Universal construction defines objects in terms of their relations with other objects.

# 1 Terminal object & initial object

• A terminal object is an object $$T$$ that for every object $$A$$ there exists a unique morphism $$f: A \to T$$ to it.
• An initial object is an object $$I$$ that for every object $$B$$ there exists a unique morphism $$g: I \to B$$ from it.

\begin{aligned} \begin{array}{rlcll} \text{terminal object:} & T & \equiv & \forall A. & \exists! f: A \to T. \\ \text{initial object:} & I & \equiv & \forall B. & \exists! g: I \to B. \end{array} \end{aligned}

A generalized element of $$A$$ is a morphism $$x: X \to A$$.

A global element of $$A$$ is a morphism $$x: 1 \to A$$.

A zero object is both a terminal object and an initial object.

A pointed category has a zero object.

# 2 Universal morphism

\begin{aligned} \begin{array}{ccccc} F: & \cC & \toot & \cD & :G \\ & A && B & \\ & f && g & \end{array} \end{aligned}

• A universal morphism from $$F$$ to $$B$$ is a terminal object $$(T, FT \xto{\epsilon} B)$$ in the comma category $$(F: \cC \to \cD) \comma (B: \cOne \to \cD)$$.
• A universal morphism from $$A$$ to $$G$$ is an initial object $$(I, A \xto{\eta} GI)$$ in the comma category $$(A: \cOne \to \cC) \comma (G: \cD \to \cC)$$.

\begin{aligned} \begin{array}{rlclll} \text{universal morphism from F to B:} & (T, FT \xto{\epsilon} B) & \equiv & \forall g: FA \to B. & \exists! \hat{g}: A \to T. & g = \epsilon \compL F\hat{g}. \\ \text{universal morphism from A to G:} & (I, A \xto{\eta} GI) & \equiv & \forall f: A \to GB. & \exists! \hat{f}: I \to B. & f = G\hat{f} \compL \eta. \end{array} \end{aligned}

• A right adjoint functor of $$F$$ is a universal morphism $$(G, \epsilon: FG \nat \id_\cD)$$ from $$F^\cD: \cC^\cD \to \cD^\cD$$ to $$\id_\cD$$.
• A left adjoint functor of $$G$$ is a universal morphism $$(F, \eta: \id_\cC \nat GF)$$ from $$\id_\cC$$ to $$G^\cC: \cD^\cC \to \cC^\cC$$.
• For each $$\cD$$-object $$B$$, there exists a universal morphism $$(GB, \epsilon_B: FGB \to B)$$ from $$F$$ to $$B$$.
• For each $$\cC$$-object $$A$$, there exists a universal morphism $$(FA, \eta_A: A \to GFA)$$ from $$A$$ to $$G$$.

$G(B \xto{g} B') := \widehat{g \compL \epsilon_B}$

$F(A \xto{f} A') := \widehat{\eta_A' \compL f}$

## 3.2 Hom-set isomorphism

$\alpha: \Hom_\cD(FA, B) \iso \Hom_\cC(A, GB) :\alpha\inv$

\begin{aligned} \begin{array}{cccc} \alpha\inv_{GB, B}: & \Hom_\cC(GB, GB) & \iso & \Hom_\cD(FGB, B) \\ & \id_{GB} & \mapsto & \epsilon_B \\ \alpha_{A, FA}: & \Hom_\cD(FA, FA) & \iso & \Hom_\cC(A, GFA) \\ & \id_{FA} & \mapsto & \eta_A \end{array} \end{aligned}

\begin{aligned} \begin{array}{rll} \text{isomorphism:} & \id_\cC = GF & FG = \id_\cD \\ \text{equivalence:} & \id_\cC \iso GF & FG \iso \id_\cD \\ \text{adjunction:} & \id_\cC \nat GF & FG \nat \id_\cD \end{array} \end{aligned}

$\cC \adjto{F}{G} \cD$

An adjunction $$F \adj G$$ consists of a pair of adjoint functors $$F: \cC \toot \cD: G$$ with

• counit $$\epsilon: FG \nat \id_\cD$$
• unit $$\eta: \id_\cC \nat GF$$

such that

• $$(\epsilon F) \compL (F \eta) = \id_F$$
• $$(G \epsilon) \compL (\eta G) = \id_G$$

## 3.4 Galois connection

$F: (A, \leq_A) \adj (B, \leq_B) :G$

\begin{aligned} Fa := \min \set{b \in B \mid a \leq_A Gb} \\ Gb := \max \set{a \in A \mid Fa \leq_B b} \end{aligned}

### 3.4.1 Multiplication and division

$(\R, \leq) \adjto{- / c}{- \times c} (\R, \leq)$

$a / c \leq b \equiv a \leq b \times c$

• $$\eta_a: a \leq a / c * c = \id_a$$
• $$\epsilon_a: a * c / c \leq a = \id_a$$

### 3.4.2 Union and intersection

$(PA, \subseteq) \adjto{- \cap X}{\overline{X} \cup -} (PA, \subseteq)$

$Z \cap X \subseteq Y \equiv Z \subseteq \overline{X} \cup Y$

• $$\eta_A: A \subseteq \overline{X} \cup (A \cap X)$$
• $$\epsilon_A: (\overline{X} \cup A) \cap X \subseteq A$$

### 3.4.3 Floor and ceiling

$(\R, \leq) \adjto{\ceil{-}}{} (\Z, \leq) \adjto{}{\floor{-}} (\R, \leq)$

$\ceil{x} \leq_\Z n \equiv x \leq_\R n$

$n \leq_\R x \equiv n \leq_\Z \floor{x}$

• $$\eta_x: x \leq_\R \ceil{x}$$
• $$\epsilon_n: \ceil{n} \leq_\Z n = \id_n$$
• $$\eta_n: n \leq_\Z \floor{n} = \id_n$$
• $$\epsilon_x: \floor{x} \leq_\R x$$

### 3.4.4 Maximum and minimum

$\cOne \adjto{\min A}{} (A, \leq) \adjto{}{\max A} \cOne$

• $$\eta_* = \id_*$$
• $$\epsilon_a: \min A \leq a$$
• $$\eta_a: a \leq \max A$$
• $$\epsilon_* = \id_*$$

$(PA, \subseteq) \adjto{\max}{A_{\leq -}} (A, \leq) \adjto{A_{\geq -}}{\min} (PA, \supseteq)$

$\max A \leq b \equiv A \subseteq A_{\leq b}$

$A_{\geq a} \supseteq B \equiv a \leq \min B$

• $$\eta_A: A \subseteq A_{\leq \max A}$$
• $$\epsilon_b: \max A_{\leq b} \leq b = \id_b$$
• $$\eta_a: a \leq \min A_{\geq a} = \id_a$$
• $$\epsilon_B: A_{\geq \min B} \supseteq B$$

$(PA, \subseteq) \adjto{\min}{A_{\geq -}} (A, \geq) \adjto{A_{\leq -}}{\max} (PA, \supseteq)$

$\min A \geq b \equiv A \subseteq A_{\geq b}$

$A_{\leq a} \supseteq B \equiv a \geq \max B$

• $$\eta_A: A \subseteq A_{\geq \min A}$$
• $$\epsilon_b: \min A_{\geq b} \geq b = \id_b$$
• $$\eta_a: a \geq \max A_{\leq a} = \id_a$$
• $$\epsilon_B: A_{\leq \max B} \supseteq B$$

$\cC \adjto{F\text{ree}}{U\text{nderlying}} \cD$

# 4 Kan

• $$F: \cA \to \cC$$
• $$G: \cA \to \cB$$
• $$H: \cB \to \cC$$
• $$\cC^G: \cC^\cB \to \cC^\cA$$ precomposition
• $$H^\cA: \cB^\cA \to \cC^\cA$$ postcomposition
• $$\Lan_G, \Ran_G: \cC^\cA \to \cC^\cB$$ Kan extension
• $$\Lift_H, \Rift_H: \cC^\cA \to \cB^\cA$$ Kan lift

## 4.1 Kan extension

• A right Kan extension of $$F$$ through $$G$$ is a universal morphism $$(\Ran_G F: \cB \to \cC, \epsilon_F: \Ran_G F G \nat F)$$ from $$\cC^G$$ to $$F$$.
• A left Kan extension of $$F$$ through $$G$$ is a universal morphism $$(\Lan_G F: \cB \to \cC, \eta_F: F \nat \Lan_G F G)$$ from $$F$$ to $$\cC^G$$.

$\Lan_G \adj \cC^G \adj \Ran_G$

## 4.2 Kan lift

• A right Kan lift of $$F$$ through $$H$$ is a universal morphism $$(\Rift_H F: \cA \to \cB, \epsilon_F: H \Rift_H F \nat F)$$ from $$H^\cA$$ to $$F$$.
• A left Kan lift of $$F$$ through $$H$$ is a universal morphism $$(\Lift_H F: \cA \to \cB, \eta_F: F \nat H \Lift_H F)$$ from $$F$$ to $$H^\cA$$.

$\Lift_H \adj H^\cA \adj \Rift_H$

# 5 Limit & colimit

A diagram of type $$\cI$$ in a category $$\cC$$ is a functor

\begin{aligned} D: \cI & \to \cC \\ i & \mapsto D_i \\ ij & \mapsto D_{ij} \end{aligned}

$$\cI$$ is called the index category.

Given the diagonal functor $$\Delta: \cC \to \cC^\cI$$ and a diagram $$D: \cI \to \cC$$:

• A limit is a universal morphism $$(\lim D, \epsilon_D: \Delta \lim D \nat D)$$ from $$\Delta$$ to $$D$$.
• A colimit is a universal morphism $$(\colim D, \eta_D: D \nat \Delta \colim D)$$ from $$D$$ to $$\Delta$$.

\begin{aligned} \begin{array}{rlclll} \text{limit:} & \epsilon_D: \Delta \lim D \nat D & \equiv & \forall c: \Delta C \nat D. & \exists! \hat{c}: C \to \lim D. & c = \epsilon_D (\Delta \hat{c}). \\ \text{colimit:} & \eta_D: D \nat \Delta \colim D & \equiv & \forall c: D \nat \Delta C. & \exists! \hat{c}: \colim D \to C. & c = (\Delta \hat{c}) \eta_D. \end{array} \end{aligned}

$\colim \adj \Delta \adj \lim$

• A (finitely) (co)complete category has all (finite) (co)limits.
• A (finitely) bicomplete category is (finitely) complete and (finitely) cocomplete.

## 5.1 Cone & cocone

A cone $$(C, c)$$ over a diagram $$D: \cI \to \cC$$ consists of

• a $$\cC$$-object $$C$$
• a collection of $$\cC$$-morphisms from $$C$$ indexed by $$\cI$$-objects $$c_i: C \to D_i$$

such that for all $$\cI$$-morphisms $$ij: i \to j$$, $$c_j = D_{ij} \compL c_i$$.

A cone morphism $$f: (C, c) \to (C', c')$$ is a $$\cC$$-morphism $$f: C \to C'$$ that for all $$\cI$$-objects $$i$$, $$c_i = c'_i \compL f$$.

A cone category $$\cCone(D)$$ consists of all cones over a diagram $$D$$ and all cone morphisms.

A limit $$\lim D$$ of a diagram $$D$$ is a terminal object in $$\cCone(D)$$.

• A limit is a cone: (a $$\cC$$-object $$\lim D$$, a collection of $$\cC$$-morphisms $$\epsilon_i: \lim D \to D_i$$)
• A limit is a terminal object: for all cones $$(C, c)$$, there exists a unique cone morphism $$f: (C, c) \to (\lim D, \epsilon)$$ such that $$c_i = \epsilon_i \compL f$$.

The concepts of cocone, cocone morphism, cocone category and colimit are defined dually.

• cone: $$c: \Delta C \nat D$$, cocone: $$c: D \nat \Delta C$$
• cone/cocone morphism: $$\Delta f: \Delta C \nat \Delta C'$$
• cone category: $$\Delta \comma D$$, cocone category: $$D \comma \Delta$$
• limit: $$\epsilon_D: \Delta \lim D \nat D$$, colimit: $$\eta_D: D \nat \Delta \colim D$$

## 5.2 Preservation

For a diagram $$D: \cI \to \cC$$ and a functor $$F: \cC \to \cD$$,

• $$F$$ preserves a limit $$(\lim D, \epsilon_D)$$ in $$\cC$$ if $$(F \lim D, F \epsilon_D)$$ is a limit in $$\cD$$
• $$F$$ preserves a colimit $$(\colim D, \eta_D)$$ in $$\cC$$ if $$(F \colim D, F \eta_D)$$ is a colimit in $$\cD$$

A (finitely) (co)continuous functor preserves all (finite) (co)limits.

• $$\Hom(A, -)$$ preserves limits

$\Hom(A, B \times C) \iso \Hom(A, B) \times \Hom(A, C)$

• $$\Hom(-, C)$$ sends colimits to limits

$\Hom(A + B, C) \iso \Hom(A, C) \times \Hom(B, C)$

$$F$$ is a left adjoint functor $$\equiv$$ $$F$$ preserves colimits. $$G$$ is a right adjoint functor $$\equiv$$ $$G$$ preserves limits.

# 6 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.

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

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

• A (finitely) (co)cartesian category has all (finite) (co)products.
• A (finitely) bicartesian category is (finitely) cartesian and (finitely) cocartesian.

## 6.1 Properties

### 6.1.1 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.

### 6.1.2 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}

### 6.1.3 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: $$0 + A \iso A \iso A + 0$$.

### 6.1.4 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}

### 6.1.5 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}

### 6.1.6 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.

## 6.2 Tensor product

The category $$\cVect$$ of vector spaces and linear maps has products, 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}

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

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

\begin{aligned} (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) &= \lambda u \otimes v = u \otimes \lambda v\\ \end{aligned}

The internal hom $$[V, W]$$ is the space of linear maps from $$V$$ to $$W$$.

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

# 7 Equalizer & coequalizer

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

finite limits $$\equiv$$ finite products and equalizers

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

## 7.1 Regular mono & epi

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

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

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

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

## 7.2 Split mono & epi

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

$\exists r: B \to A. r \compL f = \id_A$

$$r$$ is a retraction of $$f$$.

$$f: \Eq(f \compL r, \id_B) \mono B$$.

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

$\exists s: B \to A. f \compL s = \id_B$

$$s$$ is a section of $$f$$.

$$f: A \epi \Coeq(s \compL f, \id_A)$$.

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

# 8 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 $$\Pull(f, b)$$.

binary products and pullbacks $$\limp$$ equalizers

finite limits $$\equiv$$ pullbacks and a terminal object

## 8.1 Mono & epi

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

$\forall g, h: C \to A. (f \compL g = f \compL h) \to (g = h)$

$$\Pull(f, f) = A$$.

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

$\forall g, h: B \to C. (g \compL f = h \compL f) \to (g = h)$

$$\Push(f, f) = B$$.

Every iso is an epi and a mono, but not vice versa.

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

## 8.2 Base change

A morphism $$f: A \to B$$ induces a base change functor $$f^*: \cC \comma B \to \cC \comma A$$ between slice categories:

\begin{aligned} \begin{array}{cccc} f^*: & \cC \comma B & \to & \cC \comma A \\ & C \xto{o} B & \mapsto & \Pull(f, o) \xto{o^*} A \\ & C_1 \xto{m} C_2 & \mapsto & \Pull(f, o_1) \xto{m^*} \Pull(f, o_2) \end{array} \end{aligned}

$\id_A \times_B m: A \times_B C_1 \to A \times_B C_2$

## 8.3 Subobject

For a category $$\cC$$, a subobject of an object $$A$$ is a mono $$s: S \mono A$$.

For two subobjects of $$A$$, a subobject morphism is a morphism in the slice category $$\cC \comma A$$.

• inclusion of two subobjects $$s$$ and $$t$$:

$s \subseteq t: \exists f: s \to t$

• equivalence of two subobjects $$s$$ and $$t$$:

$s \equiv t: s \subseteq t \land t \subseteq s$

• local membership of a generalized element $$x: X \to A$$:

$x \in_A s: \exists f: X \to S. x = s \compL f$

For a category $$\cC$$ with a terminal object $$1$$, a subobject classifier is an object $$\Omega$$ together with a morphism $$\ltru: 1 \to \Omega$$, such that for each mono $$s: S \mono A$$, there exists a mono $$\chi_s: A \mono \Omega$$, called the classifying morphism that forms a pullback.

## 8.4 Power object

For a category $$\cC$$ with binary products, a power object of an object $$A$$ is an object $$PA$$ together with a mono $$\in_A \mono A \times PA$$, such that for each mono $$r: R \mono A \times B$$, there exists a unique morphism $$\chi_r: B \to PA$$ that forms a pullback.

# 9 Exponential object

For a category $$\cC$$ with binary products, an exponential object from $$A$$ to $$B$$ is a universal morphism $$(B^A, B^A \times A \xto{\epsilon_A} B)$$ from $$(-) \times A$$ to $$B$$.

$\begin{array}{rlclll} \text{exponential:} & (B^A, B^A \times A \xto{\epsilon_A} B) & \equiv & \forall f: C \times A \to B. & \exists! \hat{f}: C \to B^A. & f = \epsilon_A \compL (\hat{f} \times \id_A). \end{array}$

• $$\epsilon_A: B^A \times A \to B$$ is called evaluation
• $$\hat{f}: C \to B^A$$ is called the exponential transpose of $$f: C \times A \to B$$
• $$\bar{g}: C \times A \to B := \epsilon_A \compL (g \times \id_A)$$ is the exponential transpose of $$g: C \to B^A$$

$(-) \times A \adj (-)^A$

$\Hom(C \times A, B) \iso \Hom(C, B^A)$

-- function application
(\$) :: (a -> b) -> a -> b
-- reverse function application
(&) :: a -> (a -> b) -> b

A cartesian closed category has all finite products and exponentials.

In a cartesian closed category, the functor $$(-) \times A$$ is a left adjoint, which necessarily preserves all colimits, including all coproducts.

## 9.1 Composition

$A \xto{f} B \xto{g} C \xto{h} D$

### 9.1.1 Composition

$\compL: C^B \times B^A \to C^A$

-- function composition
(.) :: (b -> c) -> (a -> b) -> (a -> c)

### 9.1.2 Postcomposition

$h^B: C^B \to D^B := h \compL (-)$

\begin{aligned} \begin{array}{cccc} (-)^B: & \cC & \to & \cC \\ & C & \mapsto & C^B \\ & C \xto{h} D & \mapsto & C^B \xto{h^B} D^B \end{array} \end{aligned}

### 9.1.3 Precomposition

$C^f: C^B \to C^A := (-) \compL f$

\begin{aligned} \begin{array}{cccc} C^{(-)}: & \cC\op & \to & \cC \\ & B & \mapsto & C^B \\ & B \xto{f\op} A & \mapsto & C^B \xto{C^f} C^A \end{array} \end{aligned}

## 9.2 Currying

### 9.2.1 Partial application

$\partf: C^{A \times B} \times A \to C^B$

### 9.2.2 Currying and uncurrying

$\curry: C^{A \times B} \iso (C^B)^A :\curry\inv$

$\Hom(A \times B, C) \iso \Hom(A, C^B)$

curry   :: ((a, b) -> c) -> a -> b -> c
uncurry :: (a -> b -> c) -> (a, b) -> c 

## 9.3 Product

### 9.3.1 Product codomain

$\angles{p_1^A, p_2^A}: (B \times C)^A \iso B^A \times C^A$

$\Hom(A, B \times C) \iso \Hom(A, B) \times \Hom(A, C)$

### 9.3.2 Coproduct domain

$\angles{C^{i_1}, C^{i_2}}: C^{A + B} \iso C^A \times C^B$

$\Hom(A + B, C) \iso \Hom(A, C) \times \Hom(B, C)$

## 9.4 Tarski’s high school algebra

Bicartesian closed category:

1. $$A + B \iso B + A$$
2. $$(A + B) + C \iso A + (B + C)$$
3. $$A \times 1 \iso A$$
4. $$A \times B \iso B \times A$$
5. $$(A \times B) \times C \iso A \times (B \times C)$$
6. $$A \times B + A \times C \iso A \times (B + C)$$
7. $$1^A \iso 1$$
8. $$A^1 \iso A$$
9. $$A^{B + C} \iso A^B \times A^C$$
10. $$(A \times B)^C \iso A^C \times B^C$$
11. $$(A^B)^C \iso A^{B \times C}$$

# 10 Examples

## 10.1 Set

 terminal object singleton $$\set{*}$$ initial object empty set $$\varnothing$$ product cartesian product $$A \times B$$ coproduct disjoint union $$A + B$$ equalizer $$\Eq(f, g: A \to B) = \set{a \in A \mid f(a) = g(a)}$$ coequalizer $$\Coeq(f, g: A \to B) = B / \sim$$ where $$b_1 \sim b_2$$ if $$\exists a \in A. f(a) = b_1 \land g(a) = b_2$$ pullback $$\Pull(f: A \to C, g: B \to C) = \set{(a, b) \in A \times B \mid f(a) = g(b)}$$ pushout $$\Push(f: C \to A, g: C \to B) = (A + B) / \sim$$ where $$a \sim b$$ if $$\exists c \in C. f(c) = a \land g(c) = b$$ fiber inverse image $$\Pull(f: A \to B, b: 1 \to B) = \set{a \in A \mid f(a) = b}$$ mono injective function epi surjective function subobject subset subobject classifier indicator function power object power set exponential object the set of functions and function evaluation

## 10.2 Cat

 terminal object $$\cOne$$ initial object $$\cZero$$ product product category $$\cC \times \cD$$ exponential object functor category $$[\cC, \cD]$$

## 10.3 Poset

$$(A, \leq)$$

 terminal object greatest element / top $$\top$$ initial object least element / bottom $$\bot$$ product greatest lower bound / meet $$a \meet b$$ coproduct least upper bound / join $$a \join b$$ equalizer $$\id_a: a \leq a$$ coequalizer $$\id_b: b \leq b$$ pullback $$p_1: a \meet b \leq a$$, $$p_2: a \meet b \leq b$$ pushout $$i_1: a \leq a \join b$$, $$i_2: a \leq a \join b$$ exponential object $$a \ihom b := \bigjoin\set{c \mid c \lcon a \leq b}$$ ($$c \lcon a \leq b \equiv c \leq a \ihom b$$)

A bounded poset has a terminal object $$\top$$ and/or an initial object $$\bot$$.

• A meet-semilattice is a poset with all binary meets $$a \meet b$$.
• A join-semilattice is a poset with all binary joins $$a \join b$$.
• A lattice is a meet-semilattice and a join-semilattice.
• A bounded meet-semilattice is a finitely cartesian poset.
• A bounded join-semilattice is a finitely cocartesian poset.
• A bounded lattice is a finitely bicartesian poset.

Because all (co)equalizers exist in a poset, finitely (co)cartesian = finitely (co)complete in a poset.

• A Heyting algebra is a bounded closed lattice.
• A Boolean algebra is a Heyting algebra satisfying the law of excluded middle.

A Heyting algebra is finitely distributive because it is finitely bicartesian and closed.

• A inflattice is a cartesian poset.
• A suplattice is a cocartesian poset.
• A complete lattice is a bicartesian poset.

By the adjoint functor theorem for posets, all joins $$\equiv$$ all meets, i.e., inflattice = suplattice = complete lattice.

A inflattice/suplattice homomorphism is a monotone preserving meets/joins.

### 10.3.1$$(\N, \leq)$$

 terminal object $$-$$ initial object $$0$$ $$0 \leq a$$ product $$\min$$ $$c \leq a, c \leq b \equiv c \leq \min(a, b)$$ coproduct $$\max$$ $$a \leq c, b \leq c \equiv \max(a, b) \leq c$$ exponential object $$-$$

### 10.3.2$$(\N, \mid)$$

 terminal object $$0$$ $$a \mid 0$$ initial object $$1$$ $$1 \mid a$$ product $$\gcd$$ $$c \mid a, c \mid b \equiv c \mid \gcd(a, b)$$ copoduct $$\lcm$$ $$a \mid c, b \mid c \equiv \lcm(a, b) \mid c$$ exponential object $$-$$

### 10.3.3$$(\set{\lfal, \ltru}, \lfal \lproves \ltru)$$

 terminal object $$\ltru$$ initial object $$\lfal$$ product $$\lcon$$ $$C \lproves A, C \lproves B \equiv C \lproves A \lcon B$$ copoduct $$\ldis$$ $$A \lproves C, B \lproves C \equiv A \ldis B \lproves C$$ exponential object $$\limp$$ $$C \lcon A \lproves B \equiv C \lproves (A \limp B)$$

### 10.3.4$$([0, \infty], \geq)$$

 terminal object $$0$$ $$a \geq 0$$ initial object $$\infty$$ $$\infty \geq a$$ product $$\max$$ $$c \geq a, c \geq b \equiv c \geq \max(a, b)$$ copoduct $$\min$$ $$a \geq c, b \geq c \equiv \min(a, b) \geq c$$ exponential object $$\begin{cases}0 & a \geq b \\ b & \text{otherwise}\end{cases}$$ $$\max(c, a) \geq b \equiv c \geq \begin{cases}0 & a \geq b \\ b & \text{otherwise}\end{cases}$$

## 10.4 Met

metric spaces $$(A, d_A: A \times A \to [0, \infty])$$ and metric maps $$d_A(a, a') \geq d_B(f(a), f(a'))$$.

 terminal object singleton $$(\set{*}, d)$$ initial object empty set $$(\varnothing, d)$$ product product metric $$d_{A \times B}((a, b), (a', b')) := \max(d_A(a, a'), d_B(b, b'))$$ coproduct disjoint metric $$d_{A + B}(a, a') := d_A(a, a')$$, $$d_{A + B}(b, b') := d_B(b, b')$$, $$d_{A + B}(a, b) = d_{A + B}(b, a) := \infty$$