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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4
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2answers
123 views

Is there a “partial function” approach to subobjects in category theory?

Given a relation $f : X \rightarrow Y$, lets define that the source of $f$ is $X$, and that the domain of $f$ is the set of all $x$ such that there exists $y \in Y$ satisfying $(x,y) \in f$. Thus the ...
16
votes
8answers
494 views

Examples of categories where morphisms are not functions

Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
4
votes
1answer
150 views

Pullbacks and transpose map

Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
1
vote
1answer
93 views

Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction

Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, ...
7
votes
2answers
432 views

Products and pullbacks imply equalizers?

I was reading Herrlich & Strecker's Category Theory, and there is a theorem called The Canonical construction of Pullbacks which states that if a category has products and equalizers, then it has ...
5
votes
0answers
102 views

Freyd's Geometric Finiteness : An Example Computation

In his paper "Numerology in Topoi" available here: http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html Peter Freyd defines an object $A$ in a topos $\mathcal{E}$ to be geometrically finite if ...
0
votes
1answer
84 views

Sections, Transversals and Quotient Maps

Here I read: Given a quotient space $\bar X$ with quotient map $\pi\colon X \to \bar X$, a section of $\pi $ is called a transversal. I asked myself how such sections are possible. It must be a ...
4
votes
1answer
268 views

why split epi and mono implies iso?

I was doing some exercises on the definitions of epics, monos, split monos, etc..., and I asked myself that if you could take, for instance an epi which is mono, and then deduce it is an iso, which is ...
2
votes
2answers
106 views

MacLane's Abelian Categories Chapter

I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of ...
1
vote
3answers
100 views

Is there a category-theoretic perspective on induced functions?

Let $X$ and $Y$ denote sets. Given a function $f : X \rightarrow Y$ and a natural number, there is an induced function $g : X^n \rightarrow Y^n$ defined by $g(x_1,\cdots,x_n) = (f x_1,\cdots,f x_n).$ ...
5
votes
3answers
231 views

In what sense is the forgetful functor $Ab \to Grp$ forgetful?

One sometimes hears about "the forgetful functor $Ab \to Grp$." Given that the image of an object under this functor is still abelian, in what sense is this "forgetful"?
4
votes
1answer
85 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...
2
votes
1answer
143 views

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := ...
6
votes
1answer
110 views

How we can understand one category is small

"A category is said to be small if its objects form a set." Now one question is in my mind and that is although we know lots of sets and always working with them, but how we can show a class of ...
2
votes
1answer
105 views

Is there a categorical construction of the general linear group?

This question is related to the answer of Qiaochu in this one. Since the object $X=\mathbb{F}_2^2$ generates the category of vector spaces of dimension $2^n$ over $\mathbb{F}_2$, and since we know ...
7
votes
1answer
149 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
3
votes
1answer
67 views

What does it mean for a Category to have equalizers or/and pullbacks?

I know the definitions of what pullbacks and equalizars mean, but I don't know what it means that a given category $\mathfrak C$ has pullbacks or equalizers. Thanks
5
votes
0answers
169 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
1
vote
0answers
42 views

Good reference for co-groups, perspective of co-algebra applications

There are lot of applications of state transition systems STS (computer science, planning problems in robotics and so on) and lot of algorithms are devised, but the mathematical background for STS is ...
2
votes
1answer
150 views

A pedantic question about defining new structures in a path-independent way.

Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of ...
0
votes
1answer
104 views

Name/notation for the subgroup generated by all stabilizers

Say we have a group $G$ acting on a set $X$. I'm interested in the subgroup generated by all isotropy groups $G_x$, and looking for a designation for it. Thanks in advance! PS1: I thought about the ...
1
vote
1answer
109 views

Definition of monoids in slice category

Can someone please tell me what would be an appropriate definition of an internal monoid in the slice category? Or better yet, suppose you have an object $p : X \rightarrow A \in \mathcal{C}/{A}$ ...
2
votes
2answers
158 views

Does an adjoint pair fix a unit/counit pair?

From Ravi Vakil, Fundations of Algebraic Geometry. I want to ask if anyone can give a hint in how to prove Execrise 1.5.B(page 43). I tried to draw the diagram for half an hour but the resulting ...
4
votes
2answers
233 views

If $gf$ is an equalizer , is $f$ an equalizer?

Suppose $gf$ is an equalizer in a category $\mathfrak C$, I think that $f$ not necessarly is an equalizer, but I don't know how to come up with a counterexample; i've really tried it so hard. Thanks ...
5
votes
1answer
130 views

What is the minimum required background to understand articles in the nLab?

I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not ...
1
vote
0answers
59 views

Are finitely additive measures 'topological'?

The category of measurable spaces are topological over $Set$ in that they support initial & final structures similarly to that topological spaces. A measurable space is a set supporting ...
2
votes
1answer
85 views

Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?

Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$. Why: (1) ...
3
votes
2answers
142 views

Category-Theory and the modelling of $n$-ary functions (especially the $0$-ary functions)

Hi have a questioning regarding the modelling of $n$-ary functions and constants. First in the category of sets I know that the empty set $\emptyset$ is initial because for every other set $X$ there ...
4
votes
2answers
163 views

Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
12
votes
0answers
240 views

Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
0
votes
1answer
65 views

Mono/Epi of sections of presheaves

Let $\mathsf{C}$ be a category with initial and terminal objects, and $\phi:\mathscr{F}\to\mathscr{G}$ a morphism of presheaves on $X$ taking values in $\mathsf{C}$. I have a rather messy proof that ...
1
vote
3answers
170 views

Reference request - being rigorous about a common abuse of notation.

I've completely rewritten this question, in accordance with this advice. As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
12
votes
5answers
2k views

Real world applications of category theory

I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if ...
6
votes
3answers
176 views

Saying $a \in b$ in category theory

Suppose I have a category $C$ of sets, and $a,b \in C$. How can I express, in the language of category theory, that $a \in b$? (To clarify: the objects of $C$ are actually sets, and I want to express ...
2
votes
1answer
63 views

Is there any non-trivial relationship between kernels & kernel pairs?

Kernels are inspired by group theory, and kernel pairs by a similar concept in monoids where kernels aren't sufficient to capture the information necessary for the first isomorphism theorem. When a ...
0
votes
1answer
111 views

When is the relationship between kernel pairs and kernels an isomorphism?

Kernel pairs can be taken in any category with pullbacks, when there is a zero object we also have kernels. Then there is a morphism from the kernel to the kernel pair (via pullback uniqueness). What ...
4
votes
2answers
135 views

intersections in abelian category

Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow ...
10
votes
1answer
173 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
7
votes
4answers
249 views

Categorial definition of subsets

I cannot see why this definition http://en.wikipedia.org/wiki/Subobject is equivalent to subsets in the category of sets. I am confused by the following facts, 1) the partial order is defined ...
6
votes
2answers
1k views

Why gives a commutative diagram a proof?

I am thinking about a proof of the following: Suppose a map $f: A \to B$ has a retraction. Then for any set $T$ and for any pair of maps $x_1 : T \to A$, $x_2 : T \to A$ from any set $T$ to $A$ $$ ...
3
votes
1answer
131 views

Are faithful functors just monics in Cat?

I was looking for a characteristics of faithful functors without involving setS and then tried to see if they are just monics in Cat. Let $F: A\rightarrow B$ be a faithful functor, and $G_1,G_2:T ...
2
votes
1answer
118 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
16
votes
1answer
363 views

Examples of universal constructions in probability theory

I am looking for more examples of universal constructions in probability theory. Like the construction a of Gaussian space from a real Hilbert space, or a Poisson jump process from a measurable space ...
2
votes
1answer
106 views

Is the category of sheaves on a site always abelian?

Let $\mathcal{C}$ be a site and $\mathcal{A}$ be an abelian category. Suppose that the category of presheaves $$ Psh(\mathcal{C},\mathcal{A}) = \operatorname{Fun}(\mathcal{C}^{op},\mathcal{A}) $$ is ...
5
votes
1answer
102 views

Existence of a functor $\mathsf{Sets} \to \mathsf{Groups}$ that admits a left adjoint

The forgetful functor $\mathsf{Groups} \to \mathsf{Sets}$ admits a left adjoint, namely, "forming the free group" functor. I was wondering if this has a left adjoint. This seems unlikely, but I ...
2
votes
1answer
70 views

What are the $2$-morphisms in the $2$-category of “categories over $\mathfrak{Sch}$”?

Let $\mathfrak S$ be the category of schemes. My goal is just to visualize the $2$-morphisms in the $2$-category consisting of the following objects: categories over $\mathfrak S$, i.e. $$ \textrm{ ...
1
vote
1answer
69 views

Above the identity there must lie an isomorphism.

Let $F:\mathcal C\to \mathcal D$ be a (covariant) functor between two categories. Of course, a functor sends isomorphisms to isomorphisms. And "over" an isomorphism in $\mathcal D$ there might be an ...
0
votes
0answers
118 views

Nonsymmetric monoidal product on $[\mathbb N,\mathbf{Sets}]$

Let $\mathbf P$ be the category with objects the natural numbers and $\hom(m,n)=Sym(n)$ if $n=m$, and the empty set otherwise. It is symmetric monoidal wrt the sum of natural numbers, and has $0$ as a ...
2
votes
2answers
113 views

definition of left (right) Exact Functors

Let $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually): $F$ is left exact if $0\to A\to B\to C\to 0$ is exact ...
1
vote
0answers
139 views

Is $\mathbb{AB}$ an additive (or even abelian) category?

Notation. Let $U_0 \in U_1 \in U_2$ be Grothendieck universes, each containing $\mathbb N$. Let $\mathbf{Cat}_{U_0}$ be the ($U_1$-small) 2-category of all $U_0$-small categories, ...