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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The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
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23 views

Size of Hom-Sets in A Functor Category

I am trying to prove the following presumably easy fact: if $B$ is a category with small hom-sets and $C$ is a small category, then $B^{C}$ has small hom-sets. I am assuming the standard Z-F axioms ...
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2answers
82 views

Natural Transformations Without Objects

So I've been thinking about the definition of categories as just arrows with a defined composition (i.e. without objects). I understand this is silly, but it's fun and I have a question about it: ...
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1answer
85 views

Every Abelian group is canonically a $\mathbb{Z}$-module. Is this just a coincidence?

Every Abelian group is canonically a $\mathbb{Z}$-module, where $\mathbb{Z}$ is the initial monoid in the monoidal category of Abelian groups. And every Abelian monoid is canonically an ...
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168 views

why are the category of pointed sets and the category of sets and partial functions “essentially the same”?

I'm reading "an introduction to category theory" by Harold Simmons. In this books, exercise 1.2.7 wants us to show that $\mathcal{Set}_\bot$ (the category of pointed sets) and $\mathcal{Pfn}$ (the ...
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35 views

how would you define the term “elementary” in the context of categories and sets?

I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I ...
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21 views

example of algebraic theory,free product completion,graphs

Let us denote by $\def\Graph{{\sf Graph}}\Graph$ the category of directed graphs $G$ with multiple edges: they are given by a set $G_v$ of vertices, a set $G_e$ of edges, and two functions from $G_e$ ...
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55 views

Generalizing a statement about direct limits in the category of $A$-modules to other categories

The original problem is from Atiyah and Macdonald's Introduction to Commutative Algebra, Ex 2.15: Suppose $(M_\alpha,\mu_{\beta\gamma})$ is a direct system of modules over a unital commutative ...
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60 views

Tensor product of arbitrary categories

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...
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1answer
105 views

How to prove that : $ \mathrm{Hom} ( A(G), H) \simeq \mathrm{Hom} (G , I(H)) $?

How do we show that the functor $ A : \mathrm {Gr} \to \mathrm {Ab} $ defined by $ A (G) = G / [G, G] $ is a left adjoint functor of the inclusion functor : $ I : \mathrm {Ab} \to \mathrm {Gr} $ ?. ...
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38 views

surjection between sets which are defined through a functor

I'm facing the following problem and have no idea how to deal with it. We consider a functor $T:\underline{Set}\rightarrow\underline{Set}$ and two sets $X,Y$. We can build the product $X\times Y$ ...
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36 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
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2answers
76 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
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85 views

Why are those objects initial or final obejcts? [on hold]

$ 1) $ In the category of sets, show that $ \emptyset $ is a initial object and $ \{* \} $ is a final object. $ 2) $ In the category of groups, show that $ \{e \} $ is a initial and final object. $ ...
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34 views

Presentation of an object in an Eilenberg Moore category by generators and relations

Let $\cal C$ be a category and let $T \colon \cal C \longrightarrow C$ be a monad. An $T$-algebra $A$ is presented by generators $G \in \cal C$ and relations $R \in \cal C$ if it is the coequaliser of ...
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1answer
61 views

Relations between Theories and Categories

I'm just toying around with some thoughts, trying to grock some concepts: It seems that every formal theory induces a locally small category via interpretations: its objects are structures that ...
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1answer
64 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
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1answer
64 views

Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
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51 views

Functorial bijection between to functors [closed]

Let $A$ be a ring and $n$ be an integer and $(P_i)_i$ a family of polynomials in $A [X_1, \dots, X_n]$. Let $G$ be the functor which sends an $A$-algebra $B$ on $ G (B) = \{ \ (b_1, \dots, b_n) \in ...
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2answers
98 views

“Equivalent” definitions of the gluing axioms

I tried to convince myself that the two caracterizations of a presheaf that is a sheaf given in wikipedia are equivalent but I couldn't. (F presheaf and notations from wiki) Let's take a simple ...
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48 views

Continuations vs. Yoneda

There is this observation (worldpress blog) that the continuation monad (Wikipedia) stems from the Yoneda embedding. For fixed data types $R$, the monad maps data types $T$ to types $(T\to R)\to R$. ...
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1answer
53 views

Notation for the subcategory of commutative $R$-algebras

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 ...
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1answer
39 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 ...
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2answers
59 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 ...
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1answer
40 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 ...
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1answer
93 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 $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, ...
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1answer
69 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 ...
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2answers
84 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 ...
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1answer
33 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 ...
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66 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 ...
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78 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 ...
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333 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 ...
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1answer
173 views

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 ...
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1answer
21 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 ...
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1answer
61 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 ...
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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 ...
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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 ...
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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 ...
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1answer
123 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 ...
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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 ...
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3answers
96 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 ...
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2answers
49 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 ...
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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 ...
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70 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 ...
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58 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 ...
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1answer
189 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 ...
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48 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!
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2answers
95 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 ...
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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 ...
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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 ...