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0

One reason why categories of mathematical structures (where the morphisms are structure-preserving maps) are nice is that universal constructions like limits and colimits (1) exist and (2) are easy to describe. This usually boils down to the fact that limits and colimits in the category $\mathsf{Set}$ of sets are easy to describe, and for most "categories of ...


2

This is an instance of Kan extensions. Let $\mathbf G$ be the one object category associated to the group $G$, and $\mathbf 1$ the one associated to the trivial group. Then $\mathbf{G{-}Sets}$ is nothing else than $[\mathbf G,\mathbf{Sets}]$ and the forgetful functor is the functor $$ i^\ast \colon [\mathbf G,\mathbf{Sets}] \to [\mathbf 1,\mathbf{Sets}] ...


2

[...] an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. [...] This is a really restricted point of view on category theory. Of course, the simplest examples that come to mind when talking about categories are such, but if category theory was only about those, it would be nothing else than a ...


4

Let $S$ be a set. Then $G \times S$ can be equipped with a $G$-action by having $G$ act by left-multiplication on itself and trivially on $S$. This is a left-adjoint to the forgetful functor -- for $T$ an arbitrary $G$-set, one has $$ \text{Hom}_\text{G-set}(G \times S, T) = \text{Hom}_\text{set}(S, T) $$ by the rule $f \mapsto \left[s \mapsto f(1, ...


1

For a good concrete example, I would suggest that you look into synthetic differential geometry. This is an axiomatic approach to differential geometry which takes place in a smooth topos. The theory is very beautiful and intuitive, and allows you to rigorously reason using infinitesimals. Since this is a purely axiomatic theory, you can come up with a ...


4

The interplay between CT and MT is pretty well established. The term to search for is locally accessible categories. Another subject to look at may be topos theory, again with plenty of material online. A complete, or even a very partial list of applications of CT and MT will require a lot of bytes. MT has applications in algebra and in analysis, and that ...


-1

I am not sure one can give a justification from scratch. However, an important algebraic motivation of adjunction is to formalize the "free construction" on a set in a given algebraic theory. Leaning on the basic free constructions (groups, abelian groups, modules, rings, etc.) that your student should know, you have a panel of adjunctions $F \dashv U$ to ...


5

I hope what follows will clear up your confusion. A lax monoidal functor consists of the following data (satisfying some axioms that I won't spell out): One functor $\color{red}{F : \mathsf{C} \to \mathsf{D}}$. This means that for all objects $A \in \mathsf{C}$ you have an object $F(A) \in \mathsf{D}$, and for all morphisms $f : A \to B$ in $\mathsf{C}$ ...


1

An answer to your last question: If $\mathbb Z \to N$ is onto, then any homomorphism $f:\mathbb Q \to N$ is easily shown to be trivial. In particular we can always chose $h = 0$ (the only homomorphism $\mathbb Q \to \mathbb Z$ anyway) to satisfy $g \circ h = f$. This shows (a boring fact, since the involved maps are trivial, so nothing happens here) that ...


1

Categories of modules and opposites of such categories can be distinguished by the behavior of the natural map $\bigoplus_i M_i \to \prod_i M_i$ from a countable coproduct to the corresponding countable product. In categories of modules this map is a monomorphism but usually not an epimorphism, while in their opposites this map is an epimorphism but usually ...


3

Suppose $f: A\to B$ is an epimorphism. If $A\neq 0$, then $A$ has a point by assumption, and hence the axiom of choice guarantees the existence of $g: B\to A$ such that $A\xrightarrow{f} B\xrightarrow{g} A\xrightarrow{f} B = A\xrightarrow{f} B$. Since $f$ is epi, we may cancel it from the left, hence $B\xrightarrow{g} A\xrightarrow{f} B = \text{id}_B$. In ...


3

It is not claimed that the category of $A$-schemes has a zero object. But the category of group $A$-schemes has a zero object. This has nothing to do with schemes. If $C$ is any category with finite products, then $\mathsf{Grp}(C)$ has a zero object, the "trivial group" $T$. The underlying object is $1$, the final object, and the multiplication is the unique ...


1

This is not exactly an answer to your question, except that it is relevant to the question of homotopical invariants preserving at least some colimits. For example, the fundamental group of based spaces does not preserve all colimits: the usual Seifert-van Kampen Theorem determines the fundamental group of a union $X=U \cup V$ of based spaces if $U,V$ are ...


1

There are two possible meanings: For any pullback square as below, $$\require{AMScd} \begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \end{CD}$$ if $X \to Y$ is a monomorphism, then $X' \to Y'$ is also a monomorphism. For any commutative diagram of the form below, $$\begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \\ @VVV ...


9

In fact, "Roughly speaking, category theory is graph theory with additional structure to represent composition." is a good summary of the connection between the notion of a graph (meaning here: directed multigraph) and the notion of a category. Here is a way of how to make this precise: Let $O$ be a set ("set of vertices"). Then we have a category ...


3

Category theory and graph theory are similar in the sense that both are visualized by arrows between dots. After this the similarities quite much stop, and both have different reason for their existence. In category theory, we may have a huge amount of dots, and these dots do often represent some abstract algebraic structure or other object with some ...


2

The category of Banach spaces is locally $\aleph_1$-presentable.


2

The notion of distributor is not new and has been used in near-ring theory for a long time. It was apparently introduced by Fröhlich [2, 3] in 1958. Here are a few other relevant references. [1] L. Esch, Commutator and Distributor Theory in Near-Rings, Doc. Diss. (Boston Univ., 1974). [2] A. Fröhlich, Distributively Generated Near-Rings: (I. Ideal Theory) ...


3

I'm no expert in category theory, but here are some easy examples I can think of: Categories with closed monoidal structure are really great, because they give you a nicely behaved notion of a "mapping object". A closed monoidal category $\mathsf{C}$ is one where the functor $(-)\otimes X$ has a right adjoint $[X, (-)]$ called the internal hom. This ...


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Here are some very basic examples, but which you don't find so often mentioned. 1. For real numbers $r$ and integers $z$ we have $$\iota(z) \leq r \Leftrightarrow z \leq \lfloor r \rfloor,$$ where $\iota$ is the inclusion map from integers to real numbers and $\lfloor - \rfloor$ is the floor function. That means that $\iota : (\mathbb{Z},\leq) \to ...


1

As seems to be my habit, I'm going to say a lot of things here whose details I have not checked. Use at your own risk :). Given a metric space $(X,d)$, there is a geodesic remetrization $(X,d_G)$, equipped with a bijective short map $(X,d_G) \to (X,d)$. The geodesic metric is given by $d_G(p,q) = \sup_{\epsilon>0} \inf_{\{p = p_0,p_1,\dots,p_n=q ...


7

Consider the category $\mathsf{C}$ with a single object $x$ and two morphisms: $\operatorname{id}_x$ and $f : x \to x$ such that $f \circ f = f$. Then $x$ is of course the product of all the objects in $\mathsf{C}$. But it's not an initial object, because there are two morphisms $x \to x$.


3

If I understand you correctly, you want the pushout of this diagram: $$\require{AMScd} \begin{CD} H @>{\iota}>> G \\ @V{\iota}VV \\ G \end{CD}$$ It's possible to describe it "explicitly": it is the quotient of the free product $G * G$ by the normal subgroup $N$ generated by the relations $\iota_1(h) \equiv \iota_2(h)$ where $\iota_1$ is the ...


0

Pushouts (and colimits in general) can always be constructed from coproducts and coequalizers whenever these are available (see Awodey's text on category theory, for example). So the pushout of an arrow $ H \to K$ along along the inclusion $H \hookrightarrow G$ would be the quotient of the coproduct of $G$ and $K$ by the appropriate equivalence relation. ...


2

First, let us write $UG$ for the underlying set of a group $G$. Well, as ZhenLin pointed, a group as a pair in the set theoretic means $G=(UG,\,\circ)=\{\{UG\},\{UG,\circ\}\}$ has not much to do with the elements of $UG$. In this sense ${\bf Grp}$ is not strictly a subcategory of ${\bf Set}$. There exists, however, an injective functor ${\bf Grp}\to{\bf ...


-1

Firstly, as tcamp also commented, there also exists a similar construction as adjunctions in any bicategory where the 'unit' and 'counit' go in the same direction, and these are the (dual) Morita contexts, and it is also worth to study, having already wide literature, at least in its origin world of rings. (Note that adjoint equivalences are immediately ...


5

This is false in the category of abelian groups. Consider the composition $\mathbb Z \hookrightarrow \mathbb Q \to \mathbb Q / \mathbb Z$.


5

Perhaps the exercise was meant for the category of rings. Then the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is indeed an epimorphism. See also the discussion on MSE here. For the category of abelian groups, epimorphism and surjective morphism is the same.


2

Your question, really, is about necessary conditions for a ($\mathbf{Set}$-valued) presheaf to be representable. I think that business with automorphisms is actually a red herring – let me illustrate. Let $A$ be a set and let $\mathcal{F} : \mathbf{Set}^\mathrm{op} \to \mathbf{Grpd}$ be defined by $\mathcal{F} (X) = (\mathbb{B} \mathrm{Aut} (A))^X$, where ...


1

The unit-counit definition of an adjunction makes sense in any 2-category, and so it makes sense in any monoidal category, where it recovers the definition of a dualizable object. So you can motivate this somewhat more general notion using the familiar special case of vector spaces: when a vector space $V$ has a dual $V^{\ast}$, there is a unit map $1 \to ...


0

tcamps gave what I think is the correct answer (nice pasting diagrams justify the choice formally, examples justify it's usefulness), but here's an alternate approach which might have some pedagogical merit, if you don't think it's too roundabout. Use the adjunction isomorphism as the definition, or state that it's equivalent to the (unit, counit) one you ...


0

Maybe this veers too close to talking about the triangle equations, but I find it suggestive that it's possible to paste the unit and the counit in some way when the counit goes the right way around, whereas if it goes the wrong way around, there's simply no pasting that you can possibly do. With no pasting, there wouldn't be anything you could "do" with the ...


1

Hint 1: You are working with the universal property of limits in general. For most people, this certainly isn't the best starting point. Try writing out what the universal property means in this particular case (discrete domain) explicitly, it simplifies considerably. You can check what you got or (if you can't do it) what you're supposed to get by searching ...


2

Allegedly, one can prove this directly, but I have never seen the details. (For instance, the same fact is asserted without proof as Theorem 1.9 in [Moore, Semi-simplicial complexes and Postnikov systems].) Here is a more high-level proof. Recall that the class of anodyne extensions of simplicial sets is defined by induction: Any horn inclusion is an ...


2

If you want there to exist at least one $S$ for every possible $P$, you will need very restrictive hypotheses. For example, you might require $\mathcal T$ to be the free category on some graph. This includes the case where $\mathcal T$ is the poset of natural numbers, but not the nonnegative positive reals. I think the correct thing to say is that ...


0

A universal element for a functor $X : \mathcal{A} \to \mathbf{Set}$ is precisely an initial object in the comma category $(1 \downarrow X)$. The connection with adjunctions is this: a functor $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if, for each object $C$ in $\mathcal{C}$, the functor $\mathcal{C} (C, G -) : \mathcal{D} \to ...


0

Let $\mathsf{LCH}$ be the category of locally compact Hausdorff spaces with proper continuous maps and $\mathsf{CH}$ be the full subcategory of compact Hausdorff spaces with continuous maps. We extend it to the category $\mathsf{CH}_*$ of pointed compact Hausdorff spaces with pointed continuous maps. The Alexandrov compactification is a functor $$A : ...


1

I think part of the confusion is that the sentence below the definition - "In the above diagram, $G := \Psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \Psi(V)$" - is not meant to be part of the definition, rather it is a clarification. The definition given is that $F$ is smooth iff there exists $G$ (which, from its position in the diagram, ...


1

"The diagram commutes" means exactly what it always means: that the map produced by following any path through the diagram is the same. The notes you are reading made a mistake and forgot to require that $G$ be smooth.


2

To summarize the definition: an object of the "universal cover" $\hat{\mathcal{C}}$ of a category $\mathcal C$ is an indiscrete subcategory of $\mathcal{C}$, and a morphism $(A,(\phi_{a,a'})_{a,a' \in A}) \to (B,(\psi_{b,b'})_{b,b' \in B})$ is a collection $(f_{ab})_{a \in A,b \in B}$ of arrows which are natural in the obvious way. Composition is actually ...


3

I hesitate to say it -- Take any functor $\mathsf{Int} \times \mathsf{Int} \to \mathsf{Set}$ which doesn't factor through a projection (e.g. take the underlying set of the product ring), and then follow it with the free ring functor $\mathsf{Set} \to \mathsf{Int}$ (the free ring on a set is a domain). Ugh. I hope there's something nicer. If you want to ...


2

Here is an algebraic version (and generalization) of Qiaochu's counterexample: Let $A$ be a commutative ring and $a : A \to A$ be a homomorphism of $A$-modules (free of rank $1$), corresponding to some element $a \in A$. It is a monomorphism in the category of f.g. projective $A$-modules iff it is a monomorphism in the category of all $A$-modules iff $a$ ...


5

$\text{Vect}(X)$ isn't abelian, but it is still close enough that a morphism of vector bundles is a monomorphism iff it has trivial kernel. The problem is that the fiberwise kernel of a map of vector bundles can fail to be a vector bundle. Very explicitly, let $X = \mathbb{R}$, let $V$ be the trivial line bundle over $X$, and let $f : V \to V$ be the ...


2

For $X : J \to C$, $Y : J \to D$ we have $$\hom(F \circ X,Y) \cong \int_{j \in J} \hom(F(X(j)),Y(j)) \cong \int_{j \in J} \hom(X(j),G(Y(j)) \cong \hom(X,G \circ Y).$$ Here, $\int$ refers to an end. If you don't know this formalism, you might as well translate the proof as follows: A natural transformation $F \circ X \to Y$ consists of a family of maps ...


4

Note that the objects of $\def\Set{\mathsf{Set}}\Set^I$ are tuples $(A_i)_{i\in I}$ of sets $A_i \in \def\Ob{\mathord{\rm Ob}}\Ob(\Set)$. A morphism $f \colon (A_i) \to (B_i)$ in $\Set^I$ consists is a tuple $f = (f_i)_{i\in I}$ of $\Set$-morphims (=maps) $f_i \colon A_i \to B_i$. An object of $\Set/I$ is a pair $(A, \pi_A)$, where $A\in \Ob(\Set)$ and ...


3

The sheaf condition in the definition of a diffeological space should imply that the inclusion functor $\mathsf{Man} \hookrightarrow \mathsf{DiffLo}$ preserves finite coproducts and Hausdorff pushouts along open embeddings. It also preserves infinite coproducts, if you replace "second countable" by "paracompact" in the definition of a manifold, which is ...


4

You are right that you need to consider the objects as not just vector spaces but rather as as a vector space equipped with a distinguished endomorphism. The natural definition of morphisms in this category will be linear maps that are compatible with the endomorphisms. That is, given two pairs $(V_1, T_1)$ and $(V_2,T_2)$ of vector spaces with ...


9

The grammar is "a category is tensored over a monoidal category"; this is a generalization of a set being equipped with an action of a monoid, or an abelian group being equipped with an action of a ring. In full generality you should provide the tensoring, but sometimes if you require enough it exists uniquely. The general pattern of the uniqueness results ...


1

For the sake of an answer, In order for $f$ to be an epimorphism, it has to be right-cancellative for all morphisms $B\to C$ for all objects $C$. Take $C=\{3,4\}$. Define $f(0)=1$, as above, which is not surjective. Define $g_1,g_2\in\operatorname{Mor}(B,C)$ by $g_1(1)=3, g_1(2)=3$, and $g_2(1)=3,g_2(2)=4$. Then $g_1\circ f=g_2\circ f$ as both are the ...


4

I'm not sure what exactly you are asking, but your function $λa. λb. (a, b)$ is an element of type $a → (b → a × b)$, or more conventionally $((a × b)^b)^a$. If you apply it to a specific $a ∈ A$, you get an element of type $b → a × b$, which is a function from $b$ to $a × b$. In categorical language, this is the setting of a cartesian closed category, ...



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