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 ...

learn more… | top users | synonyms (2)

1
vote
1answer
23 views

The subcategory of commutative $R$-algebras is denoted by ???

Let $R$ be a commutative ring (with identity) and let $R\mathbf{Alg}$ denote the category of $R$-algebras. My question: is there a suitable notation for the full subcategory of commutative ...
2
votes
1answer
29 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
2
votes
2answers
55 views

Image of a category under a functor need not be a category? [duplicate]

I've been trying to understand the following counterexample posted here: http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics?page=2&tab=votes#tab-top "You only ...
1
vote
1answer
30 views

Strong epimorphic counit iff conservative right adjoint?

On page 13 of Lack and Street's Combinatorial Categorical Equivalences, it is written (but not proven) that: A right adjoint is conservative if and only if the components of the counit are strong ...
2
votes
1answer
61 views

Realization (in the sense of homotopy coherent nerve) of $\partial\Delta^n$

I need some help understanding the working of the homotopy coherent nerve (as described in Lurie's HTT). Let $i < j$, then $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
0
votes
1answer
58 views

are there examples of “category-like” structures where distinct pairs of objects have hom-sets that aren't disjoint?

I understand (based on the relatively few examples of categories I have at my disposal), why distinct pairs of objects should have disjoint hom-sets, but I wanted to know of any structures that are ...
3
votes
2answers
70 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
1
vote
1answer
31 views

Why are duals in a rigid/autonomous category unique up to unique isomorphism?

I'm having trouble understanding the following statement: "In a rigid category, duals are unique up to unique isomorphism." It seems to me that this isomorphism is not unique. Let me try to give a ...
5
votes
1answer
63 views

Basic Notions of Categorification

In this post John Baez defines a categorification of a set $S$ as a map $$ p:\DeclareMathOperator{Decat}{Decat}\Decat(\mathscr C)\to S $$ where $\Decat(\mathscr C)$ is the set of isomorphism classes ...
5
votes
1answer
67 views

Is there a formal way to describe classical logic as a reflective subcategory of constructive logic?

Working informally, we can take any proof $P$ in constructive (or intuitionistic) logic and turn it into a classical proof $cP$ by 'copying' it, since all the rules of constructive logic reappear in ...
9
votes
5answers
322 views

Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
2
votes
0answers
108 views
+50

A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
2
votes
1answer
20 views

Intersections versus Multiple Pullbacks

In an arbitrary category $\mathcal C$, a subobject of an object $X$ is a monomorphism $m\colon X'\to X$. The intersection of a class of subobjects $\langle m_i\colon X_i\to X\rangle_{i\in I}$ is ...
1
vote
1answer
59 views

Concepts like a pushout or pullback but slightly different

I'm currently reading these short lecture notes and had a question regarding example 2.6(d) (also I think there is a typo in there, but I'm not sure. Anyway...) In the given category $J$, consisting ...
3
votes
1answer
47 views

in which conditions the following holds: If the pullback of a morphism is an isomorphism, then this morphism is an isomorphism?

It is easy to see that, if p is a split epimorphism, the the following proposition is true: "If the pullback of q along p is an isomorphism, then q is an isomorphism". Is there an weaker condition ...
4
votes
1answer
72 views

Whatever Happened to Nearness Spaces?

I came across this paper about Nearness Spaces. It seemed to be at the time (1970-80s) a promising approach to general topology via category theory. I have found no posts at all on stackexchange ...
2
votes
1answer
46 views

Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
2
votes
1answer
83 views

Direct limits and pullbacks

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow ...
1
vote
0answers
41 views

Examples and definition of cocompact objects

An object $X$ of a locally small category $C$ that admits filtered colimits is called compact if $$ \operatorname{Hom}_{C}(X,-) $$ preserves filtered colimits. Let $C$ be a locally small category ...
3
votes
3answers
95 views

Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
6
votes
2answers
48 views

Algebras of the environment monad

What are the algebras of the environment monad $-^E$ in $\mathbf{Set}$? Abstractly, I see that an algebra of $-^E$ is a set $X$ with an operation $f : X^E \to X$ that obeys two laws: an "idempotence ...
3
votes
1answer
58 views

Does category-theory have an interesting perspective on the phrase 'under the induced operations'?

We often make statements like: "the set $X$ becomes a [whatever] under the induced operations." For example: Given an algebraic theory $T$ and a $T$-algebra $X$, the set of all functions $k ...
4
votes
1answer
69 views

Can the complement of a subset be realized as a limit or colimit?

Let $X$ and $Y$ be two sets, and let $f:X\rightarrow Y$. Now consider the posets $(\mathcal{P}(X),\subseteq)$ and $(\mathcal{P}(Y),\subseteq)$ as categories. The induced functions $f^*$ (preimage ...
5
votes
1answer
53 views

Does pullback of schemes by monomorphism produce topological pullback?

Suppose I have scheme maps $X\to Z,Y\to Z$. It is not in general true that the fibre product $X\times_{Z}Y$ has the same underlying topological space (or even the same underlying set) as the fiber ...
6
votes
1answer
177 views

Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
1
vote
0answers
38 views

Is the (type)class `Functor` itself a functor? [closed]

Simple yes or no question, but one that's hard to google/search for due to repetition of terms: Since Functor is the set (ok, class) of all types for which an ...
1
vote
1answer
47 views

initial and final objects in category of sets

Could someone please tell me what are the initial and final objects in the category of sets and topological spaces? I had been very confused by this question. Thanks!
5
votes
2answers
92 views

Are the categories ${\bf{Sets}}/2$ and ${\bf{Sets}} \times {\bf{Sets}}$ isomorphic? Awodey's exercise

Let $2=\{a,b\}$ be any set with exactly $2$ elements $a$ and $b$. Define a functor $F: {\bf{Sets}}/2\rightarrow{\bf{Sets}}\times{\bf{Sets}}$ with $F(f:X\rightarrow 2)=(f^{-1}(a),f^{-1}(b))$. Is ...
6
votes
0answers
109 views

Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
2
votes
0answers
40 views

Definition of Hochschild homology in terms of Tor functor (bar resolutions)

I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor: 1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology ...
3
votes
0answers
25 views

Localization and Direct limit [duplicate]

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
2
votes
0answers
86 views

Category theory as a foundation for mathematics

Can Category theory form a foundation for mathematics like set theory and mathematical logic and if it can is there a way to know if that theory will be both consistent and complete
1
vote
1answer
108 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
3
votes
1answer
66 views

Exponential objects in a category of abstract automata.

I'm working with a more or less standard definition of the category Aut(C) of automata over a category C (where C has finite products) which has tuples $$ A=\langle I_{A},O_{A},S_{A},\sigma_{A}, ...
2
votes
1answer
43 views

Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...
3
votes
1answer
93 views

Morphisms of adjunctions in 2-categories

Let $\mathcal{C}$ be a $2\!-$category and let $(F,G,\eta,\varepsilon)$, $(F',G',\eta',\varepsilon')$ be two paralell adjunctions $A-\!\!\!\rightharpoonup B$ in $\mathcal{C}$. Let $F\xrightarrow{\ ...
0
votes
0answers
37 views

ind-completion and ordinals

Given a category $C$, we have its ind-completion $Ind(C)$ whose objects are filtered diagrams in $C$. Assuming the axiom of choice, is any object in $Ind(C)$ isomorphic, in $Ind(C)$, to an ordinal ...
1
vote
0answers
33 views

Can one talk about equivalences classes in a class(which might not be a set). [duplicate]

equivalence relation on a set breaks it into disjoint sets. does 'disjointness' make sense in classes. specifically, given a category, isomorphism classes of objects makes sense or not. can it be so ...
3
votes
2answers
46 views

Is there a universal property in this proposition (which regards field extensions)?

Since learning a bit of category theory, I am trying, as an exercise, to state results that I come across in categorical language. I am trying to do this with the following: Let ...
1
vote
1answer
22 views

Proving that finite direct and inverse limits exist in an additive category having kernels and cokernels

I have attempted to prove this but am unable to complete the proof. Below is my attempt. Let $\mathcal{A}$ be a category satisfying the conditions in the title and $\{M_i,\phi^i_j\}$ be a finite ...
0
votes
0answers
65 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
2
votes
1answer
48 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
0
votes
0answers
28 views

induced functor between weighted limits

I'm working on some special weighted limits in the category of categories. And I have to prove that the induced functor between two special weighted limits is faithful. But,before working in the ...
5
votes
1answer
39 views

If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive

Let $$ F : \text{A-Mod} \to \text{A-mod} $$ be an additive functor. Then if $0$ is the zero-object $F(0) $ is the zero object. Why this is true ? The definition of additive functor that I know is ...
3
votes
2answers
85 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
3
votes
0answers
45 views

Objects without extensions

How do you call an object $X$ for which every monomorphism $i : X \hookrightarrow Y$ has a retract (i.e.\ a morphism $r : Y \rightarrow X$ such that $r \cdot i = 1_X$)? I think of Y as an extension ...
3
votes
1answer
43 views

Help defining the $\mathrm{supp}$ function on free algebraic structures.

Given an algebraic structure $F(K)$ freely generated by a set $K$ with underlying set $U(F(K))$, I'm trying to define a "support" map $\mathrm{supp} : U(F(K)) \rightarrow \mathcal{P}_\mathrm{fin}(K).$ ...
4
votes
0answers
45 views

Adjunction between cocomplete categories

Let $C$ be a small category. Let $D,E$ be cocomplete categories. Let us denote by $\hom$ (resp. $\hom_c$) the category of (cocontinuous) functors. Then there is an equivalence of categories ...
1
vote
1answer
73 views

An example of when a product in a category may not exist.

My question is about the category of finitely generated Abelian groups; in particular, I want to show, by definition, that there exists a set of objects $T$ in this category for which there is no ...
7
votes
1answer
64 views

Is there a name for those commutative monoids in which the divisibility order is antisymmetric?

Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows. $$x \mid y \leftrightarrow \exists a(ax=y)$$ Is there a name for those commutative monoids such ...