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<fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="true"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>985</fr:anchor><fr:addr
type="user">bib-0001</fr:addr><fr:route>bib-0001.xml</fr:route><fr:title
text="Mathematical bibliography">Mathematical bibliography</fr:title><fr:taxon>Bibliography</fr:taxon><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>855</fr:anchor><fr:addr
type="user">bib-0002</fr:addr><fr:route>bib-0002.xml</fr:route><fr:title
text="Double category theory">Double category theory</fr:title><fr:taxon>Bibliography</fr:taxon><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>Double category theory:

<fr:ul><fr:li>Textbook reference: <fr:link
type="local"
href="grandis-2019.xml"
addr="grandis-2019"
title="Higher dimensional categories: From double to multiple categories">Higher dimensional categories: From double to multiple categories</fr:link></fr:li>
  <fr:li>Equipments: <fr:link
type="local"
href="shulman-2008.xml"
addr="shulman-2008"
title="Framed bicategories and monoidal fibrations">Framed bicategories and monoidal fibrations</fr:link></fr:li></fr:ul></fr:p><fr:p>Double-categorical logic:

<fr:ul><fr:li><fr:link
type="local"
href="cartesian-double-theories-2024.xml"
addr="cartesian-double-theories-2024"
title="Cartesian double theories: A double-categorical framework for categorical doctrines">Cartesian double theories: A double-categorical framework for categorical doctrines</fr:link></fr:li>
  <fr:li><fr:link
type="local"
href="double-products-2026.xml"
addr="double-products-2026"
title="Products in double categories, revisited">Products in double categories, revisited</fr:link></fr:li>
  <fr:li><fr:link
type="local"
href="double-instances-2026.xml"
addr="double-instances-2026"
title="Presheaves on lax double functors; or, Instances of models of double theories">Presheaves on lax double functors; or, Instances of models of double theories</fr:link></fr:li></fr:ul></fr:p><fr:p>Background on categorical logic:

<fr:ul><fr:li>Treatment of categorical logic using multicategories: <fr:link
type="local"
href="shulman-2016.xml"
addr="shulman-2016"
title="Categorical logic from a categorical point of view">Categorical logic from a categorical point of view</fr:link></fr:li></fr:ul></fr:p><fr:p>Double-categorical systems theory:

<fr:ul><fr:li><fr:link
type="local"
href="dots-2025.xml"
addr="dots-2025"
title="Towards a double operadic theory of systems">Towards a double operadic theory of systems</fr:link></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>856</fr:anchor><fr:addr
type="user">bib-0003</fr:addr><fr:route>bib-0003.xml</fr:route><fr:title
text="Modeling frameworks">Modeling frameworks</fr:title><fr:taxon>Bibliography</fr:taxon><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:addr
type="machine">#244</fr:addr><fr:route>unstable-244.xml</fr:route><fr:title
text="Knowledge and data">Knowledge and data</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
type="local"
href="ologs-2012.xml"
addr="ologs-2012"
title="Ologs: A categorical framework for knowledge representation">Ologs: A categorical framework for knowledge representation</fr:link></fr:li>
<fr:li><fr:link
type="local"
href="alg-databases-2017.xml"
addr="alg-databases-2017"
title="Algebraic databases">Algebraic databases</fr:link>; see also <fr:link
type="local"
href="acsets-2022.xml"
addr="acsets-2022"
title="Categorical data structures for technical computing">Categorical data structures for technical computing</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>516</fr:anchor><fr:addr
type="machine">#245</fr:addr><fr:route>unstable-245.xml</fr:route><fr:title
text="Dynamical systems">Dynamical systems</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>Regulatory networks and causal loop diagrams: <fr:link
type="local"
href="reg-nets-2024.xml"
addr="reg-nets-2024"
title="A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks">A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks</fr:link></fr:li>
<fr:li>Stock-flow diagrams: <fr:link
type="local"
href="stock-flow-2023.xml"
addr="stock-flow-2023"
title="Compositional modeling with stock and flow diagrams">Compositional modeling with stock and flow diagrams</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title
text="References">References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>986</fr:anchor><fr:addr
type="user">double-instances-2026</fr:addr><fr:route>double-instances-2026.xml</fr:route><fr:title
text="Presheaves on lax double functors; or, Instances of models of double theories">Presheaves on lax double functors; or, Instances of models of double theories</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2026</fr:year></fr:date><fr:authors><fr:author>Kevin Carlson</fr:author><fr:author>Evan Patterson</fr:author></fr:authors><fr:meta
name="doi">10.1007/s10485-026-09866-y</fr:meta><fr:meta
name="arxiv">2510.08861</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>989</fr:anchor><fr:addr
type="user">double-products-2026</fr:addr><fr:route>double-products-2026.xml</fr:route><fr:title
text="Products in double categories, revisited">Products in double categories, revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2026</fr:year></fr:date><fr:authors><fr:author>Evan Patterson</fr:author></fr:authors><fr:meta
name="arxiv">2401.08990</fr:meta><fr:meta
name="external">http://www.tac.mta.ca/tac/volumes/45/16/45-16abs.html</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>991</fr:anchor><fr:addr
type="user">dots-2025</fr:addr><fr:route>dots-2025.xml</fr:route><fr:title
text="Towards a double operadic theory of systems">Towards a double operadic theory of systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2025</fr:year></fr:date><fr:authors><fr:author>Sophie Libkind</fr:author><fr:author>David Jaz Myers</fr:author></fr:authors><fr:meta
name="arxiv">2505.18329</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>994</fr:anchor><fr:addr
type="user">reg-nets-2024</fr:addr><fr:route>reg-nets-2024.xml</fr:route><fr:title
text="A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks">A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year></fr:date><fr:authors><fr:author>Rebekah Aduddell</fr:author><fr:author>James Fairbanks</fr:author><fr:author>Amit Kumar</fr:author><fr:author>Pablo S. Ocal</fr:author><fr:author>Evan Patterson</fr:author><fr:author>Brandon T. Shapiro</fr:author></fr:authors><fr:meta
name="arxiv">2301.01445</fr:meta><fr:meta
name="doi">10.32408/compositionality-6-2</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1001</fr:anchor><fr:addr
type="user">cartesian-double-theories-2024</fr:addr><fr:route>cartesian-double-theories-2024.xml</fr:route><fr:title
text="Cartesian double theories: A double-categorical framework for categorical doctrines">Cartesian double theories: A double-categorical framework for categorical doctrines</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year></fr:date><fr:authors><fr:author>Michael Lambert</fr:author><fr:author>Evan Patterson</fr:author></fr:authors><fr:meta
name="doi">10.1016/j.aim.2024.109630</fr:meta><fr:meta
name="arxiv">2310.05384</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1004</fr:anchor><fr:addr
type="user">stock-flow-2023</fr:addr><fr:route>stock-flow-2023.xml</fr:route><fr:title
text="Compositional modeling with stock and flow diagrams">Compositional modeling with stock and flow diagrams</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2023</fr:year></fr:date><fr:authors><fr:author>John Baez</fr:author><fr:author>Xiaoyan Li</fr:author><fr:author>Sophie Libkind</fr:author><fr:author>Nathaniel Osgood</fr:author><fr:author>Evan Patterson</fr:author></fr:authors><fr:meta
name="doi">10.4204/EPTCS.380.5</fr:meta><fr:meta
name="arxiv">2205.08373</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1010</fr:anchor><fr:addr
type="user">acsets-2022</fr:addr><fr:route>acsets-2022.xml</fr:route><fr:title
text="Categorical data structures for technical computing">Categorical data structures for technical computing</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2022</fr:year></fr:date><fr:authors><fr:author>Evan Patterson</fr:author><fr:author>Owen Lynch</fr:author><fr:author>James Fairbanks</fr:author></fr:authors><fr:meta
name="external">https://github.com/AlgebraicJulia/ACSets.jl</fr:meta><fr:meta
name="doi">10.32408/compositionality-4-5</fr:meta><fr:meta
name="arxiv">2106.04703</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1014</fr:anchor><fr:addr
type="user">grandis-2019</fr:addr><fr:route>grandis-2019.xml</fr:route><fr:title
text="Higher dimensional categories: From double to multiple categories">Higher dimensional categories: From double to multiple categories</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2019</fr:year></fr:date><fr:authors><fr:author>Marco Grandis</fr:author></fr:authors><fr:meta
name="doi">10.1142/11406</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1016</fr:anchor><fr:addr
type="user">alg-databases-2017</fr:addr><fr:route>alg-databases-2017.xml</fr:route><fr:title
text="Algebraic databases">Algebraic databases</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2017</fr:year></fr:date><fr:authors><fr:author>Patrick Schultz</fr:author><fr:author>David I. Spivak</fr:author><fr:author>Christina Vasilakopoulou</fr:author><fr:author>Ryan Wisnesky</fr:author></fr:authors><fr:meta
name="external">http://www.tac.mta.ca/tac/volumes/32/16/32-16abs.html</fr:meta><fr:meta
name="arxiv">1602.03501</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1021</fr:anchor><fr:addr
type="user">shulman-2016</fr:addr><fr:route>shulman-2016.xml</fr:route><fr:title
text="Categorical logic from a categorical point of view">Categorical logic from a categorical point of view</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2016</fr:year></fr:date><fr:authors><fr:author>Michael Shulman</fr:author></fr:authors><fr:meta
name="external">https://mikeshulman.github.io/catlog/catlog.pdf</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1023</fr:anchor><fr:addr
type="user">ologs-2012</fr:addr><fr:route>ologs-2012.xml</fr:route><fr:title
text="Ologs: A categorical framework for knowledge representation">Ologs: A categorical framework for knowledge representation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2012</fr:year></fr:date><fr:authors><fr:author>David I. Spivak</fr:author><fr:author>Robert E. Kent</fr:author></fr:authors><fr:meta
name="doi">10.1371/journal.pone.0024274</fr:meta><fr:meta
name="arxiv">1102.1889</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1026</fr:anchor><fr:addr
type="user">shulman-2008</fr:addr><fr:route>shulman-2008.xml</fr:route><fr:title
text="Framed bicategories and monoidal fibrations">Framed bicategories and monoidal fibrations</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2008</fr:year></fr:date><fr:authors><fr:author>Michael Shulman</fr:author></fr:authors><fr:meta
name="arxiv">0706.1286</fr:meta><fr:meta
name="external">http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html</fr:meta></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="false"
numbered="false"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title
text="Backlinks">Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="false"
show-heading="true"
show-metadata="true"
expanded="false"
root="true"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>1028</fr:anchor><fr:addr
type="user">mat-0001</fr:addr><fr:route>index.xml</fr:route><fr:title
text="CatColab: Mathematical Documentation">CatColab: Mathematical Documentation</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:addr
type="machine">#251</fr:addr><fr:route>unstable-251.xml</fr:route><fr:title
text="Introduction">Introduction</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>As the name suggests, CatColab is based on mathematical ideas from category
  theory. It is a specific design goal that the system be usable <fr:em>without</fr:em>
  any knowledge of such ideas. These docs are for people who want to know about
  the mathematics, perhaps because they want to develop the core package
  <fr:link
type="external"
href="/dev/rust/catlog/">catlog</fr:link> or just out of curiosity. These pages are
  currently far from being a self-contained introduction, serving mainly to
  collect new bits of CatColab-related mathematics as they are created. For
  proper exposition, see the <fr:link
type="local"
href="bib-0001.xml"
addr="bib-0001"
title="Mathematical bibliography">published literature</fr:link>.</fr:p><fr:p>In mathematical terms, CatColab is an editor for categorical structures and
  their morphisms and higher morphisms. The meta-logical framework organizing
  these structures is based on <fr:link
type="external"
href="https://mathoverflow.net/q/476936">double category
  theory</fr:link>. More precisely, the
  <fr:strong>domain-specific logics</fr:strong> in CatColab are defined by <fr:link
type="local"
href="cartesian-double-theories-2024.xml"
addr="cartesian-double-theories-2024"
title="Cartesian double theories: A double-categorical framework for categorical doctrines">double
  theories</fr:link>, and the <fr:strong>models</fr:strong> in CatColab
  are models of double theories.</fr:p><fr:p>The library of domain-specific logics in CatColab is inspired by a wide body
  of research in applied category theory and beyond. Incomplete bibliographies
  are in <fr:link
type="local"
href="bib-0003.xml"
addr="bib-0003"
title="Modeling frameworks">these docs</fr:link> and the <fr:link
type="external"
href="https://next.catcolab.org/dev/rust/catlog/refs"><fr:code>catlog</fr:code>
  docs</fr:link>.</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>530</fr:anchor><fr:addr
type="machine">#252</fr:addr><fr:route>unstable-252.xml</fr:route><fr:title
text="Doctrines">Doctrines</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>Systems of categorical logic, or <fr:strong>doctrines</fr:strong>, whose <fr:em>free</fr:em> objects
have a systems interpretation:</fr:p><html:table
xmlns:html="http://www.w3.org/1999/xhtml">
  <html:tr><html:th>Doctrine</html:th> <html:th>Free objects</html:th> <html:th>Systems interpretation</html:th></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0001.xml"
addr="dct-0001"
title="Categories">Categories</fr:link></html:td> <html:td>Graphs</html:td> <html:td>[numerous]</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0006.xml"
addr="dct-0006"
title="Profunctors">Profunctors</fr:link></html:td> <html:td>Bridges between graphs</html:td> <html:td>Database schemas</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0002.xml"
addr="dct-0002"
title="Signed categories">Signed categories</fr:link></html:td> <html:td>Signed graphs</html:td> <html:td>Regulatory networks/causal loop diagrams (CLDs)</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000B.xml"
addr="dct-000B"
title="Delayable signed categories">Delayable signed categories</fr:link></html:td> <html:td>Signed graphs with delays</html:td> <html:td>CLDs with delays</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0007.xml"
addr="dct-0007"
title="Nullable signed categories">Nullable signed categories</fr:link></html:td> <html:td>Nullable signed graphs</html:td> <html:td>CLDs with indeterminates</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0003.xml"
addr="dct-0003"
title="Categories with links">Categories with links</fr:link></html:td> <html:td>Graphs with links</html:td> <html:td>Stock-flow diagrams</html:td></html:tr>
  <html:tr><html:td>Symmetric (or commutative) monoidal categories</html:td> <html:td>Hypergraphs</html:td> <html:td>Petri nets/reaction networks</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000A.xml"
addr="dct-000A"
title="Multicategories">Multicategories</fr:link></html:td> <html:td>Multigraphs</html:td> <html:td>Term graphs/diagrammatic equations</html:td></html:tr>
</html:table><fr:p>Doctrines whose objects have interpretations as process theories:</fr:p><html:table
xmlns:html="http://www.w3.org/1999/xhtml">
  <html:tr><html:th>Doctrine</html:th> <html:th>As process theories</html:th></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0004.xml"
addr="dct-0004"
title="Promonads">Promonads</fr:link> / <fr:link
type="local"
href="dct-000K.xml"
addr="dct-000K"
title="Graded categories">Graded categories</fr:link></html:td> <html:td>Hierarchy of morphisms</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-0005.xml"
addr="dct-0005"
title="Generalized monoidal categories">Generalized monoidal categories</fr:link> / <fr:link
type="local"
href="dct-000E.xml"
addr="dct-000E"
title="Generalized PROs">Generalized PROs</fr:link> / <fr:link
type="local"
href="dct-000P.xml"
addr="dct-000P"
title="Unnormalized PROs">Unnormalized PROs</fr:link></html:td>
    <html:td>Monoidal theory, algebraic theory, ...</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000A.xml"
addr="dct-000A"
title="Multicategories">Multicategories</fr:link> / <fr:link
type="local"
href="dct-000F.xml"
addr="dct-000F"
title="Generalized multicategories">Generalized multicategories</fr:link> / <fr:link
type="local"
href="dct-000N.xml"
addr="dct-000N"
title="Unnormalized multicategories">Unnormalized multicategories</fr:link> / <fr:link
type="local"
href="dct-000O.xml"
addr="dct-000O"
title="Representable multicategories">Representable multicategories</fr:link></html:td>
    <html:td>Operad, algebraic theory, ...</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000M.xml"
addr="dct-000M"
title="Cartesian categories">Cartesian categories</fr:link></html:td> <html:td>Algebraic theory</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000G.xml"
addr="dct-000G"
title="Monoidal promonads">Monoidal promonads</fr:link> / <fr:link
type="local"
href="dct-000H.xml"
addr="dct-000H"
title="Relative monoidal promonads">Relative monoidal promonads</fr:link></html:td> <html:td>Hierarchy of processes</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000I.xml"
addr="dct-000I"
title="Markov categories">Markov categories</fr:link></html:td> <html:td>Probabilistic processes/statistical models</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000J.xml"
addr="dct-000J"
title="Categories of partial maps">Categories of partial maps</fr:link></html:td> <html:td>Fallible processes/processes with constraints</html:td></html:tr>
  <html:tr><html:td><fr:link
type="local"
href="dct-000L.xml"
addr="dct-000L"
title="Partial Markov categories">Partial Markov categories</fr:link></html:td> <html:td>Fallible, probabilistic processes</html:td></html:tr>
</html:table><fr:p>Additional doctrines: <fr:link
type="local"
href="dct-0008.xml"
addr="dct-0008"
title="Companions">Companions</fr:link>, <fr:link
type="local"
href="dct-0009.xml"
addr="dct-0009"
title="Conjoints">Conjoints</fr:link>, <fr:link
type="local"
href="dct-000C.xml"
addr="dct-000C"
title="Fibrations">Fibrations</fr:link>, <fr:link
type="local"
href="dct-000D.xml"
addr="dct-000D"
title="Categories with pullbacks">Categories with pullbacks</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
toc="true"
numbered="true"
show-heading="true"
show-metadata="false"
expanded="true"
root="false"
xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:addr
type="user">dbl-0001</fr:addr><fr:route>dbl-0001.xml</fr:route><fr:title
text="Double category theory">Double category theory</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>Double category theory:
<fr:ul><fr:li><fr:link
type="local"
href="dbl-0002.xml"
addr="dbl-0002"
title="Tabulators">Tabulators</fr:link></fr:li></fr:ul></fr:p><fr:p>Double-categorical logic:
<fr:ul><fr:li><fr:link
type="local"
href="dbl-0008.xml"
addr="dbl-0008"
title="Double Lawvere theories">Double Lawvere theories</fr:link></fr:li>
<fr:li><fr:link
type="local"
href="dbl-0005.xml"
addr="dbl-0005"
title="Discrete models of double theories">Discrete models</fr:link> and <fr:link
type="local"
href="dbl-000D.xml"
addr="dbl-000D"
title="Free models of double theories">free models</fr:link></fr:li>
<fr:li><fr:link
type="local"
href="mdt-0008.xml"
addr="mdt-0008"
title="Modal double theories">Modal double theories</fr:link></fr:li></fr:ul>

<fr:p>Bibliography: <fr:link
type="local"
href="bib-0002.xml"
addr="bib-0002"
title="Double category theory">Double category theory</fr:link></fr:p></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>