Category Theory

‎

1. Notation
2. Basic definitions
3. Limits and colimits; definition and basic examples
- 3.1. Colimits
  - 3.1.1. Formal definition of a colimit
- 3.2. Weighted limits: What is a weighted limit?
4. Weighted limits, colimits, Kan extensions as weighted limits
5. The nerve-realization pairing with weighted colimits; the dual of Kan's theorem
- 5.1. TODO Find examples of this dual nerve-realization !!
6. Simplifying a weighted colimit, with special attention to the case of simplicial Abelian groups
- 6.1. Realization
- 6.2. TODO Give a technique for analyzing a category and showing how one can replace it by a simpler, Morita equivalent one; especially in the Dold-Kan case.
7. Day convolution in a promonoidal category
8. Split idempotents, absolute limits and colimits
- 8.1. Karoubi Completion
9. Yoneda as an end, Density lemma (Co-yoneda) as a coend

1 Notation

2 Basic definitions

2.1 Diagram

The word "diagram", formally, is syntactic sugar for "functor." That is, when $\mathcal{C}$ is a category, and we speak of a diagram in $\mathcal{C}$ - say, a commutative square with objects $A, B, C, D$ and maps $f : A\to B, g : B\to D, h : A\to C, k : C\to D$ satisfying $g\circ f = k\circ h$, by this we mean (if we intend to speak perfectly formally, restricting ourselves to only the basic concepts of category theory) that we introduce a category $\mathcal{I}$ (called the "shape of the diagram"), which has four objects, $A_I, B_I, C_I$ and $D_I$, and five non-identity morphisms, $f_I : A_I\to B_I,g_I : B_I\to D_I, h_I : A_I\to C_I, k_I : C_I\to D_I$, and $k_I\circ h_I = g_I\circ f_I$, and a functor $F : \mathcal{I}\to \mathcal{C}$ carrying $A_I$ to $A$, $f_I$ to $I$, and so on.

2.2 Discrete

A category $\mathcal{C}$ is discrete if, for $i,j\in \operatorname{Obj}(\mathcal{C})$, $\operatorname{Hom}(i,j) =\varnothing$ for $i\neq j$ and $\operatorname{Hom}(i,i) = \left\{ \operatorname{id}_i \right\}$.

2.3 Constant

Let $\mathbf{1}$ denote a category with one object and one morphism. $\mathbf{1}$ is a terminal object in $\mathbf{Cat}$. Functors $\mathbf{1}\to \mathcal{D}$ are in one-to-one correspondence with objects of $\mathcal{D}$. The canonical functor $\mathcal{C}\to \mathbf{1}$ is denoted $!$. A functor $F: \mathcal{C}\to \mathcal{D}$ is constant if it factors as $d\circ ! : \mathcal{C}\to \mathbf{1}\to \mathcal{D}$.

2.4 Presheaf

If $\mathcal{C}$ is a category, a presheaf on $\mathcal{C}$ is a contravariant functor $X : \mathcal{C}^{\rm op}\to \mathbf{Sets}$.

3 Limits and colimits; definition and basic examples

3.1 Colimits

Colimits are a kind of gluing construction. Given a diagram of objects and maps between them, the colimit of the diagram is the object constructed by "gluing the objects together along the maps." In the category of $\mathbf{Sets}$ we can make this precise as follows: if $\mathcal{I}$ is a small category and $F :\mathcal{I}\to \mathbf{Sets}$ is a diagram of sets and functions indexed by the arrows and objects of $\mathcal{I}$, then the colimit of $F$ is constructed as follows: first, form the disjoint union $X = \coprod_{i\in \operatorname{Obj}(\mathcal{I})}F(i)$ of all sets in the diagram, i.e., the set of all ordered pairs $(a,i)$ with $i\in \operatorname{Obj}(\mathcal{I})$ and $a\in F(i)$. Let $\sim$ be the smallest equivalence relation on $X$ such that $(a,i)\sim (b,j)$ such that $t : i\to j$ is a morphism in $\mathcal{I}$ and $F(t)(a)=b$. Then $X/\sim$, the set of equivalence classes of $X$ under the equivalence relation $\sim$, is the colimit of $F$.

If we replace $\mathbf{Sets}$ with $\mathbf{Top}$, the category of topological spaces, then this description also gives the colimit of (the underlying set of) a diagram of topological spaces and continuous maps. The topology on $X/\sim$ is defined as follows: for each $i$ in $\operatorname{Obj}(\mathcal{I})$, there is a canonical map $\lambda_i: F(i)\to X/\sim$ sending $a$ to the equivalence class of $(a,i)$; we impose on $X/\sim$ the weakest topology such that each $\lambda_i$ is continuous, i.e. $U\subset X/\sim$ is open iff the preimage $\lambda^{-1}_i(U)$ is open in the space $F(i)$ for every $i$.

The two most important cases to consider are :

when $\mathcal{I}$ is a discrete category; in this case the equivalence relation $\sim$ is the trivial one, and the colimit is simply the disjoint union of the spaces $F(i)$; we call this the coproduct of the spaces $F(i)$.
when $\mathcal{I}$ has two objects $i,j$ and two distinct non-identity morphisms $f, g : i\to j$; then a functor $F : \mathcal{I}\to \operatorname{Top}$ is a pair of spaces $A,B$ together with continuous maps $f,g: A\to B$; in this case the colimit of the diagram is the quotient space $B/\sim$ where $\sim$ is the equivalence relation generated by requiring that $b\sim b'$ whenever $\exists a\in A$ with $f(a)=b, g(b)=b'$; the topology on $B/\sim$ is the weakest one such that $B\to B/\sim$ is continuous. We call this space the coequalizer of $f,g$.

Actually, all colimits can be constructed out of these basic building blocks of colimits and coequalizers.

3.1.1 Formal definition of a colimit

We give a few equivalent definitions.

Let $\mathcal{C}$, $\mathcal{D}$ be categories; let $F: \mathcal{C}\to \mathcal{D}$ be a functor. A cocone out of $F$ is a pair consisting of

a constant functor $d: \mathcal{C}\to \mathcal{D}$, and
a natural transformation $\lambda: F\to d$.

A morphism of cocones $\alpha : (d,\lambda)\to (d',\lambda')$ ($F$ is regarded as fixed) is a natural transformation $\alpha : d\to d'$ such that $\lambda'=\alpha\circ \lambda$. This defines the category of cocones out of $F$. A colimit of $F$ is an initial object in the category of cocones out of $F$.

Another definition: Let $\left[ !;\mathcal{D} \right]$ be the functor $\left[ 1;\mathcal{D} \right]\to \left[ \mathcal{C};\mathcal{D} \right]$ sending $d$ to $d\circ !$, i.e. "precompose with $!$". Then a colimit of $F$ is a universal arrow $\left( F\downarrow \left[!;\mathcal{D} \right] \right)$.

We will see later that a colimit is one of the simplest examples of a left Kan extension; it is the left Kan extension of $F: \mathcal{C}\to \mathcal{D}$ along $! : \mathcal{C}\to \mathbf{1}$. If you don't know what this means, it suffices to read it as saying: a colimit is a canonical choice of extension of $F: \mathcal{C}\to \mathcal{D}$ along $! : \mathcal{C}\to \mathbf{1}$ making the diagram of functors commute up to natural transformation.

3.2 Weighted limits: What is a weighted limit?

We consider only the case $\mathcal{V}=\mathbf{Sets}$ here for the sake of building intuition; i.e. we do not consider any concepts of enriched category theory.

A weighted colimit, or an indexed colimit, is a kind of colimit with multiplicity. Consider a simplicial complex: it is built out of only a few basic pieces, the simplices $\Delta^0,\Delta^1,\dots$. We can agree that this is a kind of gluing construction, as the simplices are glued together along their faces, but the construction is somewhat complicated to be rendered as an ordinary colimit, as there may be many copies of the $n$-simplex embedded in the complex for each $n$. What we want is a construction that allows us to say

what the basic building blocks are that will be glued together in the construction
how many of each kind of building block we need
along what maps will the building blocks be glued
which building blocks will be glued to which other building blocks

To illustrate this concept we will examine the delta complexes of Hatcher. What are the building blocks? The simplices, $\Delta^0,\Delta^1,\Delta^{2},\dots$. Along what maps are they to be glued? The inclusions which embed the $n$ simplex as a face of the $n+1$ simplex.

More precisely, let $\Delta_d$ be the semisimplex category defined as follows. The objects of the category are the finite ordinals $0=\varnothing, 1= \left\{ 0 \right\}, 2 = \left\{ 0,1 \right\},\dots$. The morphisms of the category are the injective, monotonic maps between ordinals; thus there are ${{n+k} \choose n}$ maps from $n$ to $n+k$. $0$ is initial and there are no maps from $n+k$ to $n$ for $k\geq 1$.

Let $F : \Delta_d\to \mathbf{Top}$ be the functor which sends $n$ to the $(n-1)$ simplex $\Delta^{n-1}$, the convex hull of $n$ points $\left[ v_0,v_1,\dots, v_{n-1} \right]$ in an $n$ dimensional real vector space in general position. $F$ sends the ordinal map $f : n \to m$ to the unique linear map $\Delta^{n-1}\to \Delta^{m-1}$ carrying $v_i$ to $v_{f(i)}$ for each $i\in n$. We associate to $0$ the empty topological space.

The diagram $F$ contains half the information we want: it says what the building blocks are which will be glued together (the simplices $F(n)$) and along what maps they will be glued (the linear embeddings $F(f)$)

How many of each simplex will appear in the construction? In order to answer this, we will associate to each ordinal $n$ a set $X_n$. The cardinality of $X_n$ determines how many copies of the simplex $F(n) = \Delta^{n-1}$ appear in the final complex; each element of $X_n$ corresponds to one copy of $\Delta^{n-1}$, and they can be thought of as 'labels' for copies of $\Delta^{n-1}$.

Which simplices will be glued to which other simplices? As it stands now, the family of sets $\left\{ X_n \right\}_{n\in \omega}$ is insufficient to describe how the complex should be built. Each element of of $X_{n+1}$ names some copy of $\Delta^n$, and this copy of $\Delta^n$ has in turn $n+1$ $n-1$ dimensional faces, one for each vertex which can be deleted from $\Delta^n$. So, since each of these faces will be an $n-1$ simplex which appears in the final complex, they should have corresponding labels in $X_n$.

If $x\in X_n$, let $\Delta^{n-1}_x$ denote the copy of the $n-1$ simplex labelled by $x$. For each $n$ and each $i, 0\leq i\leq n$, we want to add a relation on $X_n$ and $X_{n+1}$, $R^i\subset X_n\times X_{n+1}$, where $R^i(x,y)$ means "$x$ is the $i$ th face of $y$". Actually this is a functional relation $\partial^i : X_{n+1}\to X_n$, as each $n$ simplex has one and only one $i$ th face. On the other hand, if $x$ is the label of an $n-1$ simplex, then this may be a face of no $n$ simplex at all, or of more than one $n$ simplex (if two $n$ simplices are glued together along this common face).

This data can be summarized by saying that we want $X$ to form a functor $\Delta^{\rm op}_d\to \mathbf{Sets}$ sending $n$ to the set $X_n$ of labels for copies of $n-1$ simplices. Whenever $f : n\to m$ is a morphism in $\Delta_d$, and $x\in X_m$, then $X(f)(x)$ tells which label in $X_n$ corresponds to the $n-1$ simplex embedded in $\Delta^{m-1}_x$ containing the vertices $\left[ v_{f(0)},\dots, v_{f(n-1)} \right]$.

Thus, the functor $F$ being fixed, a combinatorial schema for a delta complex is exactly a presheaf on $\Delta_d$.

Now, how do we carry out the gluing?

We first give an explicit construction. First, form the space which is the disjoint union of all the simplices referenced in any of the sets $X_n$,

\begin{equation} \coprod_{n\in \Delta_d, x\in X_n}\Delta^{n-1}_x \end{equation}

We write $a\in \Delta^{n-1}_x$ as $(n,x,a)$ when we need to explicit about which simplex it belongs to.

Let $\sim$ be the equivalence relation on this space generated by requiring that if $x\in X_m$ and $a\in \Delta^{n-1}_x$, and $f : n\to m$ is a morphism in $\Delta_d$, then we identify $(n,X(f)(x),a) \sim (m, x, F(f)(a))$. Then the simplicial complex associated to $X$ is given by

\begin{equation} \left( \coprod_{n\in \Delta_d, x\in X_n}\Delta^{n-1}_x \right)/\sim \end{equation}

where the quotient space is equipped with the usual identification topology.

In other words, whenever the mapping data of $X$ says that $y=X(f)(x)$X is the label for the face of $\Delta^{m-1}_x$ given by deleting all vertices in $\Delta^{m-1}_x$ except for those in the image of $f$, we glue $\Delta^{n-1}_y$ to $\Delta^{m-1}_x$ along the inclusion map $F(f)$. Let us call this space $\mathbf{R}_F(X)$, the geometric realization of $X$ (with respect to $F$).

Now let's look at what the universal property of the gluing construction should be. In general, the universal property of a gluing construction is given by explaining how to give a map out of the construction. How does one give a map out of a simplicial complex? The simplest way to say this is: If $X$ is a simplicial complex, and $Z$ is an arbitrary space, then to build a map from $X$ to $Z$ we should give, for each simplex $\sigma$ in $X$, a continuous map $g_{\sigma}: \sigma\to Z$; subject (of course) to the requirement that these play nice with taking face maps, i.e. if $\tau$ is the $i$ th face of $\sigma$, then the restriction of $g_{\sigma}$ to the face $\tau$ should be $g_{\tau}$. And that turns out to be perfectly sufficient as a definition. (Note that if $\sigma,\sigma'$ are two $n$ simplices glued together as a common face $\tau$, then saying that $g_{\sigma},g_{\sigma'}$ agree on $\sigma\cap \sigma'$ is the same as saying that $g_{\tau} = g_{\sigma}\mid_{\tau} = g_{\sigma'}\mid_{\tau}$. So if $g$ respects restricting along face maps, it will make sure that the two functions defined on two simplices agree when they are glued together along a common face.

The data we are giving then amounts to a family of maps $X_n\to \operatorname{Hom}(\Delta^{n-1}, Z)$ - i.e., for each element $\sigma$ of $X_n$ (each copy of the $n-1$ simplex that occurs in the complex), we give a continuous map $g_{\sigma}: F(n)\to Z$. Asserting that these play nicely with restriction-of-domain maps should mean that whenever $f : n\to m$, and $\sigma\in X_m$, $\tau = X(f)(x)$, the map $g_{\sigma}$ restricted to $\tau$ (i.e., precomposed with $F(f)$) should give exactly $g_{\tau}$. This is equivalent to saying that the maps $\sigma\mapsto g_{\sigma}$ are natural in the variable $n$, i.e. that we have a natural transformation

\begin{equation} g : X\to \operatorname{Hom}(F(-),Z) \end{equation}

Our assertion is that such a $g$ should then define a continuous map $\tilde{g} : \mathbf{R}_F(X)\to Z$ from the geometric realization. Conversely, any map from the geometric realization $k : \mathbf{R}_F(X)\to Z$ determines functions from each $n$-simplex into $Z$, simply by restricting $k$ to the embedded simplices $\Delta^{n-1}_x$. Thus, $\mathbf{R}_F(X)$ is defined by the universal property that there is a natural isomorphism

\begin{equation}\label{univ-weighted-colimit} \operatorname{Nat}_{[\Delta_d^{\rm op},\mathbf{Sets}]}(X, \operatorname{Hom}_{\operatorname{Top}}F(-),Z) \cong \operatorname{Hom}_{\mathbf{Top}}(\mathbf{R}_F(X), Z) \end{equation}

natural in $Z$. This is the universal property of the weighted colimit.

Actually there is some flexibility in how we choose to characterize the universal property. We could also describe the maps from $\mathbf{R}_F(X)$ into $Z$ as follows. A family of maps $X_n\to \operatorname{Hom}(\Delta^{n-1},Z)$ can just as well be described as a single map $\coprod_{x\in X_n}\Delta^{n-1}_x\to Z$; this is exactly the universal property of the coproduct of spaces. This operation of taking the coproduct of copies of a single object comes up decently often in category theory, sufficiently often that we give it its own name and refer to it as the $\mathbf{copower}$. We concisely denote it as $X_n\cdot \Delta^{n-1}$. In what follows it will be necessary to understand how $\cdot$ behaves as a functor $\mathbf{Sets}\times \mathbf{Top}\to \mathbf{Top}$; it is left as an exercise to determine how this operates on morphisms.

Thus, the data of maps defined on the simplices can be defined as a family of maps $X_n\cdot \Delta^{n-1}\to Z$ for each ordinal $n$. The coherence condition that these maps must satisfy subtle, as $X$ is a contravariant functor and $F$ is a covariant functor. Therefore, we cannot say that the maps $X_n\cdot F(n)\to Z$ determine a natural transformation. We must give a more general definition of naturality which accounts for this mixed-variance. Let $\left\{ g_n \right\}_{n\in \omega}$ be a family of maps $X_n\cdot F(n)\to Z$; then we say that the maps $g_n$ form a cowedge if they satisfy the following coherence condition: for each map $f : n\to m$ in $\Delta_d$, the following two maps $X_m\cdot \Delta^n \to Z$ agree:

\begin{equation} g_m\circ \left( \operatorname{id}_{X_m}\cdot F(f) \right) = g_n\circ \left( X(f)\cdot \operatorname{id}_{\Delta^n} \right) \end{equation}

Then we ask that $\mathbf{R}_{F}(X)$ have the universal property that cowedges from $X\cdot F$ into $Z$ are in one-to-one correspondence with continuous maps $\mathbf{R}_F(X)\to Z$, naturally in $Z$; we say $\mathbf{R}_F(X)$ is a coend of $X\cdot F$. (Note that if we set $Z:= \mathbf{R}_F(X)$ and look at the identity morphism $\mathbf{R}_F(X)\to \mathbf{R}_F(X)$, this corresponds to a kind of "universal" cowedge $X\cdot F\to \mathbf{R}_F(X)$; this is really what we mean when we talk about the coend of $X\cdot F$.

Theorem Let $n$ be an ordinal in $\Delta_d$, and $y$ the Yoneda embedding. Then $\mathbf{R}_F(y(n)) \cong F(n)$, naturally in $n$.

Thus, each object in the diagram $F$ can be expressed as a "gluing together" of objects in $F$; we would naturally hope that this is true. To understand what this says, we can visualize the system of weights that $y(n)$ assigns to the objects and maps of $F$. First of all, there is a distinguished copy of $F(n)$; the one associated to the identity map $n\to n$. Then, for every object $m$ and every map $f : m\to n$, we add to the diagram a copy of the object $F(m)$, which is glued to $F(n)$ along the map $F(f)$. Intuitively it is clear that when we take the colimit, each of these copies of $F(m)$ will simply be "absorbed" into $F(n)$ along the gluing maps. There can exist maps $F(m)\to F(m')$ but only those that commute with the gluing maps.

A rigorous proof can be given by the universal property. We want

\begin{equation} \operatorname{Hom}(\mathbf{R}_F(y(n)), Z)\cong \operatorname{Nat}(y(n), \operatorname{Hom}(F(-),Z)) \cong \operatorname{Nat}(F(n),Z) \end{equation}

where the second isomorphism is by an application of the Yoneda lemma. Thus it is clear we can take $\mathbf{R}_F(y(n)):=F(n)$ and it will have the desired universal property. By the Yoneda lemma, again, $\mathbf{R}_F(y(n))$ is uniquely determined by this universal property, so we can conclude $\mathbf{R}_F(y(n))\cong F(n)$.

Actually, $\mathbf{R}_F : \left[\Delta^{\rm op}_d, \mathbf{Sets} \right]\to \mathbf{Top}$ is a kind of canonical choice of extension of $F: \Delta_d\to \mathbf{Top}$ along the Yoneda embedding $y : \Delta_d\to [\Delta_d^{\rm op}; \mathbf{Sets}]$; it is the left Kan extension of $F$ along $y$.

Theorem $\mathbf{R}_F$ is functorial in the presheaf $X$ and is left adjoint to the functor $\mathbf{N}_F : \mathbf{Top}\to \left[ \Delta^{\rm op}; \mathbf{Sets} \right]$ defined by

\begin{equation} \mathbf{N}_F(Z) = n\mapsto \operatorname{Hom}(F(n),Z) \end{equation}

Proof: Functoriality is left as an exercise. We want to prove a natural correspondence

\begin{equation} \operatorname{Hom}(\mathbf{R}_F(X),Z)\cong \operatorname{Nat}(X,\mathbf{N}_F(Z)) \end{equation}

but this is precisely what is given by \ref{univ-weighted-colimit}.

$\mathbf{N}_F$ is the singular nerve functor. See https://ncatlab.org/nlab/show/nerve+and+realization for more details.

4 Weighted limits, colimits, Kan extensions as weighted limits

Let $K: M\to C$, $T : M\to A$.

Then the pointwise left Kan extension of $T$ along $K$, if it exists, is given by

\begin{equation} L(c) := C(K-,c)\ast T \end{equation}

and similarly the pointwise right Kan extension is given by

\begin{equation} R(c) := \left\{C(c,K-),T \right\} \end{equation}

Proof: Let $S$ be arbitrary.

\begin{multline} [C,A](LT,S):= \int_c A(C(K-,c)\ast T,Sc) = \int_c\hat{M}(C(K-,c),A(T-,Sc)) \\=\int_{c,m} \mathbf{Ens}(C(Km,c),A(Tm,Sc))\cong\int_m\hat{C}(y(Km),A(Tm,S-))=\int_mA(Tm,SKm)=[M,A](T,SK) \end{multline}

as desired. The proof for right Kan extensions is dual.

5 The nerve-realization pairing with weighted colimits; the dual of Kan's theorem

In the case where $K$ is the Yoneda embedding, the left Kan extension formula above simplifies to

\begin{equation} L(X) = X\ast A = \int^c X(c)\otimes A(c) \end{equation}

which is as simple as we can ask for.

Likewise,if $M$ is a small category and $\mathcal{A}$ is complete, and $F : M\to \mathcal{A}$ is a functor, then there is a right Kan extension of $F$ along the covariant Yoneda embedding $y : M\to \left[ M, \mathbf{Sets} \right]$, and it is given by

\begin{equation} R(X) = \left[ X;A \right] = \int_m A(m)^{X(m)} \end{equation}

This right Kan extension is right adjoint (as a functor $\left[ M,\mathbf{Sets} \right]^{\rm op}\to \mathcal{A}$) to the functor $\mathcal{A}\to \left[M,\mathbf{Sets} \right]^{\rm op}$ sending $a$ to $m\mapsto \operatorname{Hom}(a,F(m))$, the functor that Kelly calls $\hat{F}$. The monad on $\mathcal{A}$ associated to the adjunction is the density monad of $F$.

5.1 TODO Find examples of this dual nerve-realization !!

It must surely be the case that there are examples of this dual construction, considering how important the nerve-realization theorem is. It's a bit weird as it is a contravariant adjunction, meaning that both functors are somehow 'left adjoint.' It is probably the case that the induced monad on

6 Simplifying a weighted colimit, with special attention to the case of simplicial Abelian groups

Let $X$ be a simplicial Abelian group, and $F$ a cosimplicial object in a cocomplete category $\mathcal{A}$. Then $X\star F$ is defined by the universal property

\begin{equation} \mathcal{A}(X\star F,A)\cong \mathbf{SAb}(X,\mathcal{A}(F-,A)) \end{equation}

natural in $A$.

Ok. After thinking about this for a while, I've come to the following conclusion.

Let $G : M\to \mathcal{A}$, with $M$ small and $\mathcal{A}$ cocomplete, and let $F : M^{\rm op}\to \mathcal{V}$. Fix $K: M\to M'$.

Assume that both $F$ and $G$ have left Kan extensions along $K$, say $F', G'$. Then in the weighted colimit $F'\star G'$ we simplify its characteristic property:

\begin{equation} \mathcal{A}(F'\star G',A)\cong \hat{M}'(F',\mathcal{A}(G'(-),A))\cong \hat{M}(F,\mathcal{A}(G'K(-),A)) =\int_m \mathcal{V}(Fm,\mathcal{A}G'Km,A) \end{equation}

and on the other hand, $F\star G$ is characterized by

\begin{equation} \mathcal{A}(F\star G,A)\cong \hat{M}(F,\mathcal{A}(G(-),A)) \end{equation}

If we unfold the definition of the pointwise left Kan extension,

\begin{equation} \mathcal{A}(G'Km,A) \cong \mathcal{A}(M'(K-,Km)\star G,A) \cong \hat{M}(M'(K-,Km),\mathcal{A}(G(-),A)) \end{equation}

\begin{equation} \int_m \mathcal{V}(Fm,\mathcal{A}G'Km,A)=\int_m\mathcal{V}(Fm,\int_n\mathcal{V}(M'(Kn,Km),\mathcal{A}(Gn,A)))=\int_{m,n}\mathcal{V}(Fm\otimes M'(Kn,Km),\mathcal{A}(Gn,A)) \end{equation}

6.1 Realization

Evaluation of weighted limits is a kind of composition of profunctors. Adding in a Morita equivalence and its inverse can be used to simplify the calculation. A Morita equivalence is an equivalence in the bicategory of profunctors.

6.2 TODO Give a technique for analyzing a category and showing how one can replace it by a simpler, Morita equivalent one; especially in the Dold-Kan case.

7 Day convolution in a promonoidal category

See math.se answer by Alexander Campbell. Let $\mathcal{C}$ be a category. A promonoidal category is a monoid object in the bicategory of categories and profunctors, i.e. a category $\mathcal{C}$ together with a multiplication $P : \mathcal{C}\times \mathcal{C}\to \mathcal{C}$ (really $P : \mathcal{C}^{\rm op}\times\mathcal{C}^{\rm op}\times \mathcal{C}\to \mathcal{V}$), $I : 1\to \mathcal{C}$ (really $I : \mathcal{C}\to \mathcal{V}$), etc satisfying the usual axioms. Then the Day convolution of presheaves is

\begin{equation} (X\otimes Y)(c) = \int^{a,b}P(a,b,c)\otimes X(a)\otimes Y(b) \end{equation}

8 Split idempotents, absolute limits and colimits

An idempotent morphism in a category $\mathcal{C}$ is an endomorphism $e$ of an object satisfying $e^2=e$.

A split idempotent in $\mathcal{C}$ is a morphism $e : A\to A$ which has a retract, i.e. there is some object $B$ and morphisms $r : A\to B, s : B\to A$, with $sr=e$, $rs = \operatorname{id}_B$.

A split idempotent is an idempotent as $srsr=sr$.

In an Abelian category, a split idempotent (by the split exact sequence lemma) is a projection onto a direct summand. In a semi-Abelian category, split exact sequences have similar nice properties.

Theorem: Let $\mathcal{I}$ be the category with one object, and one (nontrivial) endomorphism $e$, subject to the requirement $e^2=e$. Let $\mathcal{C}$ be a category. Let $r : B\to A, s : A\to B$ with $rs=\operatorname{id}_A$. Let $F: \mathcal{I}\to \mathcal{C}$ be the functor sending $e$ to $sr$. Then the morphism $s : A\to B$ is a limit cone for $F$, and the morphism $r : B\to A$ is a colimit cocone for $F$.

Proof: $srs=s$, so $s$ is a cone. Let $f : C\to B$ with $srf=f$. Then $f$ factors through $s$ by $rf$. Moreover if $sg=f$ then $rsg=g=rf$. So there is a unique morphism of cones from $f$ to $s$.

$rsr=r$, so $r$ is a cocone. is If $k : B\to D$ is a cocone, i.e. satisfies $ksr=k$, then $k$ factors through $r$ by $ks$. Moreover if $j : A\to D$ satisfies $jr=k$ then $jrs=j=ks$ so $ks$ is the unique morphism of cocones $r\to k$.

Theorem: If $G: \mathcal{C}\to \mathcal{D}$ is any functor, the limit and colimit of $F$ are both preserved by $G$; they are absolute limits and colimits.

This is obvious.

8.1 Karoubi Completion

https://ncatlab.org/nlab/show/Karoubian+category Let $\mathcal{C}$ be an Ab-enriched cat. $\mathcal{C}$ is Karoubian if every idempotent endomorphism has a kernel. Here, by kernel we simply mean $e$ and $0$ have an equalizer.

Proposition: If $\mathcal{C}$ is Karoubian, every idempotent endomorphism has a cokernel.

Proof: Note that if $e$ is idempotent, then $(1-e)e = e-e^2=0$, and similarly with $e(1-e)$. So $1-e$ factors through the kernel $s : A\to B$ by some $t : B\to A$.

$sts=(1-e)s = s-es =s$. But $s$ is monic, so $ts = \operatorname{id}$, and $s$ is split monic, with retraction $t$, and $1-e$ is split idempotent. Mutatis mutandis, $e$ is split idempotent.

Note that $ste= (1-e)e=0$. But $s$ is monic, so $te=0$. If $q : A\to C$ with $qe=0$, then $qst=q(1-e) = q-qe=q$. Moreover if $f : A\to C$ with $ft=q$ then

9 Yoneda as an end, Density lemma (Co-yoneda) as a coend

Here $P$ is assumed contravariant. Yoneda:

\begin{equation} \hat{\mathcal{C}}(y(c),P) \cong P(c) \end{equation}

i.e.,

\begin{equation} \int_d\hat{\mathcal{V}}(\mathcal{C}(d,c),P(d)) = P(c) \end{equation}

Co-Yoneda:

\begin{equation} P(d) = \int^c \mathcal{C}(d,c)\cdot P(c) \end{equation}

Author: Patrick Nicodemus

Created: 2021-08-21 Sat 17:25

Validate

Search This Blog

Unity of Opposites