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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 ...


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 $x \to x$.


6

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 ...


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.


5

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


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 $\mathsf{C}$ ...


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, ...


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 ...


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 ...


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 ...


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

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 ...


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 ...


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 ...


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) ...


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

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 ...


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 ...


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

[...] 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 ...


1

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 ...


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 ...


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 ...


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 ...


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 ...


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 ...


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 ...


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

Assume that there exists a natural transformation $\alpha : {\bf Sym} \to {\bf Ord}$, draw its naturality diagram, and apply it to the identity permutation over a simple set like $B = \{ 0, 1 \}$. Let $f : B \to B$ be defined by $f(0) = 1$, $f(1) = 0$. Look at $\alpha_B \circ {\bf Sym}(f)$ and ${\bf Ord}(f) \circ \alpha_B$ applied to the identity ...



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