Category Theory (Lecture)

I followed this course by, Prof. Benno van den Berg during my 5th Semester at the ILLC Amsterdam for my M.Sc. Logic (UvA). A full description of the course can be found in the MasterMath website at link. I also collect other works I do in Category Theory.

Assignments

I use https://q.uiver.app to draw the diagrams.

Lecture Notes

Functor

A functor $F$ , is $F_{0} : C_{0} \to D_{0}$ and $F_{1} : C_{1} \to D_{1}$ s.t.:
1. $\forall_{f : X \to Y} F_{1} (f) : F_{0} (X) \to F_{0} (Y)$
2. $\forall_{f, g \in C_{1}} ((X \to^{f} Y \to^{g} Z) \to F_{1} (g f) = F_{1} (g) F_{1} (f))$
3. $\forall_{X \in C_{0}} F_{1} (1_{X}) = 1_{F_{0} (X)}$
Functors preserve isomorphisms, 2.3
A Functor preserves a property $P$ if $\forall_{x \in C_{0} \cup C_{1}} P (F (x)) \to P (x)$ , 2.4
for $F$ let $\forall_{A, B \in C_{0}} F_{A, B} : C (A, B) \to D (F (A), F (B))$ :
1. if surj. $F$ is full
2. if inj. $F$ if faithful
  - if $F$ faithful, then $F$ reflects epsi and monos. 2.6
3. if bij. $F$ is fully faithful
Product Category: $C \times D$
1. $O b (C \times D) := {(C_{0}, D_{0}) : C_{0} \in O b (C) \land D_{0} \in D}$
2. $Ho m_{C \times D} ((C_{0}, D_{0}), (C_{1}, D_{1})) := Ho m_{C} (C_{0}, C_{1}) \times Ho m_{D} (D_{0}, D_{1}))$
3. $(g_{0}, g_{1}) \circ (f_{0}, f_{1}) := (g_{0} \circ f_{0}, g_{1} \circ f_{1})$
Opposite Category: $C^{o p}$
1. $O b (C^{o p}) := O b (C)$
2. $Ho m_{C^{o p}} (A, B) := Ho m_{C} (B, A)$
3. $g \circ_{C^{o p}} f := f \circ_{C} g$
Duality principle: of any definition, I can always find a dual (co-)
- initial / terminal object
  - Terminal objects are unique up to unique isomorphism
- $f : A \to B$ is a mono iff. $f g = f h \to g = h$
  - mono are “often” inj.func
    - proved for $Set$
    - in $Cat$ : $F$ mono iff. $F$ inj.on.obj. and faithful, ex.1
- $f : A \to B$ is epi iff. $g f = h f \to g = h$
  - epi are “often” surj.func
    - proved for $Set$
    - in $Cat$ : $F$ surj.onobj. and full $\Rightarrow$ $F$ full., ex.1
Functorial Structures
- Example: $R$ a ring, $M_{2} (R)$ the $2 \times 2$ matrixes with components in $R$ ,
  - is $M_{2} : Ring \to Ring$ , $f : R \to S$ , $M_{2} (f) : M_{2} (R) \to M_{2} (S), M_{s} (f) (a, b c, d) = (f (a), f (b) f (c), f (d))$
- $F : C \to D$ , if $α \in C_{1}$ iso, then $F (α)$ iso.

Natural Transformation

for $C, D$ ex. $[C, D]$ cat. s.t. $[C, D]_{0} = Ho m_{Cat} (C, D)$ , $[C, D]_{1}$ are natural transformations
- for $F, G : C \to D$ a nat.tr. $σ : F \Rightarrow G$ is a fam. of morph. $\forall_{C \in C_{0}} σ_{C} : F (C) \to G (C)$ s.t.
  1. for $\forall_{C, C^{'} \in C_{0}} \exists_{α \in Ho m_{C} (C, C^{'})} D ia g . C o mm .$ , (Naturality Condition)
    1. $D ia g$ : $F (C) \to^{F (α)} F C^{'} \to^{σ_{C^{'}}} G (C^{'})$ and $F (C) \to^{σ_{C}} G (C) \to^{G (α)} G (C^{'})$
- if $σ : F \Rightarrow G$ in $[C, D]$ is iso. iff $\forall_{C \in C_{0}} i so . (σ_{C})$
- $\circ : [D, E] \times [C, D] \to [C, E]$ , functor composition is the object part of a functor (see diag), 3.3

Equivalence of Categories

$C ≃ D$ if there are functors $F : C \to D$ , $G : C \to D$ s.t.
1. $G \circ F ≅ 1_{C} \in [C, C]_{0}$
2. $F \circ G ≅ 1_{D} \in [D, D]_{0}$
- it is called equivalence and $G$ is the pseudo-inverse
- also $≅$ is natural isomorphism of categories
Equivalence preserves and reflects terminal objects (and most other properties)
$F : C \to D$ is an equivalence iff:, 3.6
1. $F$ is essentially surjective and…
  - that is: $\forall_{D \in D_{0}} \exists_{C \in C_{0}} F_{0} (C) ≅ D$
2. … $F$ is fully faithful
  - $F$ is full and faithful
$C, D, E$ sm.cat., $D ≃ E \to [C, D] ≃ [C, E]$ , ex.2.2

Limits

for $F : C \to D$ , a cone is $(D, μ)$ s.t. $D \in D_{0}$ , nat.tr. $μ : Δ_{D} \Rightarrow F$ (cone), s.t.
1. $(μ_{C} : D \to F (C) : C \in C_{0})$ with nat.req., 4.1
- $Δ_{D} : C \to D$ , if $\forall_{C \in C_{0}} (Δ_{D})_{0} (C) = D$ and $\forall_{f \in C_{1}} (Δ_{D})_{0} = 1_{D}$
- for $D \in D_{0}$ , $Δ_{g} : Δ_{D} \to Δ_{D^{'}}$ s.t. $(Δ_{g})_{C} = g$
  - hence $Δ : D \to [C, D]$
$(D, μ) \to (D^{'}, μ^{'})$ , is a map $g : D \to D^{'}$ s.t. $μ^{'} \circ Δ_{g} = μ$ (map of cones), 4.1
a lim.cone for $F$ is term.obj of $C o n e (F)$ (limiting cone). 4.2
a lim.con for $! : 0 \to D$ is term.obj. in $D$
- $C o n e (!) ≅ D$
Product Category 4.1.2
- let $F : 2 \to D$ , a product is the vertex of the limiting cone, called $(A \times B, π)$ .
- $2$ only has elements $A, B$ and only reflexive arrows.
Equalizer 4.1.3
- for $F : \hat{2} \to D$ let $(D, μ)$ be a cone, if it is limiting $μ_{A}$ is an equalizer.
  - $D \to^{μ_{A}} A \to^{f, g} B$ .
- $\hat{2}$ is $A \to^{f, g} b$
Pullbacks: 4.1.4
- for $F : J \to D$ , a let $(W, p)$ be the lim.cone, its diagram is the pullback diagram
- $J$ has $x, y, z$ as objects and arrows $y \to^{b} z$ and $x \to^{a} z$ .
$C$ has lim.of.sh. $I$ if $\forall_{F : I \to C} \exists_{(D, μ \in C o n e (F))} lim.cone. (D, μ)$ , 4.3
if $C$ is s.t. $\forall_{sm.cat (I)} C lim.of.sh (I)$ it is said complete.
- if only on all $I$ s.t. $∣ I_{0} ∣ < ℵ_{0}$ it is fin.compl.
  - fin.compl.cat are also called lex, left exact or cartesian.
- $C$ has …
  - prod. if complete on all $F : 2 \to C$
  - equal. if complete on all $F : \hat{2} \to C$
  - pullb. if complete on all $F : J \to C$
- for $M : I \to C$ , and $(C, μ)_{M}$ lim.cone, say it preserved by $F : C \to D$ if $(F (C), F μ)$ lim.con, 4.4
  - $F μ := {F (μ_{I}) : I \in I_{0}}$
  - $F$ pres.lim.of.shape. $I$ if it preserves every lim.cone of $G : I \to C$
    - it preserves small/finite limits if it preserves limits of shape $I$ for each $I$ small/finite.
complete categories:
- sets, top, grps, ring, ab.gr, vs. over a fixed field, cat. (fields are not!)
Colimits 4.2
a colimiting cocone for $F$ is a limiting cone for $F^{o p}$ .
- $(ν, D)_{F}$ s.t. $ν : F \Rightarrow Δ_{D}$ init.obj. in $C oco n e (F)$ .
- a colimiting cocone for $F : 2 \to C$ , denoted $A + B$ or $A ⊔ B$
- $C$ is cocomplete if all $I$ small have colimits of for diagrams $I \to C$ .
coprod. of proj. is proj., ex.3.1.b

Complete Categories

if $C$ has small products and equalizers, then $C$ is complete, 5.1
- also for $< k$ for $k$ a cardinal
- also co- version
if $C$ has term.obj and pullb. it has binary prod. and equal, 5.3
tfae: 5.4
- $C$ fin.compl.
- $C$ has fin.prod. and equil.
- $C$ has pullb. and a term.obj.
$D$ has lim.of.sh. $I$ , then so does any $[C, D]$ , 5.5
- $\forall_{C} compl. (D) \to compl. ([C, D])$
for $F : C \to D$ an equiv. and $I$ a cat., $F$ preserves and reflects limits of $I$ , 5.6
if $D$ complete has lim.of.sh. $I$ , does $C$
- $compl. (D) \land equiv. (F : C \to D) \to compl. (C$
if $small. (C) \land compl. (C)$ , then $C$ is a preorder

Cartesian Closed Categories

in $Set$ , for $C_{1} ∋ f : X \to Y$ mono:
- $X, Y \in C_{0}$ are iso. if $\exists_{f, g \in C_{1}} f : X \to Y \land g : Y \to X \land mono. (f) \land mono. (g)$
- for $f : X \to Y$ , there is a coproduct diagram with $f$ and $g : Z \to Y$
- given the AC, every epi is split.
Exponentials: for $C$ and $X, Y \in C_{0}$ , $X \times Y$ and $1_{X}$ are distinguished
- $X^{Y} \in C_{0}$ , $C_{1} ∋ e v : X^{Y} \times Y \to X$ s.t. $\forall_{A \in C_{0}} \forall_{C_{1} ∋ h : A \times Y \to X} \exists!_{C_{1} ∋ H : A \to X^{Y}} e v \circ (H \times 1_{Y}) = h$ , 6.1
  - equivalently $Hom_{C} (A, X^{Y}) \to Hom_{C} (A \times Y, X) : H \mapsto e v \circ (H \times 1_{y})$ is a bij.
- $C$ is ccc if it has fin.prod. and every $X^{Y}$ exists.
- in $Cat$ ccc holds with $D^{C} := [C, D]$ , $e v_{0} (F, C) = F (C)$ , $e v_{1} (σ, f) = σ_{C} (f)$
  - (note, nat.tr. are precisely such $σ$ )
- for $C$ with fin.prod., is called exponentiable if all exponentials exists
- 6.6?
Natural Numbers Object
- in $C$ with term.obj. $1$ , a nat.num.obj. is $(0, N, S) \in C_{1} \times C_{0} \times C_{1}$ s.t.:
  1. $1 \to^{0} N$ , $N \to^{S} N$
  2. $\forall_{1 \to^{x} X \to^{f} X} \exists!_{φ : N \to X} diag.comm.$
- in $C$ with ccc and nat.num.obj. $(0, N, S)$ call $f : N^{k} \to N$ a num.th.fun. corresponding to $F : N^{k} \to N$ if $\forall_{(n_{1}, ..., n_{k}) \in N^{k}} f \circ ⟨ S^{n_{1}}, ..., S^{n_{k}} ⟩ = S^{F (n_{1}, ..., n_{k})}$
  - PA can be done within this frame

Presheaves

for $C$ sm.cat., the presh. $C$ is $\hat{C} := [C^{o p}, Set]$ , 7.1
- $\hat{C}$ is complete and cocomplete, limits calculated pointwise
- for $D$ with pullb., $σ \in [C, D]_{0}$ is mono iff $\forall_{C \in C_{0}} mono. (σ_{C})$ , 7.3
  - for $D$ with pusho., $σ \in [C, D]_{0}$ is epi iff $\forall_{C \in C_{0}} epi. (σ_{C})$
Yoneda Lemma: 7.2
- for $C$ loc.sm.cat, $Hom_{C} : C^{o p} \times C \to Set$ , hence $Hom_{C} (A, B) \in Set$
- also, for $(C^{o p} \times C)_{1} ∋ (f, g) : (A, B) \to (A^{'}, B^{'})$ , $Hom_{C} (A, B) \to Hom_{C} (A^{'}, B^{'})$ s.t. $(h : A \to B) \mapsto g \circ h \circ f : A^{'} \to B^{'}$ .
- since $Set^{C^{o p}}$ is an exp.cat., $Hom_{C}$ corresponds to $y : C \to Set^{C^{o p}}$
  - since $y : C \to [C^{o p}, Set]$ , first take $C \in C$ , then $D \in C^{o p}$ .
  - so, for $C \in C_{0}$ , we have $(y_{C}) (D) := (y (C)) (D) = Hom_{C} (D, C)$
  - and, for $C_{1} ∋ f : D \to D^{'}$ , get $(y_{C}) (D^{'}) \to (y_{C}) (D)$ , comp.w. $f$
  - and, for $C_{1} ∋ g : C \to C^{'}$ , get $C_{2} ∋ y_{g} : y_{C} \to y_{C^{'}}$ s.t. $(y_{g})_{D} : (y_{C}) (D) \to (y_{C^{'}}) (D)$ , com.w. $g$
- $\forall_{F \in [C^{o p}, Set]_{0}} \forall_{C \in C_{0}} bij. (α_{C, F} : Set^{C^{o p}} (y_{C}, F) \to F (C))$ , $(σ : y_{C} \to F) \to (σ_{C}) (1_{C})$ s.t. (Yoneda)
  1. for $C_{1} ∋ g : C^{'} \to C$ , $Set^{C^{o p}} ∋ μ F \Rightarrow F$ , diag.com. Yoneda_Naturality.jpeg (Naturality)
  2. $C (C, C^{'}) := {f \in C_{1} : f : C \to C^{'}}$ , 7.4
  - hence, $α$ os a nat.iso. s.t. $(C^{o p} \times Set^{C^{o p}} \to^{e v} Set) ≅^{α} C^{o p} \times Set^{C^{o p}} \to^{y \times 1} (Set^{C^{o p}})^{o p} \times Set^{C^{o p}} \to^{Hom} Set$
  - $X \in \hat{C}$ is repr. if $\exists_{C \in C} X ≅ y_{C}$ , 7.5
  - for $X \in \hat{C}$ , $y ↓ X$ is a comm.cat. if
    - $(y ↓ X)_{0} := {(C, μ) : C \in C_{0} \land \hat{C}_{1} ∋ μ : y_{C} \to X}$
    - $(y ↓ X)_{1} := {(C, μ) \to (C^{'}, ν) ∣ f : c \to C^{'} \land ν \circ y_{f} = μ}$
  - $U_{X} : y ↓ X \to C, (C, μ) \mapsto C, f \mapsto f$
  - $y \circ U_{X} : y ↓ X \to Set^{C^{o p}}$ , recall $Δ_{X} : y ↓ X \to \hat{C}$
  - $ρ_{X} : y \circ U_{X} \to Δ_{X}$
  - $(X, ρ_{X})$ is colimiting cocone in $\hat{C}$ , 7.6
    - every presheaf is a colimit of representables
  - $\hat{F} := F \circ U_{X}$ , so $\hat{F} : y ↓ X \to D$
  - $y$ is inj. and fully faithful, 7.7
- $\hat{C}$ is ccc
  - as an example of limit, take the exponentials
    - for $C \in C_{0}$ , $Y^{X} (C) := \hat{C} (y_{C} \times X, Y)$
    - for $C_{1} ∋ f : C \to C^{'}$ , $Y^{X} (f) : Y^{X} (C) \to Y^{X} (C^{'})$ ?
- $\forall_{X \in \hat{C}_{0}} \exists_{C \in C_{0}} X ≅ y_{C} \to proj. (X)$ , ex.3.1.c
- a presh is proj. iff it is a retract of a copr. of repres.

Presheaves as a Topos

a topos is a ccc.w.fin.lim with a subobject classifier
for $E$ cat.fin.lim. a subo.cl. is a $t : T \to Ω$ s.t.:
1. $t$ is mono
2. $\forall_{m \in E_{1}} mono. (m) \to \exists!_{E ∋ ϕ : B \to Ω} t \circ ψ = ϕ \circ m$
  - $ϕ$ is classifies $m$ , or the subobject represented by $m$
  - since $t$ is mono, there is a unique such $ψ$ too.
- hence, $T$ is terminal, 8.3
- call $t : T \to Ω$ subo.cl. or also just $Ω$
for $X \in E_{0}$ , def. ${E_{1} ∋ f : A \to X ∣ A \in E_{0} \land mono. (f)}$ and order
1. $(m : A \to X) \leq (n : B \to X) := \exists_{E_{1} ∋ k : A \to B} n \circ k = m$
  - $k$ is then unique and mono, 8.4
- for $k$ iso, we get eq.classes which we call subo., write $Sub_{E} (X)$
  - if $Sub_{E} (X)$ is a set, call $E$ well-powered, this we assume for $E$ .
- in $Set$ , if $m : A \to X$ inj., $A ≅ I m (m) \subseteq X$
- for $E_{1} ∋ f : Y \to X$ , $f^{*} : Sub_{E} (X) \to Sub_{E} (Y)$
  - $f^{*} \in Cat_{1}$ and monotone
  - if $E$ small, $Sub_{E} : E^{o p} \to Set$
  - having a subo.cl. is eq. to saying that $Sub_{E}$ is repr.
  - hence, $Sub (-) : E^{o p} \to Set, A \mapsto Sub (A), f \mapsto f^{*}$ lec.10.1.5:36
  - $E$ has subo.cl. if $Sub (-)$ is representable lec.10.1.6:19
    - ex. $Ω \in E_{0}$ s.t. $E (-, Ω) ≅ Sub_{E} (-)$ nat.iso.
for $A \in E_{0}$ , $A$ is pow.obj. of $X \in E_{0}$ if $E (Y, A) ≅ Sub_{E} (X \times Y)$ s.t.
1. $E_{1} ∋ f : Y \to A, g : Z \to Y$ and $f$ corr. to the subo. $U$ of $X \times Y$ , then $f g : Z \to A$ corr. to the subo. $(i d_{X} \times g)^{*} (U)$ of $X \times Z$ .
  - correspondence is given by the subo.cl.,
- note $E (Y, A), Sub_{E} (X \times Y) \in Set_{0}$ , hence " $≅$ " only requires a bij.
- pow.obj. are unique up to iso., usually denoted with $P (X)$
Subobject Classifiers in Presheaves
if $X \in \hat{C}_{0}$ , a subpresh $Y$ of $X$ is s.t.:
1. $\forall_{C \in C_{0}} Y (C) \subseteq X (C)$
2. $\forall_{f \in C_{1}} Y (f) = X (f) ∣_{Y (co d (f))}$
- let $SubPSh_{C} (X)$ be the set of all supresh. on $C$ of $X$
$Sub_{C} (X) ≅ SubPSh_{C} (X)$
let $Ω$ s.t. $Ω (C) \in SubPSh_{\hat{C}} (y_{C})$ , $Ω (f)$ is defined by pulling back along $y_{f}$
- if $R$ is a sieve on $C$ , then $R ∋ f : C^{'} \to C \to \forall_{g : C^{''} \to C^{'}} f g = y_{C} (g) (f) \in R$
- $R ↭ Ω (C) \in SubPSh_{\hat{C}} (y_{C})$
- for $R ∋ f : C^{'} \to C$ , $f^{*} (R) = {g : D \to C^{'} ∣ f g \in R_{C}}$
- hence $Ω = {R : sieve.on. C (R)}$ , $Ω (f) (R) = f^{*} (R)$
- $t : 1 \to Ω, 1 (C) \mapsto max.sieve.on. C$
  - $max.sieve.on. C$ is the unique one containing $i d_{C}$
for $E$ a topos, we have $Hom_{E} (Y, Ω^{X}) ≅ Hom_{E} (X \times Y, Ω) ≅ Sub_{E} (C, \times Y)$ natuarally in $E$ .

Adjunctions

$Cat_{1} ∋ F : C \to D, G : D \to C$ , an adj. is a nat.bij $m$ or $F ⊣ G$ s.t.:
1. $\forall_{C \in C_{0}} \forall_{D \in D_{0}} m_{C, D} : D (FC, D) \to C (C, G D)$
  - hence maps $C_{1} ∋ f : F D \to C$ , $D_{1} ∋ g : D \to GC$ are called transposes of each others
  - write $\overline{f} := m (f) = g$ , note $\overline{\overline{f}} = \overline{g} = f$ .
2. $\forall_{C_{1} ∋ f : C \to C^{'}} \forall_{D_{1} ∋ g : D \to D^{'}} \forall_{D_{1} ∋ α : FC \to D} G g \circ \overline{α} \circ f = \overline{g \circ α \circ F f}$ (Naturality)
  - this is eq. to $f = 1$ , $g = 1$ s.t. $G g \circ \overline{α} = \overline{g \circ α}$ and $\overline{α} \circ f = \overline{α \circ F f}$ .
for adj. as above, fix $D \in C_{0}$ , by nat. $C^{o p} \to Set : C \mapsto D (FC, D)$ is repr. by $G D$ .
- by Yon. $D_{1} ∋ ε_{D} = m_{G D, D}^{- 1} : FG D \to D$ , s.t. $m_{C, D}^{- 1} (β) = ε_{D} \circ Fβ$
- dually: for $C \in C_{0}$ , $η_{C} = m_{C, FC} (1_{FC} : C \to GFC)$ s.t. $m_{C, D} (α) = G α \circ η_{C}$
${η_{C} : C \in C_{0}} ⇝ (1_{C} \Rightarrow GF)$ and ${ε_{D} : D \in D_{0}} ⇝ (FG \Rightarrow 1_{D})$ , 9.2
- call $η$ the unit and $ε$ the
- if $m_{C, D}$ is inv. of $m_{C, D}^{- 1}$ iff $G \Rightarrow^{η ⋆ G} GFG \Rightarrow^{G \circ ε} G = 1_{G}$ and $F \Rightarrow^{F \circ η} FGF \Rightarrow^{ε ⋆ F} F = 1_{F}$
  - $η ⋆ G := {η_{G D} ∣ D \in D_{0}}$ and $G \circ ε := {G (ε_{D}) ∣ D \in D_{0}}$
for $Cat_{1} ∋ F : C \to D, G : D \to C$ , $η : 1_{C} \Rightarrow GF$ and $ε : FG \Rightarrow 1_{D}$ s.t.
1. $(G \circ ε) \cdot (η ⋆ G) = 1_{G}$
2. $(ε ⋆ F) \cdot (F \circ η) = 1_{F}$
- then: $F ⊣ G$ with $η$ unit and $ε$ counit, 9.3
$F : C \to D$ eq.cat. with $G$ ps.inv, for $μ : 1_{C} \Rightarrow GF$ , $ν : FG \Rightarrow 1_{D}$ , then $F ⊣ G$ with unit $μ$ and one with counit $ν$ (not nec. at the same time), 9.4
if $F ⊣ G$ then $F$ pres.colim., $G$ pres.lim.
- $F$ pres. init.obj.
- if $F$ has right.adj., let $M : E \to C$ , then $\hat{F} : Cocone (M) \to Cocone (FM)$ has right.adj. too, 9.7

Monads and Algebras

for $F : C \to D$ and $F ⊣ G$ , let $T = GF : C \to C$ ,
1. note nat.tr. $η : 1_{C} \Rightarrow T$
2. also $μ : T^{2} \Rightarrow T$ , for $μ_{C} = T^{2} (C) = GFGFC \to^{G_{(ε_{FC})}} GFC = T (C)$
3. see diag. 10.1
for $C$ cat, a monad is $(T, η, μ)$ s.t.:
1. $T : C \to C$
2. $η : 1_{C} \Rightarrow T$ nat.tr.
3. $μ : T^{2} \Rightarrow T$
4. and s.t. 10.1 holds
- note that it is in fact a monoid ?
for $(T, η, μ)$ on a cat. $C$ , the cat. $T$ -Alg is:
1. $(Alg_{C, T})_{0} := {(X, h) ∣ X \in C_{0} \land C_{1} ∋ h : (T (C) \to X \land T^{2} (C) \to^{T (h)}$ $T (X) \to^{h} X = T^{2} (X) \to^{μ_{X}} T (X) \to^{h} X \land X \to^{η_{X}} T (X) \to^{h} X = 1_{X}}$
2. $(Alg_{C, T})_{1} := {(X, h) \to (Y, k) ∣ f : X \to Y \land f \circ h = k \circ T (f)}$
there is an adj. $F ⊣ G$ between $C$ and $Alg_{C, T}$ and it is ge. by $T = GF$ ., 10.4
for $F : C \to D$ and $F ⊣ G$ , let $K : D \to Alg_{C, T}$ s.t.
1. $K_{0} (D) = (GFG (D) \to^{G (ε_{D})} G (D))$
2. $K_{1} (f : D \to D^{'}) = G_{1} (f)$
- nat. of $ε$ proves well-definability.
- call $G : D \to C$ monadic (or that $D$ over $C$ is) if $K$ is an equiv.

Presheaves Revisited

for $C$ sm.cat, let $X \in \hat{C}_{0}$ recall $\int_{C} X := y ↓ X$ , call it category of elements.
$x : (y_{C} \to X) ↭ x \in X (C)$ , we can then think of $(C, x) \in (\int_{C} X)_{0}$ with $x \in X (C)$
for $U : y ↓ X \to C$ forg.fun., def. $y \circ U : y ↓ X \to Psh (C)$
- $(X, ρ)$ is a lim.cocone on $y \circ U$ , 11.2
- $\forall_{(C, x) \in (y ↓ X)_{0}} \exists_{(y ↓ X)_{1} ∋ y U (C, x)} y U (C, x) = y_{C} \to X = x$
- every presheaf is a colimit of representables
Kan Extensions
- for $C$ sm.cat., $y : C \to PSh (C)$ is s.t.
  1. $\forall_{E \in Cat_{0}} loc.sm.cat (E) \land cocom. (E) \to \forall_{Cat_{1} ∋ F : C \to E} \exists_{Cat_{1} ∋ F_{!} : PSh (C) \to E} F_{!} \circ y ≅ F$ , 11.3
  - call $F_{!}$ the left Kan extension
- for $f : C \to D$ for $C, D$ sm.cat., the prec.fun. $f^{*} : \hat{D} \to \hat{C}$ , $f^{*} (Q) (C) = Q (f C)$ , has both adj., 11.5

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