# 1 Product & coproduct

A (co)product is the (co)limit of the discrete diagram $$\cdot \hphantom{ {}\to{} } \cdot$$.

The unique morphisms $$\angles{f, g}$$ and $$\bracks{f, g}$$ are called the paring and coparing of $$f$$ and $$g$$, respectively.

• $$\angles{f, g} \compL h = \angles{f \compL h, g \compL h}$$
• $$h \compL \bracks{f, g} = \bracks{h \compL f, h \compL g}$$
• A (finitely) (co)cartesian category has all (finite) (co)products.
• A (finitely) bicartesian category is (finitely) cartesian and (finitely) cocartesian.

finite products $$\equiv$$ binary products and a terminal object

finite coproducts $$\equiv$$ binary coproducts and an initial object

## 1.1 Bifunctor

For two morphisms $$f: A \to C$$ and $$g: B \to D$$, the (co)product morphism is defined as

\begin{aligned} f \times g &: A \times B \to C \times D := \angles{f \compL p_1, g \compL p_2} \\ f + g &: A + B \to C + D := \bracks{i_1 \compL f, i_2 \compL g} \end{aligned}

which makes $$\times: \cC \times \cC \to \cC$$ and $$+: \cC \times \cC \to \cC$$ bifunctors.

## 1.2 Diagonal & codiagonal

For an object $$A$$, the (co)diagonal morphism is defined as

\begin{aligned} \Delta_A &: A \to A \times A := \angles{\id_A, \id_A} \\ \nabla_A &: A + A \to A := \bracks{\id_A, \id_A} \end{aligned}

• $$(f_1 \times f_2) \compL \Delta_A = \angles{f_1, f_2}$$
• $$\nabla_B \compL (f_1 + f_2) = \bracks{f_1, f_2}$$

## 1.3 Unitality

• The terminal object $$1$$ is the unit of the product: $$1 \times A \iso A \iso A \times 1$$.
• The initial object $$0$$ is the unit of the coproduct: $$0 + A \iso A \iso A + 0$$.

## 1.4 Associativity

\begin{aligned} \begin{array}{lcr} \alpha_{A, B, C} := \angles{p_1 \compL p_1, p_2 \times \id_C} &: (A \times B) \times C \iso A \times (B \times C) :& \angles{\id_A \times p_1, p_2 \compL p_2} =: \alpha_{A, B, C}\inv \\ \alpha_{A, B, C} := \bracks{\id_A + i_1, i_2 \compL i_2} &: (A + B) + C \iso A + (B + C) :& \bracks{i_1 \compL i_1, i_2 + \id_C} =: \alpha_{A, B, C}\inv \end{array} \end{aligned}

## 1.5 Commutativity

\begin{aligned} \begin{array}{lcr} \beta_{A, B} := \angles{p_2, p_1} &: A \times B \iso B \times A :& \angles{p_2, p_1} =: \beta_{B, A} \\ \beta_{A, B} := \bracks{i_2, i_1} &: A + B \iso B + A :& \bracks{i_2, i_1} =: \beta_{B, A} \end{array} \end{aligned}

## 1.6 Distributivity

For objects $$A$$, $$B$$, and $$C$$, there exists a distributive morphism of the product over the coproduct:

$\delta_{A, B, C}: A \times B + A \times C \to A \times (B + C):= \bracks{\id_A \times i_1, \id_A \times i_2}$

A distributive category is a category where all distributive morphisms are isomorphisms.

## 1.7 Tensor product

The product in the category $$\cVect$$ of vector spaces and linear functions is given by the direct product $$\times$$ of vector spaces.

In the product $$U \times V$$:

\begin{aligned} (u_1, v_1) + (u_2, v_2) &:= (u_1 + v_2, u_1 + v_2)\\ \lambda (u, v) &:= (\lambda u, \lambda v)\\ \end{aligned}

A bilinear function $$m: U \times V \to W$$ is a binary function that is linear in each argument:

$m(u_1 + u_2, v) = m(u_1, v) + m(u_2, v)$

$m(u, v_1 + v_2) = m(u, v_1) + m(u, v_2)$

$m(\lambda u, v) = m(u, \lambda v) = \lambda m(u, v)$

Examples: matrix multiplication, inner product

The tensor product of two vector spaces $$U$$ and $$V$$ is the initial object in the category of bilinear functions out of the product $$U \times V$$.

\begin{aligned} \begin{array}{rccc} \alpha: & U \times V & \to & U \otimes V \\ & (u, v) & \mapsto & u \otimes v \end{array} \end{aligned}

In the tensor product $$U \otimes V$$:

$(u_1 + u_2) \otimes v = u_1 \otimes v + u_2 \otimes v$

$u \otimes (v_1 + v_2) = u \otimes v_1 + u \otimes v_2$

$\lambda u \otimes v = u \otimes \lambda v = \lambda (u \otimes v)$

The tensor product of two linear functions $$f: U_1 \to U_2$$ and $$g: V_1 \to V_2$$ is the unique linear function $$f \otimes g: U_1 \otimes V_1 \to U_2 \otimes V_2$$ such that

$(f \otimes g)(u \otimes v) = f(u) \otimes g(v)$

which makes $$\otimes: \cVect \times \cVect \to \cVect$$ a bifunctor.

The tensor product of representations is the composition $$F_1 \otimes F_2: \cC \xto{\angles{F_1, F_2}} \cVect \times \cVect \xto{\otimes} \cVect$$.

The internal hom $$[V, W]$$ in the category $$\cVect$$ is the vector space of linear functions from $$V$$ to $$W$$:

$(f + g)(x) := f(x) + g(x)$

$(\lambda f)(x) := \lambda f(x)$

$\text{bilinear } U \times V \to W \iso \text{linear } U \otimes V \to W \iso \text{linear } U \to [V, W]$

# 2 Equalizer & coequalizer

A (co)equalizer is the (co)limit of the free quiver $$\cdot \rightrightarrows \cdot$$.

finite products and equalizers $$\limp$$ finite limits

$$\Eq(f, f) \iso A$$, $$B \iso \Coeq(f, f)$$.

## 2.1 Regular mono & epi

Regular monomorphism $$f: A \mono B$$:

• $$\exists g, h: B \to C. f: \Eq(g, h) \to B$$.

Regular epimorphism $$f: A \epi B$$:

• $$\exists g, h: C \to A. f: A \to \Coeq(g, h)$$.

## 2.2 Split mono & epi

Split monomorphism $$f: A \mono B$$:

• $$\exists r: B \to A. r \compL f = \id_A$$.
• $$\exists r: B \to A. f: \Eq(f \compL r, \id_B) \to B$$.
• $$r$$ is called a retraction of $$f$$.

Split epimorphism $$f: A \epi B$$:

• $$\exists s: B \to A. f \compL s = \id_B$$.
• $$\exists s: B \to A. f: A \to \Coeq(s \compL f, \id_A)$$.
• $$s$$ is called a section of $$f$$.

If $$f: A \to B$$ is a split monomorphism in a category $$\cC$$ and has a retraction $$g$$ ($$g \compL f = \id_A$$) but is not an isomorphism ($$f \compL g \neq \id_B$$), then $$(f, \id_B): f \to \id_B$$ is a pointwise retraction in the arrow category $$[\cTwo, \cC]$$ but is not a natural retraction.

# 3 Pullback & pushout

A pullback is the limit of the cospan $$\cdot \rightarrow \cdot \leftarrow \cdot$$, and a pushout is the colimit of the span $$\cdot \leftarrow \cdot \rightarrow \cdot$$.

• Pullbacks $$\Pull(f: A \to C, g: B \to C)$$ in $$\cC$$ are products $$A \times_C B$$ in $$\cC \comma C$$.
• Pushouts $$\Push(f: C \to A, g: C \to B)$$ in $$\cC$$ are coproducts $$A +_C B$$ in $$C \comma \cC$$.

The fiber of a morphism $$f: A \to B$$ over a global element $$b: 1 \to B$$ is the pullback $$\Pull(f, b)$$.

pullbacks and a terminal object $$\limp$$ products: $$A \times_1 B \iso A \times B$$

pullbacks and products $$\limp$$ equalizers: $$\Pull(\angles{f, g}, \Delta_B) \iso \Eq(f, g)$$

Pullbacks commute with products.

$\Pull(f_1 \times f_2, g_1 \times g_2) \iso \Pull(f_1, g_1) \times \Pull(f_2, g_2)$

This is a special case of limits commuting with limits, which is a result of the adjoint functor theorem.

For every pair of morphisms $$q_A: Q \to A$$ and $$q_B: Q \to B$$ such that $$(f_1 \times f_2) \compL q_A = (g_1 \times g_2) \compL q_B$$, we have $$f_1 \compL p_1 \compL q_A = g_1 \compL p_1 \compL q_B$$ and $$f_2 \compL p_2 \compL q_A = g_2 \compL p_2 \compL q_B$$, so there exist unique morphisms $$q_1: Q \to \Pull(f_1, g_1)$$ and $$q_2: Q \to \Pull(f_2, g_2)$$ such that $$q_A = \angles{{f_1}^*g_1 \compL q_1, {f_2}^*g_2 \compL q_2}$$ and $$q_B = \angles{{g_1}^*f_1 \compL q_1, {g_2}^*f_2 \compL q_2}$$. Therefore, $$\angles{q_1, q_2}: Q \to \Pull(f_1, g_1) \times \Pull(f_2, g_2)$$ is the unique morphism such that $$q_A = \pi_1 \compL \angles{q_1, q_2}$$ and $$q_B = \pi_2 \compL \angles{q_1, q_2}$$, where $$\pi_1 = (f_1 \times f_2)^*(g_1 \times g_2) = {f_1}^*g_1 \times {f_2}^*g_2$$ and $$\pi_2 = (g_1 \times g_2)^*(f_1 \times f_2) = {g_1}^*f_1 \times {g_2}^*f_2$$.

## 3.1 Mono & epi

Monomorphism $$f: A \mono B$$:

• $$\forall g, h: C \to A. (f \compL g = f \compL h) \to (g = h)$$.
• Postcomposition $$f \compL (-)$$ reflects equality.
• $$\Hom(C, f): \Hom(C, A) \to \Hom(C, B)$$ is injective.
• $$\Pull(f, f) \iso A$$.

Epimorphism $$f: A \epi B$$:

• $$\forall g, h: B \to C. (g \compL f = h \compL f) \to (g = h)$$.
• Precomposition $$(-) \compL f$$ reflects equality.
• $$\Hom(f, C): \Hom(B, C) \to \Hom(A, C)$$ is injective.
• $$\Push(f, f) \iso B$$.

Morphisms from a terminal object are monomorphisms; morphisms to an initial object are epimorphisms.

Pullbacks preserve monomorphisms.

$(n \compL g_1 = n \compL g_2) \limp (f \compL n \compL g_1 = f \compL n \compL g_2) \limp (m \compL f' \compL g_1 = m \compL f' \compL g_2) \limp (f' \compL g_1 = f' \compL g_2)$

split mono/epi $$\limp$$ regular mono/epi $$\limp$$ mono/epi

Every isomorphism is an epimorphism and a monomorphism, but not vice versa.

A category is balanced if every monic epic morphism is an isomorphism.