Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
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22 views

Adjoint functors between $Set^I$ and $Set/I$ [on hold]

I've defined the two functors $F$ and $G$ between the two categories $Set^I$ and $Set/I$, but now I have to prove that they are also adjoint left and right, i.e. : $Set/I(F((X_i)_{i \in ...
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37 views

Do finite products commute with colimits in the category of spaces?

Let $X$ be a topological space. The endofunctor $\_\times X$ of the category of all topological spaces does in general not possess a right adjoint, since the category is not cartesian closed. Is it ...
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21 views

Question on the coskeleton functor $\mathbf{coskel}'$ for pointed simplicial sets

There is an abstract construction of the coskeleton functor for simplicial set as follows: Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq ...
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25 views

Category of $\mathcal{L}$-structures

Given a purely relational language $\mathcal{L}$, the set of $\mathcal{L}$-structures forms a category under the definition of homomorphism found here. Call this category ...
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1answer
26 views

Equivalence of category of subsets and subobjects

I'm trying to show that the categories $\mathcal{P}(X)$ and $Sub(X)$ are equivalent. According to Steve Awodey's "Category Theory" I need to find two functors $ E: \mathcal{P}(X) \to Sub(X)$ $F: ...
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23 views

Empty Category and Trace [on hold]

I would like to have your assistance regarding the mathematica rationale behind the ideas of empty category principle and trace as mentiones in the following link: ...
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1answer
75 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
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21 views

How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...
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1answer
47 views

If an isomorphism can be expressed as a composition of morphisms, what can we say about its components?

Suppose $f:X\to Y$ is an isomomorphism, and $f=g\circ h$ where $h:X\to Z$ and $g:Z\to Y$. Can we infer that either of these component morphisms is an isomorphism as well? And does this change ...
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43 views

Is a topological space $X$ the colimit of an open cover $\cup U_i$ in this way?

Let $X$ be a topological space space and $X=\cup_{i\in I} U_i$ a covering of $X$ by open subsets $U_i\subseteq X$. Is it true that $$ \operatorname{colim}\left(\coprod_{(i,j)\in I\times I} ...
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58 views

$\mathsf{Top}$ with proper maps has products.

In I.M. James' General Topology and Homotopy Theory, he presents proper maps before introducing compact sets, by defining $\phi:X \to Y$ to be proper iff $\phi \times \text{Id}_T$ is closed for all $T ...
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40 views

Maclane's Coherence Theorem: why not just use the functors themselves?

I have a softball question on Maclane's proof. Suppose $B=\left ( B, \square , \alpha ,\rho ,\lambda \right )$ is a monoidal category. Fix $b\in B$. Define $W$, the (monoidal) category of binary ...
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1answer
28 views

Understanding the $\mathfrak{a}$-adic completion of an $A$-module as a functor

$\require{AMScd}$ I recently read the chapter 10 on Completions in Atiyah-MacDonald. They describe the $\mathfrak{a}$-adic completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
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1answer
32 views

integral domains and field of fractions

I've read about integral domains and their induced fields of fractions. For an integral domain $R$ its field of fractions $K$ is the "smallest" field that includes $R$, i.e. there is an injective map ...
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70 views

Definition of adjoint functor and locally small categories

In the definition of an adjoint pair of functors, is it implicit that the categories are locally small? I have searched for ages, and nowhere is this stated as an assumption, but the definition seems ...
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1answer
25 views

Show that w-complete Posets and continuous aplications between them form a category

I'm really lost with this thing that looks innocent but just can't figure out... can you help me? Show that $\omega$-complete Posets and continuous functions between them form a category. Thank ...
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2answers
125 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
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49 views

Relation between being epic and having a “full” image.

Let $g: A \to B$ be a morphism in an abelian category. Is it true that $g$ is epic iff $im(g)=B$? Context: Given a short sequence in an abelian category $0 \to A \to B \to C \to 0$ with maps $f: A ...
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1answer
109 views

Definiton of Limit and Foundational problems.

I am new to Category theory and I have a quite strong foundational problem. For example, let's start from the definition of Limit stated by wikipedia ...
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1answer
122 views

The colimit of all finite-dimensional vector spaces

Let $\mathsf{iFinVect}_K$ be the category of finite-dimensional vector spaces with injective linear maps and $X : \mathsf{iFinVect}_K \to \mathsf{Vect}_K$ be the inclusion functor. Then ...
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37 views

A localization of a topos is still a topos

I am trying to see how a localization of a topos is a topos. That is, localization of a cartesian closed category is cartesian closed, and if a category has a subobject classifier any localization of ...
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35 views

Is the Groupoid of germs associated to an orbifold a Hausdorff proper Lie groupoid?

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
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1answer
57 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
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66 views

No natural transformation

I'm having trouble with part b. Intuitively, it is clear that there is no canonical way of assigning a permutation of $X$ onto an ordering of $X$, but I've failed to prove it rigorously thus far. ...
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Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
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106 views

Left G-Sets category

Let $G$ be a group, and $\mathbf{G\text{-}Sets}$ the category whose objects are left G-Sets and whose morphisms are G-Set homomorphisms, that is functions $f:X\to Y$ such that $f(ax) = af(x)$, $a\in ...
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87 views

Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
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41 views

Pushout in $\mathsf{Set}$ where one of the maps is injective

From I.M. James' book General Topology and Homotopy Theory: Suppose we have a cotriad $$X \xleftarrow{\xi}W \xrightarrow{\eta} Y.$$ ... we might expect the pushout of the cotriad to be a ...
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1answer
45 views

Natural transformations and the definition of Monoidal lax functors

The definition of a lax monoidal functor requires the existence of a natural transformation, $\phi$ http://en.wikipedia.org/wiki/Monoidal_functor. A natural transformation relates at least 2 ...
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1answer
36 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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1answer
100 views

Elementary proof that the category of modules is not self-dual

If $R,S$ are rings such that ${}_R \mathsf{Mod}$ is equivalent to ${}_S \mathsf{Mod}^{\mathrm{op}}$, then $R$ and $S$ are trivial. This is well-known. The usual proof uses of the notions of limit and ...
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1answer
73 views

How to prove that in this context epimorphisms are 'surjective'

In article AN ELEMENTARY THEORY OF THE CATEGORY OF SETS of William Lawvere I met a proposition left to reader (poor me) and I hope someone can help me with it. It wouldn't surprise me if it is not ...
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1answer
48 views

zero object in the category of group schemes

I am currently reading Ravi's lecture notes on AG, and in the introduction of group schemes(Section 6.6), he made a comment after 6.6N that the category of group schemes has a zero object. I can ...
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263 views

Is Category Theory similar to Graph Theory?

The following author noted: Roughly speaking, category theory is graph theory with additional structure to represent composition. My question is: Is Category Theory similar to Graph Theory?
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1answer
60 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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34 views

Definition of a regular category via extremal epi

I know one definition of a regular category saying that a category is regular if it is finitely complete, every kernel pair admits a coequalizer and regular epis are stable under pullback. Now, I am ...
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2answers
82 views

What are some beautiful examples of adjunctions?

Lately I've been very interested in finding examples of adjunctions. In particular, examples that are elementary enough for an undergrad like me to grasp. So I was wondering if perhaps you could share ...
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1answer
43 views

What does it mean for pullbacks to preserve monomorphisms?

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving ...
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1answer
89 views

Is the product of all objects of a finite category an initial object?

If the product of all objects in a finite category exists, is it an initial object? I presume so, but I'm still learning this subject and I can't make a proof go through. Advice welcome. (Not a ...
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2answers
57 views

Pushout of a subgroup

Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
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1answer
100 views

Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
3
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0answers
54 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
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2answers
64 views

The inclusion $\mathbb Z \to \mathbb Q$ is an epimorphism

I am supposed to show that the inclusion of the integers in the rationals is an epimorphism in the category of abelian groups. Not only am I unable to find the right argument, I am starting to wonder ...
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1answer
69 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
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Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
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1answer
33 views

Understanding products and coproducts

I am currently trying to make sense of products and coproducts in different categories, and even in starting with the basic examples (cartesian product and disjoint union in the category of sets) I've ...
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1answer
29 views

Why is the internal hom of a Kan complex also a Kan complex?

Let $X, Y : \Delta^{op} \to \text{Set}$ be simplicial sets. I've seen it stated that if $Y$ is a Kan complex, then the internal hom $Y^X$ is a Kan complex also. How does one show this?
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Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?

Let $C'$ be the category with objects $C$ and morphism the monic morphisms of $C$. In any topos, $\phi_A: \operatorname{Sub}(A) \cong \operatorname{Hom}_C(A,\Omega)$ and $\phi_A(m) = \chi_m$. This ...
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1answer
51 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...