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In Categories for the Working Mathematician MacLane included $\lambda_I = \rho_I$ as one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual …

UPDATE: the following argument is wrong, see the comments. If $\mathcal{C}l$ admits a right adjoint then it preserves colimits, and coproducts in particular. Now, in your category of quadratic vector …

This answer builds on sdcvvc's answer and the comments below it, and in particular concerns the (non)existence of a canonical quadratic form $q$ (in sdcvvc's notation). Let me denote by $\mathcal{Q}$ …

In Anton Geraschenko's notes from Martin Olsson's course on stacks, you can find the following quote: The upshot is that if you choose a splitting, you really have no idea what’s going on. That' …

Let $\mathcal{C}$ be a small finitely complete category equipped with a Grothendieck (pre)topology $\tau$. For $\ast$ the terminal category (one object, one morphism), denote by $i: \ast \to \mathcal{ …

Suppose that the dual $M^{\mathrm{op}}$ of your original simplicial model category $M$ is combinatorial, so that its associated $\infty$-category $\mathcal{M}^{\mathrm{op}}$ is presentable. Then what …

A counterexample in which your first three conditions hold is the following: take two copies of the real line and glue them along the open subset $\mathbb{R}^\ast$. This can be realized as the colimit …

I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, bi …

The highbrow way of reformulating your question is as follows. Consider the category $Sch$ of all schemes endowed with the Zariski topology. There is a fully faithful embedding of the category of aff …