Uploaded May 2025 | Updated September 2026, 2 weeks ago

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Topos Institute Colloquium, 1st of May 2025.
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This is joint work with Kathryn Hess, Brenda Johnson and Julie Rasmusen.

Colimits (and limits) are among the most fundamental notions in category theory, and also among the most useful of the basic structures. In topology, colimits are used to “glue” spaces together. However, problems arise when we try to work with spaces as they continuously deform, because colimits are not invariant under such deformations. In this case, one uses a related notion called a homotopy colimit. But what are these, really? Homotopy colimits do not satisfy a universal property, even in the homotopy category, and are usually defined by the way they are computed in particular types of categories, such as model categories. In joint work, Hess and Johnson identified a list of properties that one would expect homotopy limits to satisfy in any homotopical category. These properties were chosen carefully because they are needed to perform certain constructions in functor calculus. Building on their work, we have identified the categorical structures that govern these properties. A distillation system relates two actions of the category of small categories on the category of categories through a lax linear functor. In this talk, I will define distillation systems and explain when they do and don’t recover homotopy colimits.