Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductiv …

I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about h …

I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is t …

In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a …

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the approp …

This question is somehow a more generalized re-edit of a former question of mine: http://mathoverflow.net/questions/119339/glueing-flat-families-of-objects-over-a-blow-up I guess …

Hey Is there a universal property for the smash product (of pointed spaces or pointed CW-complexes or something of that ilk)? I've seen the smash product of spectra defined with …

This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa …

Let $C$ be a category with filtered projective limits and coproducts (denoted by $+$). Then for all filtered projective systems $\{X_i\}$, $\{Y_i\}$ in $C$ there is a canonical mor …

Hello, I have two questions, the first less important. Let $X$ be a scheme, $x \in X$ a schematic point. What is an elegant way of defining/characterizing the map $\operatorn …

This question has not the ambition of being very precise, instead it is more philosophical. The question is the following: does the Stein factorization of a morphism have some kin …

I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. a very ni …

There is of course the standard construction of the reals by considering the set of sequences that are Cauchy with respect to the standard metric and taking the quotient by sequenc …

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the followi …

What is the universal property of normalization? I'm looking for an answer something like If X is a scheme and Y→X is its normalization, then the morphism Y→X has …