

This is well-defined, which I won’t show here. Next, for identity: with the identity of and the identity of , we have as the identity on , given that we take things componentwise. So for out of , This also allows us to find inverses, but does not guarantee their existence (since that would be axiomatic).

Exercise 1.5.4. There exists a category called , whose objects are topological spaces and whose maps are homotopy classes of continuous maps from to . What do we need to know about homotopy in order to prove that is a category? What does it mean for two objects of to be isomorphic?

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and :

Several paths between points X and Y with arrows sweeping between them, picturing one homotopy class of maps from X to Y.

a) We need to know what composition looks like: with and , so gives us .

Bundles of paths for a class α from X to Y and a class β from Y to Z, and in blue the composite class β∘α from X to Z.

b) We need to know the identity, which is just the class of the continuous identity map of to itself. Easy.

c) Associativity. Two objects in are isomorphic if all their equivalence classes …