1 Category theory

1.1 Over category

Proposition 1.1.1 Sliced adjoint functors

✓

If $a : F ⊢ G$ is an adjunction between $F : C \to D$ and $G : D \to C$ and $X : C$ , then there is an adjunction between $F / X : C / X \to D / F (X)$ and $G / X : D / F (X) \to C / X$ .

Proof ▶

Proposition 1.1.2 Limit-preserving functors lift to over categories

✓

Let $J$ be a shape (i.e. a category). Let $\tilde{J}$ denote the category which is the same as $J$ , but has an extra object $*$ which is terminal. If $F : C \to D$ is a functor preserving limits of shape $\tilde{J}$ , then the obvious functor $C / X \to D / F (X)$ preserves limits of shape $J$ .

Proof ▶

Extend a functor $K : J \to C / X$ to a functor $\tilde{K} : \tilde{J} \to C$ , by letting $\tilde{K} (*) = X$ .

Proposition 1.1.3 Essential image of a sliced functor

✓

If $F : C \to D$ is a full functor between cartesian-monoidal categories, then $F / X : C / X hom D / F (X)$ has the same essential image as $F$ .

Proof ▶

1.2 Objects

1.2.1 Group objects

Proposition 1.2.1 Fully faithful product-preserving functors lift to monoid/group objects

✓

If a finite-products-preserving functor $F : C \to D$ is fully faithful, then so is $Grp (F) : Grp C \to Grp D$ .

Proof ▶

Faithfulness is immediate.

For fullness, assume $f : F (G) \to F (H)$ is a morphism. By fullness of $F$ , find $g : G \to H$ such that $F (g) = f$ . $g$ is a morphism because we can pull back each diagram from $D$ to $C$ along $F$ which is faithful.

Proposition 1.2.2 Equivalences lift to monoid/group object categories

✓

If $e : C ⋍ D$ is an equivalence of cartesian-monoidal categories, then $Grp (e) : Grp (C) ⋍ Grp (D)$ too is an equivalence of categories.

Proof ▶

Proposition 1.2.3 Essential image of a functor on monoid/group objects

✓

If $F : C \to D$ is a fully faithful functor between cartesian-monoidal categories, then $Grp (F) : Grp (C) hom Grp (D)$ has the same essential image as $F$ .

Proof ▶

Lemma 1.2.4 A fully faithful functor product-preserving is a group isomorphism on hom sets

✓

If $F : C \to D$ is a fully faithful functor between cartesian-monoidal categories and $X, G \in C$ are an object and a group object respectively, then $Grp (F) : (X hom G) hom \backcong (F (X) hom F (G))$ is a group isomorphism.

Proof ▶

1.2.2 Module objects

Proposition 1.2.5 Pulling back a module object

Let $M, N$ be two monoid objects in a monoidal category $C$ . Let $f : M \to N$ be a monoid morphism. If $X$ is a $N$ -module object, then it is also a $M$ -module object.

Proof ▶

Define the multiplication $μ_{M} : M \otimes X \to X$ as $f \otimes 1 ≫ μ_{N}$ . All proofs follow easily. □