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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Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
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41 views

Every closed (not-necessarily symmetric) monoidal category is canonically self-enriched, right?

Here it is stated that: A closed symmetric monoidal category is canonically self-enriched. This makes sense, but I don't see why it has to be symmetric. Every closed (not-necessarily symmetric) ...
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42 views

Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
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1answer
57 views

Exponential objects and hom-sets.

Let $C$ be a cartesian closed category and $X, Y$ two objects of $C$. Is it the case that $\text{Hom}(X,Y) = Y^X$?
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78 views

Showing that morphisms transforms as claimed, Borceux and enriched natural transformations.

I am having trouble following the proof given in the images below of lemma 6.3.3. More specifically, it is claimed that diagram 6.22 is equivalent to diagram 6.23, but I can't see it. Here, the object ...
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1answer
44 views

Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
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1answer
48 views

“Nice proof” that the unit of the left Kan extension of $F$ is an isomorphism, if $F$ is fully faithful

Let $F: \mathbf C \to \mathbf D, G: \mathbf C \to \mathbf E$ be functors. Assume that $\mathbf C$ is small, $\mathbf D$ is locally small and $\mathbf E$ is cocomplete. Then, I can compute the left Kan ...
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2answers
43 views

A property of product category

This property of the product category states that the projections $P$ and $Q$ are "universal" among pairs of functors to $B$ and $C$. Can someone specify me exactly the sense of that assertion? I ...
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36 views

directed colimits,category,unary algebras,preservation

Are there some natural properties of mono-unary algebras NOT preserved by (omega)-directed colimits (unlike being connected,having no cycles...)? Formalization to logical formulas is not necessary.
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25 views

Map of monads and left adjoints

Let $(T,\eta,\mu)$, $(T',\eta',\mu')$ be two monads on a category $X$. Let $\theta:T\Rightarrow T'$ be a map of monads. Then, we have an induced functor $X^\theta:X^{T'}\rightarrow X^T$ (from the ...
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1answer
111 views

Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
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1answer
53 views

Terminal objects of the category of morphisms

I'm reading Basic Category Theory for Computer Scientists by Benjamin C. Pierce and in exercice 1.4.6, he asks what the terminal objects are in $Set^\to$. Let $C$ be a category. The category ...
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89 views

Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
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1answer
31 views

Pos,coequalizer,terminal poset

Consider the reflexive pair $u, v \colon (1 + 2) \to 2$ from the coproduct of the terminal poset $1$ and the two-element chain, where both morphisms are the identity on the second summand, and they ...
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1answer
54 views

Free object in category of groups.

Suppose $X$ is a set and $F$ is a free object on $X$ (with $i:X\rightarrow F$) in the category of groups. Prove that $i(X)$ is a set of generator for the group $F$. I have the following hint: If ...
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1answer
111 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
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93 views

ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
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56 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to ...
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1answer
41 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
2
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1answer
67 views

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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44 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
107 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
112 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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2answers
217 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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48 views

Classifying topological spaces by measures

While looking at some spaces, I happend to know, that in some spaces (like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ ...
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46 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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24 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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60 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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79 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
116 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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45 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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46 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
87 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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36 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 ...
3
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1answer
79 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 ...
3
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1answer
70 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^* ...
3
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1answer
83 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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2answers
103 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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60 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
55 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 ...
2
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1answer
42 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
65 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
51 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
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1answer
112 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
72 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
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2answers
96 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
45 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
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1answer
74 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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1answer
97 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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419 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 ...