# Tagged Questions

548 views

### Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
210 views

### Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?

The question is in the title, here is my motivation: $\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if ...
286 views

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
1k views

### What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...
220 views

### Example of a non-closed cocomplete symmetric monoidal category

Background By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all ...
203 views

### two essentially different concretizaions

It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete ...
4k views

### Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
330 views

### Examples of “inner products” of parallel morphisms in a dagger category

There is a very interesting abstract notion of the trace of an endomorphism $f : c \to c$ of an object $c$ in a braided monoidal category (although the symmetric case is easier): see, for example, ...
230 views

### Is there a category with a subobject classifier but which is not finitely complete?

This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned ...
1k views

### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that $$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$ is and exact sequence in $C$. Does it follow that ...
1k views

### Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
322 views

### Categories with products that preserve quotients

It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve ...
280 views

### Example of a Grothendieck pretopology satisfying a weak saturation condition

Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...
394 views

### Weakly initial sets - examples and nonexamples

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I. The question is then, does Fields have a weakly ...
1k views

### What category without initial object do you care about?

Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
627 views

### Is there a finitely complete category with terminal object but NO subobject classifier?

This came up today while thinking about topoi in seminar, as the title suggests my question is; Is there a finitely complete category with terminal object but NO subobject classifier? Hopefully ...
3k views

### What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of? I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory. ...
192 views

### Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories. The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...
2k views

### Simple show cases for the Yoneda lemma

I've been given a very simple motivating and instructive show case for the Yoneda lemma: Given the category of graphs and a graph object $G$, seen as a quadruple \$(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...